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The Red Viper's guide to mathematics - part 5: The Infinity Saga - Cont.

This entry will be a tad shorter than what we had so far, and will have mostly additional things that should be discussed regarding infinity. Starting off with a debt I owe you from the last entry, so without further ado, let's kick this off!

Aleph - from zero to hero:

Last time we learned what is א‎0 - the countable infinity - really is: it's the smallest cardinality of infinity, and represents infinities that can still be put in an ordered list. However, we did see that there are uncountable infinities, and we also know that the next step from א‎0 is simply called א (aleph). But, what does it exactly mean? What determines the aleph-cardinality of a set? We know that the smallest aleph cardinality is the countable infinity, which makes it easier to determine if a certain sat is א0 or not, but what about uncountable infinite sets? How do we know if the cardinality of ℝ is א, or א‎2, א‎3, or something else maybe? What distinguishes the infinite number of uncountable infinities from each other?

Well, in order to delve into that, we first need to know what a power-set is. Its definition is a tad confusing:

Let A be a set. The power set of A is the set of all subsets contained in A, and is notated P(A). That is, if we use the set-builder notation that we saw in entry number 2: P(A) = {S | S⊆A}. Note that a power set is always a set of sets.

For example, if A = {1,2,3}, then P(A) = { Ø, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }. Note that I bolded out Ø (the empty set) and {1,2,3} (the set A itself). That is because, it is important to note that the power set of any set will always have the empty set and the original set itself as elements. If you look back at the 2nd entry you will see that I mentioned the empty set is contained in every set (i.e it is a subset of every set), and every set is also a subset of itself. All the rest of the elements are non-empty, strict subsets of the original set A. Now, how are power sets important to our topic? Because of their cardinality. Let's say that |A| = n. Then |P(A)| = 2n, that is, 2 to the nth power. Why is that? Well, there's a simple answer for that using combinatorics. But we haven't discussed that yet, so we'll leave it at that for the moment. We can already see though, that for the set A = {1,2,3} that we defined earlier, |A| = 3 and |P(A)| = 23 = 8. Still don't believe me? Try thinking of different sized sets and write down their power sets. You'll see for yourselves.

So, the question we now need to ask ourselves, what is |P(ℕ)|? Is it 2א‎0? Yes, that is exactly what it is.

But wait...

In the previous entry, we discussed the continuum hypothesis, which suggests that there doesn't exist a set whose cardinality is strictly between א‎0 and א. Does that mean that 2א‎0 = א? Well, that is kinda where the continuum hypothesis was born: Georg Cantor theorized that there's nothing that's strictly between א‎0 and א, and we know that |P(ℕ)| = 2א‎0, but infinity being weird as we know it is, it wouldn't be trivial to just assume that 2א‎0 > א‎0. Luckily, we can prove that P(ℕ) is uncountable, with a very similar strategy to the one we used when showing that ℝ is uncountable:

First, assume by contradiction that P(ℕ) is countable. Therefore, the elements of P(ℕ) can be put in an ordered list. Now, what we will do, is translate each element of P(ℕ), which is a certain set of naturals - into an infinite sequence of ones and zeroes, in the following manner:

Let S∈P(ℕ) and let n∈ℕ. If n∈S, then the nth digit in the sequence representing S will be 1. Otherwise, it will be 0. Confusing, I know, so here's a few examples:

  • S = {0,1,2,3}, the sequence will be: 1111000000...
  • S = {1,3,7}, the sequence will be: 010100010000...
  • S = {0,2,4,6...} (all evens), the sequence will be: 10101010...
  • S = Ø, the sequence will be: 00000000...
  • S = ℕ, the sequence will be: 11111111...

So, let's say we have such an infinite list of infinite sequences that allegedly represent all the sets in P(ℕ). Same as we did with ℝ, we will construct a "rogue" sequence that cannot appear on the list. Looking at the first digit of the first sequence on the list, if it's 1 then the first digit in our rogue sequence will be 0, and vice versa. Then the 2nd digit in our rogue sequence will be similarly determined by the 2nd digit of the 2nd sequence on the list, and so on and so forth. That would give us the exact same result as we got when proving that ℝ is uncountable, resulting in the conclusion that P(ℕ) is also uncountable, and therefore 2א‎0 > א‎0. Q.E.D.

Now, given that the continuum hypothesis suggests that there's nothing strictly between א‎0 and א, that would mean that (allegedly, assuming the continuum hypothesis is correct) 2א‎0 = א. Generalizing this proposition, we can say: let S be an infinite set, such that |S| = א‎n for some n∈ℕ. Then |P(S)| = 2א‎n = א‎n+1. That is commonly known as the "Generalized Continuum Hypothesis". Yes, mathematicians are horrendously unoriginal when it comes to naming stuff.

In fact, it is possible to show a bijection between P(ℕ) and ℝ, but it's a pretty long proof that uses several other theorems and concepts which we haven't discussed (such as binary representations of numbers and the Cantor-Schreder-Bernstein theorem), so we'll skip it for now. In any case, that is how you determine the aleph cardinality of an uncountable set: if you have an uncountable set, say S, and you can find a bijection between S and P(ℕ), then |S| = 2א‎0. Otherwise, if you can find a bijection between S and P(P(ℕ)) (which is a set of sets of sets), then |S| = 22א‎0 (which would be equal to א‎2 according to the continuum hypothesis). Otherwise, go to P(P(P(ℕ))), and so on.

If Chuck Norris has one infinity and you have 2 infinities, Chuck Norris has more infinities than you:

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Well, not really; you actually have the same amount. Depending on the cardinality of course. OK, maybe that wasn't a perfect analogy... but I like Chuck Norris jokes! Anyway, allow me to explain what I'm getting at here; assume you have 2 countably infinite sets, and Chuck Norris has 1 countably infinite set. Now, let's say you perform a union between your two sets and create one big set. Who has the bigger set, you or Chuck? You both have sets of the same size! Reason being, that a union of 2 countably infinite sets is still countably infinite, i.e א‎0. Let us prove that.

There is one important thing to note before starting this proof, though; we all know, since entry number 3, that if there exists a bijective function between 2 sets, then these sets are equivalent. However, that conditioning works both ways: for any two equivalent sets, there exists a bijection between them. That bijection might not be an explicit function with a specific formula (well, the sets don't have to be sets of numbers to begin with), but there is always a way to have a surjective, one-to-one matching between the elements of two equivalent sets. That is a very powerful theorem: if we start off with the knowledge that 2 sets are equivalent, we can just go off with there being a certain bijection between them even if we have no idea what that bijection actually is.

Anyway, let's get to it. Assume we have 2 countably infinite sets, A and B. Let's also assume that A∩B = Ø, i.e the two sets have no elements in common (when 2 sets have no elements in common, they are said to be "disjoint").

Seeing as |A| = |B| = |ℕ| = א‎0, there exists a bijective function f: ℕ→A and a bijective function g: ℕ→B.

Now, define: C = A∪B, and a function h:ℕ→C, by:

  • If x is even, then h(x) = f(x/2).
  • If x is odd, then h(x) = g((x-1)/2).

We will prove that h is bijective, thus showing that C is a countably infinite set. Firstly, it's easy to see that h is total, as it is well defined for every x∈ℕ. Now to prove that it is injective:

  • Let x,y∈ℕ such that h(x) = h(y). We will prove that x=y.
  • Option 1: x and y are even.
    • By definition of h that means that h(x) = f(x/2), h(y) = f(y/2).
    • That is, f(x/2) = f(y/2).
    • Seeing as f is a bijection, it has to be injective, which means that necessarily x/2=y/2.
    • Multiplying both sides by 2 yields x=y.
  • Option 2: x and y are odd.
    • By definition of h that means that h(x) = g((x-1)/2), h(y) = g((y-1)/2).
    • that is, g((x-1)/2)=g((y-1)/2).
    • Seeing as g is a bijection, it has to be injective, which means that necessarily (x-1)/2=(y-1)/2.
    • Multiplying both sides by 2 and adding 1 yields x=y.
  • Option 3: x is even and y is odd, without loss of generality. We will show why this is impossible.
    • Assume by contradiction that it is possible for x to be even and y to be odd while h(x) = h(y).
    • By definition of h that means that h(x) = f(x/2), h(y) = g((y-1)/2).
    • That is, f(x/2) = g((y-1)/2).
    • We know that A and B are disjoint. We also know that, f(x) is an element in A while g(y) is an element in B, for all x,y∈ℕ.
    • Since f(x/2) = g((y-1)/2), we actually get that there is an element that belongs in both A and in B. That is a contradiction! Therefore, our assumption must be false, which means that it is impossible for x to be even and y to be odd while h(x) = h(y).

All possible cases have been covered, therefore we have proven that h is injective. Q.E.D. Now to prove it's surjective:

  • Let y∈C. We will show that there exists a x∈ℕ such that h(x)=y.
  • Option 1: y∈A.
    • Seeing as f is a bijection, it has to be surjective, which means that there exists some x∈ℕ such that f(x)=y.
    • Choosing z=2x, we get h(z) = h(2x) = f(2x/2) = f(x) = y, as required.
  • Option 2: y∈B.
    • Seeing as g is a bijection, it has to be surjective, which means that there exists some x∈ℕ such that g(x)=y.
    • Choosing z=2x+1, we get h(z) = h(2x+1) = g((2x+1-1)/2) = g(x) = y, as required.

All possible cases have been covered, therefore we have proven that h is surjective. Q.E.D. Now we see that h is indeed bijective, which means that |C| = |ℕ| = א‎0 - which teaches us that the union of 2 countably infinite sets is also countably infinite.

Now, interesting thought: what happens if we have 3 countably infinite sets? Say, A, B and D? We already know that the union of A and B, which we called C, is countably infinite. So the union of A, B and D, will actually be the union of C and D (which we can call E), which is, well, exactly the same as the union of A and B which we now proved to be countably infinite. So the union of 3 countably infinite sets is still countably infinite. The major plot twist, however, comes when we reach 4 countably infinite sets!

Just kidding, it's still just the same. And same for 5, 6, 7, or however many sets you want to throw into that union. In fact, the union of a countably infinite number of countably infinite sets, is still countably infinite.

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So, if Chuck Norris has one countably infinite set, and you have a countably infinite number of countably infinite sets... Chuck Norris still has just as much infinity as you. He's just that much of a badass.

Some more sorcery and witchcraft:

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The bijection between the naturals and the integers that we saw last time is pretty tame compared to what we're going to see right now. I mean, really, you're about to witness some real Elder-Wand-level stuff. Let's start by seeing what an interval is, which is pretty intuitive. First, we have 2 basic types of intervals: open and closed, notated (a,b) and [a,b] respectively (a<b necessarily). What these things mean, is:

  • The open interval (a,b) is the set of all reals that are strictly between a and b, ergo {x∈ℝ | a < x < b}.
  • The closed interval [a,b] is the set of the numbers a, b, and all reals that are between a and b, ergo {x∈ℝ | a ≤ x ≤ b}.

Other than that, there's semi-closed intervals (or semi-open, both names are equivalent) - [a,b) and (a,b]. The number that's next to the square bracket is included in the interval, the other one isn't. Easy.

Now, here comes the part with the Luciferian black magic: every interval - open, closed or semi-closed - is equivalent to the entire set of reals. No matter how small. Even |(0.001,0.002)| = |ℝ|. Let's go ahead and prove the equivalence of ℝ to (0,1) for example. It's a pretty long proof though, so here goes:

Define f:(0,1)→ℝ by:

  • If 0 < x < 0.5, then f(x) = (x - 0.5)/x.
  • If 0.5 ≤ x < 1, then f(x) = (x - 0.5)/(1 - x).

Easy to see that f is total, given that it's defined for all x∈(0,1). Let's show it's injective:

  • Let x,y∈(0,1) such that f(x) = f(y). We will show that x=y.
  • Option 1: 0 < x,y < 0.5.
    • By definition of f, that means that f(x) = (x - 0.5)/x, f(y) = (y - 0.5)/y.
    • That is, (x - 0.5)/x = (y - 0.5)/y.
    • Multiplying both sides by xy and simplifying yields xy - 0.5y = xy - 0.5x.
    • Subtracting xy from both sides and multiplying by -2 yields x=y, as required.
  • Option 2: 0.5 ≤ x,y < 1.
    • By definition of f, that means that, f(x) = (x - 0.5)/(1 - x), f(y) = (y - 0.5)/(1 - y).
    • That is, (x - 0.5)/(1 - x) = (y - 0.5)/(1 - y).
    • Multiplying both sides by (1 - x)(1 - y) and simplifying yields: x - 0.5 - xy + 0.5y = y - 0.5 - xy + 0.5x.
    • Adding (0.5 + xy) to both sides yields x + 0.5y = y + 0.5x.
    • That is, 0.5x = 0.5y. Multiplying both sides by 2 yields x=y, as required.
  • Option 3: 0 < x < 0.5, 0.5 ≤ y < 1, without loss of generality. We will show why this is impossible.
    • Assume by contradiction that it is possible.
    • Therefore, by definition of f, we get: f(x) = (x - 0.5)/x, f(y) = (y - 0.5)/(1 - y).
    • That is, (x - 0.5)/x = (y - 0.5)/(1 - y). That is only possible when x=y=0.5.
    • However we know that x < 0.5. We have reached a contradiction, therefore it is impossible for the equality f(x) = f(y) to hold while 0 < x < 0.5, 0.5 ≤ y < 1.

We have covered all possible options, so we deduce that f is injective. Now to prove it's surjective:

  • Let y∈ℝ. We will show that there exists some x∈(0,1) such that f(x) = y.
  • Option 1: y < 0.
    • Consider x = -1/(2y - 2). Note that this is a positive expression.
    • Since y < 0, then necessarily 2y - 2 < -2, so 1/(2y - 2) > -0.5, meaning 0 < -1/(2y - 2) < 0.5.
    • Therefore f(x) = f(-1/(2y - 2)) = (-1/(2y - 2) - 0.5)/(-1/(2y - 2)) = 1 + (2y - 2)/2 = 1 + y - 1 = y.
  • Option 2: y ≥ 0.
    • Consider x = (y + 0.5)/(y + 1). Note that y + 0.5 < y + 1, so (y + 0.5)/(y + 1) < 1.
    • Simplifying x, we get x = 1 - 1/(2y + 2).
    • Since y ≥ 0, then 2y + 2 ≥ 2, so 1/(2y + 2) ≤ 0.5, meaning 1 - 1/(2y + 2) ≥ 0.5.
    • Therefore f(x) = f(1 - 1/(2y + 2)) = (1 - 1/(2y + 2) - 0.5)/(1 - (1 - 1/(2y + 2))) = (0.5 - 1/(2y + 2))/(1/(2y + 2)) = ...
    • ... = ((2y + 2 - 2)/(4y + 4))/(1/(2y + 2)) = (y/(2y + 2))/ (1/(2y + 2)) = (y/(2y + 2))/(2y + 2) = y.

All options have been covered, therefore f is surjective. Finally, we deduce that f is bijective, and therefore |(1,0)| = |ℝ|. Q.E.D.

Now, that was a long proof with a pretty complicated function. But I think that the thought of every small interval being equivalent to the entire x-axis of real numbers, is pretty baffling. As I said, infinity is absolutely mind-blowing.

If you didn't read the proof, though, here's another way of looking at it. Consider the function f(x) = tan(π(x - 0.5)). Yes, we haven't discussed trigonometric functions such as tan yet, but here's the graph of this function (when defined as f:ℝ→ℝ):

No Caption Provided

Look at the interval (0,1), or any other interval (n,n+1) on the picture above: we can see that the red graph in that interval is unbounded from above nor from below - it shoots up to infinity and down to negative infinity, which is to say that it is surjective. We also see that no 2 points on that interval have the same y value, therefore it is also injective. In other words, the x values in the interval (x,y) are injectively and surjectively mapped onto ℝ. The image, by the way, is taken from Desmos - an online graphing calculator, which pretty much saved my butt in calculus.

In summary:

So, that closes our saga on infinity, which started pretty much at the very first entry. I hope I managed to pass some of my love for this subject, and maybe even math as a whole, onto you. It's truly fascinating in my opinion and there's always more to learn - but I think I'll leave it at that for now. In future entries I will discuss some other topics in math; calculus, combinatorics, and maybe other things too. So thanks for sticking with me so far, and hope to see you in future entries too! As always, your feedback will be appreciated and questions will gladly be answered.

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The Red Viper's guide to mathematics - part 4: The Infinity Saga

Finally, we've reached the moment when we can really discuss infinity. I'll tell you something: most people really don't know what infinity is. It's easy to think of infinity in the raw sense of the word as "something that has no end", but... it's really more than that. Much more. It's a tough subject to wrap our minds around, but it is, in fact, truly mind blowing. Let's start with a thinking exercise, widely known as "Hilbert's Hotel Paradox", to give us an idea about just how messed up the idea of infinity is:

Hilbert's Hotel Paradox:

So, let's say that somewhere in the world there's a hotel, that has an infinite number of rooms - numbered from 1 and onward. One night, the hotel is fully booked. That already is a weird suggestion: how can a hotel with an infinite number of rooms be full? But let's imagine that it is in fact full, ergo, there's a guest in each room. A weary, dusty traveler arrives at the hotel and asks for a room... but he can't have one because the hotel is full, right? Wrong. There may be a guest in each and every room of the hotel, but there's plenty room for the weary traveler! The night manager goes up to the guest in room 1, and asks him to move to room 2. He then goes to the guest in room 2, and asks him to move to room 3. The guest in room 3 is asked to move to room 4, and so on and so forth. That way, each guest moves from room 'n' to room 'n + 1',. Since there's an infinite number of rooms in the hotel, this room 'n + 1' always exists, each guest has a new room to move to, and room number 1 is now available for the weary traveler. Awesome!

What happens when, half an hour later, a group of 10 travelers arrive and ask for rooms? Easy; the night manager asks each guest to move 10 rooms up, ergo, from room 'n' to room 'n + 10', which vacates the first 10 rooms and the 10 new arrivals can happily check in! Great!

But now we have a new problem: a bus with an infinite number of weary passengers arrives at or infinite hotel. How does the night manager vacate an infinite number of rooms, to accommodate for an infinite number of guests? Not a problem! The guest in room 1 is asked to move to room 2. The guest in room 2 is then asked to move to room 4. The guest in room 3 moves to room 6, and so each guest moves from room 'n' to room '2n' - which makes it so only the even-numbered rooms are now occupied. The infinite number of new guests are then booked for all the now vacant odd-numbered rooms, and everyone's happy!

Now, I could go on with what happens when an infinite number of infinite buses with an infinite number of passengers arrive in the hotel, which has a pretty elegant solution, but you get the idea: the hotel is completely full, and yet there's always room for more. How is that possible? Because infinity is just weird like that. You can take almost everything you know about math and chuck it out the window when infinities are involved.

In the previous entry, I promised to tell you what is the cardinality of ℕ. And you know what they say...

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So, I will pay this debt. But before I do, we have to discuss a few things.

Just what is "infinity"?

As I said above, the concept of "infinity" has a lot more to it than just "something that doesn't end". Infinity has a precise mathematical definition. Now, we know from the previous entry that the "size" of a set is called "cardinality", and is notated the same way as absolute value. So, if we define the sets A = {1,2,3,4,5} and B = {10,33,980,-12,0.6}, then |A| = |B| = 5. Two (or more) sets that have the same cardinality are called equivalent. Not to be confused with "equal": two sets are equal when they have the exact same elements. So, what is an infinite set?

An infinite set is a set that is equivalent to a strict subset of itself. Ergo, if A and B are sets, and A⊂B (A is contained in but is not equal to B), and also |A| = |B|, then B is infinite (and, transitively, that makes A infinite as well).

The equivalence of two sets can be proven if there exists a bijective function between them, as seen in the previous entry. So, here's a simple example to prove ourselves that ℕ, the set of natural numbers, is infinite. We all know that it is, sure, but let's prove it - formally and mathematically.

Firstly, let's define a new set: A = ℕ\{0}, ergo, A is the set {1,2,3,4...} - just the same as , but without the number 0. Note that A⊂ℕ, since A is contained in , but they aren't equal.

Now we will define a function f: ℕ→A by f(x) = x+1. We will prove that this function is a bijection, which will show us that |A| = |ℕ|. First, note that f is total: it is well-defined for every x∈. We have already proved that this function is injective in the last entry, but here it is again:

  • Let x,y∈ℕ, such that f(x)=f(y). We will show that x=y.
  • By definition of f, that means x+1=y+1.
  • Subtracting 1 from both sides gives us x=y, as needed. Thus f is injective. Q.E.D.

And let's show that it is also surjective:

  • Let y∈A. We will show that ∃x∈ℕ such that f(x)=y.
  • Consider x=y-1. Easy to see that (y-1)∈ℕ, because it is necessarily an integer larger than or equal to 0.
  • So, f(x)=f(y-1)=(y-1)+1=y. So, for each y∈A, choosing x=y-1 yields the required result. Therefore f is surjective. Q.E.D.

So, f is bijective. That means that |A| = |ℕ|. Given that A⊂ℕ, that means that ℕ is infinite!

That still doesn't tell us what the cardinality of ℕ is, though, now does it? So what gives?

Well, you all are probably familiar with the lemniscate (aka "Sideways Eight"), the symbol used widely to represent infinity: ∞

However, there's more than just that to it, and simply saying that |ℕ|=∞ would not be correct. There are, in fact, many kinds of infinity. How many kinds? Well...

The infinite infinities:

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Yeah, don't look so surprised there Shaqnos, we already get that infinity is just totally messed up in all the right ways. So let's get down to it. Infinity indeed has, well, an infinite number of different cardinalities. We can regard them as tiers: some infinities are "stronger", or "bigger" than others. The different cardinalities of infinity are notated with the letter "א" - that letter is "aleph", the first letter of the Hebrew alphabet and also the first letter in the Hebrew word for infinity - אינסוף (pronounced "einsoph" - literally translates to "no end"). The smallest aleph cardinality of infinity is א‎0 - "aleph zero", aka "aleph null", "aleph naught", and also "countable infinity" (more on that term later). That is, in fact, the cardinality of the natural numbers: anything smaller than this is not infinite. The following tier of the aleph cardinalities is, as you may have assumed, aleph one, (or just aleph - א), which is the cardinality of ℝ - the set of real numbers, and is called the "continuum cardinality". Then comes aleph two, and so on. In fact, since the infinite cardinalities are numbered by the natural numbers, there are aleph zero such cardinalities, which is why there are, in fact, infinite infinities.

The notion that there are no "midway tiers" between the different aleph cardinalities (for example, something strictly bigger than א‎0 and strictly smaller than א), is called the "continuum hypothesis". Now, it should be said that it isn't clear cut whether the continuum hypothesis is true or false. In fact, it has been proven by mathematician Kurt Godel in 1940, that the truth of the continuum hypothesis, cannot actually be proven. Later, in 1963, it has been proven by mathematician Paul Cohen, that the continuum hypothesis cannot be refuted, either. For that reason, the continuum hypothesis is considered to be "independent" from the regular axiom system of set theory (the system is called ZFC, maybe more on that in a future entry).

So, that answers the question: |ℕ|=א‎0. Debt paid in full.

Sorcery and witchcraft:

That is, I sh*t you not, what I was thinking when I first learned what I'm about to discuss with you now.

Let's consider the set of natural numbers, ℕ, and the set of integers - ℤ. At first glance, it may seem as if the set of integers is larger than the set of naturals; twice as large, in fact, since every natural number also belongs in ℤ, along with its negative counterpart, which doesn't belong in ℕ. But, fact of the matter is, the two sets are perfectly equivalent. Let us prove it: we will define a function, f:ℤ→ℕ, as a split function, by:

  • If x ≤ 0, then f(x) = -2x
  • If x > 0, f(x) = 2x - 1

Now we will prove that this is a bijection. First, it's easy to see that this function is total, as it is defined for every integer. Now, to prove it is injective. This proof would have to be a tad longer than what we've seen till now, since it's a split function:

  • Let x,y∈ℤ, such that f(x)=f(y). We will prove that x=y.
  • Option 1: x≤0 and y≤0.
    • In that case, by the definition of f, f(x)=-2x, f(y)=-2y.
    • That is, -2x=-2y.
    • Dividing both sides by -2 yields x=y, as required.
  • Option 2: x>0 and y>0.
    • In that case, by the definition of f, f(x)=2x-1, f(y)=2y-1.
    • That is, 2x+1=2y+1.
    • Adding1 and dividing both sides by 2 yields x=y, as required.
  • Option 3: x>0 and y≤0, without loss of generality*.
    • We will show that this option is impossible, so we will assume by contradiction that it is.
    • By the definition of f, f(x)=2x+1, f(y)=-2y.
    • That is, 2x-1=-2y.
    • Dividing both sides by -2 yields y=-x+0.5.
    • Given that x is an integer, we deduce that -x+0.5 is not an integer (since it has a fractional part), which means that y is not an integer as well. We have reached a contradiction, since y∈ℤ.
    • So, that means that our assumption must be false. Therefore, if f(x)=f(y), it is impossible that x≥0 and y<0.

We have covered all possibilities, so we deduce that x=y and f is injective. Q.E.D.

*The phrase "without loss of generality" is commonly used when using some assumption in a proof that narrows it down to some specific case (in this case, it was the assumption that x≥0 and y<0), and it serves to imply that this proof is equivalent and can be applied to all other cases. Ergo, it saved us the bother of repeating the same proof for the case of y>0 and x≤0, which would have been exactly the same.

Now, proving that f is surjective.

  • Let y∈ℕ. We will show that there exists some x∈ℤ, such that f(x)=y.
  • Option 1: y is even.
    • Consider x = -y/2. Note that, since y is an even, nonnegative integer, then x is necessarily a non-positive integer (even or odd). Ergo, x≤0.
    • Therefore, by definition of f, f(x)=-2x=-2(-y/2)=y.
  • Option 2: y is odd.
    • In that case, y+1 is even. Consider x = (y+1)/2. Note that, since y+1 is a positive even, then x is necessarily a positive integer (even or odd). Ergo, x>0.
    • Therefore, by definition of f, f(x)=2x-1=2((y+1)/2)-1=y+1-1=y.

We have covered all possibilities, so we deduce that for every y∈ℕ, there exists some x∈ℤ, such that f(x)=y, as required. Q.E.D.

Now we have proven that f is a bijection. And as such, we must deduce that, in fact, |ℕ|=|ℤ|=א‎0. In other, simpler words: the amount of naturals is the same as the amount of integers. I'll tell you a secret: The cardinality of rationals is also א‎0, which means that |ℕ|=|ℚ|, but that is more complicated to prove. Like I said, sorcery and witchcraft.

To Countable Infinity and Beyond!

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Earlier, when discussing א‎0, I mentioned that it is called the "countable infinity". Let me delve into what this term really means, which is actually quite intuitive: A set that is countably infinite, basically means that the elements of the set an be systematically ordered in a certain fashion that would make us able to count them. For example, the naturals: ℕ = {0,1,2,3,4,5,...}, we can easily count from 0 upwards. As for integers, while we can't really start at the lowest number and go up, or vice versa, since there are no highest or lowest integers, we can do this: ℤ = {0,1,-1,2,-2,3,-3,...}. That way, we start at 0, count the next positive integer, and then its positive counterpart.

Now, what about the rationals? I did say that |ℚ|=א‎0, but it's rather hard to imagine putting the all fractions in a list, right? There's an infinite number of rational numbers even on the smallest of intervals, for example between 0 and 1 (it's a principle called the "density of rationals" - we will likely meet and even prove it in the future). How can you put them all in a systematically ordered list? There is quite an elegant solution for that; we will first write the fractions in an infinitely long and infinitely wide table, where the columns represent the numerator, and the rows represent the denominator:

/1234...
11/12/13/14/1
21/22/23/24/2
31/32/33/34/3
41/42/43/44/4
...

The number 202/319, for example, would appear at the 202nd column, 319th row.

Then, we begin at the upper-left corner (1/1) and traverse the table diagonally: we move to 1/2, then to 2/1, 3/1, 2/2, 1/3, 1/4, etc. Sure, there will be lots of repetitions (for example, 1/1 and 2/2 both equal 1 and represent, in fact, the exact same number), but we can simply skip each fraction that we've already seen under a different representation. Even if we don't, though, remember: repetitions in sets are meaningless either way, so we're all good! Of course, that only takes care of the positive rationals, but we can do the exact same thing with the negatives (and zero) and just stitch them together. Easy.

Now, who's to say that we can't perform a similar schtick with all real numbers? I mean, you can give me an infinitely long list of decimal numbers and tell me that you got 'em all in there. Who the hell am I to tell you that you're missing something out? Well, I might be a nobody, but Georg Cantor - the man without whom I'd have no blog to write - had already proven you wrong centuries ago. And here's how Cantor proved that the real numbers are not countable:

Let's assume, by contradiction, that the real numbers are countable. Since they are countable, they can be put in an ordered, indexed list just like we saw for the naturals, the integers and even the rationals, so let's say you gave me such an infinite list. If I will give you a number that is not on your list, I will prove you wrong, so that is what I'm gonna have to do. But wait, how do I confirm that my number isn't on your list? It would take an infinitely long time to check, now wouldn't it? That's why I will have to construct this number carefully, in such a way that would leave no shadow of doubt that it cannot be found on the list without even having to check.

The number I will construct will be an infinitely long decimal number (same as π = 3.141592654...). It will be in the range between 0 and 1 (randomly chosen range, could have chosen any 2 integers that I want).So, the number will be 0.x‎1x‎2x‎3x‎4..., such that 0 < x‎n < 9, for all n∈ℕ. Now, here comes the clever part: I will look at the 1st number on your list - more precisely, I will look at the 1st digit after the decimal point of that number. If that digit is 1, then the first digit after the decimal point in my number will be 0. If it's anything other than 1, then the first digit after the decimal point in my number will be 1. Then I do the same thing with the next digit in my number, according to the second digit after the decimal point on the second number in your list, and so on and so forth. How does that guarantee that my number cannot be found in your list? Well, if we go to any number on your list, say, the 100th number - let's call it 't', then we know that there's at least one difference between t and my number: the 100th digit after the decimal point must be different, even if everything else is, by some miraculous off-chance, exactly the same. Substitute 100 by any natural number n, and you get the same result: There has to be a difference, even if it's very small, between my number and yours - in the nth digit after the decimal point. That is a contradiction to our initial assumption of the real numbers being countable; therefore, they aren't, and |ℝ| ≠ א‎0. Q.E.D. Truly one of the most elegant proofs I have ever learned in math. Since we can very easily find a one-to-one (injective) function f: ℕ→ℝ (for example the identity function: f(x) = x), it's clear that |ℝ|≥|ℕ|, and given that we just proved that they aren't equal, we now know that |ℝ|>|ℕ|. How do we know that |ℝ| = א, and what א exactly is - well, maybe some other time. In the meantime, we will leave it at that.

So what did we have here?

All this buildup, throughout 3 increasingly boring entries, led up to this magnificent moment. Was it worth it? You tell me, but I sure did have a good time writing it all down. Infinity is something pretty messed up and mind-blowing in mathematics, in my opinion, and isn't easy to comprehend. But it is truly fascinating in my opinion and I really love it; I hope you will find it equally interesting. The three most important lessons I was trying to convey here are, that:

  1. Infinity is a much more complicated concept than just "something that has no end".
  2. There are many different kinds of infinity, and saying that something is "infinite" isn't enough to tell us just "how infinite" it really is.
  3. Many times, infinite sets that can appear to be different in size, actually aren't.

There's much and more to learn about infinity, and set theory as a whole, and I just might explore it in future entries. Thanks for reading and your feedback will be highly appreciated!

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The Red Viper's guide to mathematics - part 3: Can't spell "Function" without "Fun"!

Yeah yeah, mock me for the title later.

Anyway, it's been quiiiiiite a while since I made that last entry, but I really felt like making a new one. Let me tell you, we're really getting somewhere, and this entry will give us everything we need in order to talk about infinity and its different cardinalities, which is one of my favorite subjects in set theory and in mathematics overall.

Now, what is a function? You may remember functions from high-school as some thoroughly unpleasant piece of business that you were forced to derive, integrate, and do all other sorts of evil, macabre and even somewhat satanic things with. But functions are much more than a series of operations that map "x" to "y" and are actually quite fascinating. Let's start with the basics.

What are functions composed of?

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Well, in order to define a function, all you really need are 3 sets (which we discussed in the 2nd entry): a domain, a co-domain, and a graph, which is a set of ordered pairs.

Before we continue, let's take a moment to see what an ordered pair is. Pretty intuitive, really; an ordered pair is a pair of 2 elements, with importance to their order of appearance. Ordered pairs are notated with either rounded parentheses - for example (x,y), or with angle brackets - for example ⟨x,y⟩ (angle brackets are used so that ordered pairs aren't confused with open intervals, which are traditionally notated with rounded parentheses as well). The main thing about ordered pairs, and what differentiates them from a set with 2 elements, is, well - the order. for example, the sets {1,2} and {2,1} are identical. But, the ordered pairs (1,2) and (2,1) aren't. Moreover, while a set has no repetitions, an ordered pair can have repetitions. For example, the set {1,1} has in fact just one element and is just the same as {1}, whereas the ordered pair (1,1) is still a pair and has two elements - both of which just happen to be identical.

Now, let's talk about each of the three components mentioned above:

  • The domain of the function is the set of elements we wish to map.
  • The co-domain of the function is the set of elements to which we map the elements of the domain.
  • And the graph is the set of ordered pairs where each ordered pair has an element from the domain (its left component) and an element from the co-domain (its right component). The graph represents the mapping performed by the function.

Let's look at an an example. Let's say that our domain is the set of the following superheroes: {Superman, Wolverine, Spider-Man, Aquaman}, and the co-domain is the set of the following superpowers: {Flight, Healing, Spider-Sense, Talking to fish}. We would like to map each superhero from the domain, to their respective superpower from the co-domain. So, this is what the graph would look like: { (Superman, Flight), (Wolverine, Healing), (Spider-Man, Spider-Sense), (Aquaman, Telepathy) }. Note that the left component of each ordered pair is an element from the domain, and the right component is from the co-domain.

Now, a function's graph can just be a set of ordered pairs that have no sense to their mapping (such as some mathematical formula, or mapping a superhero to a superpower). The following illustration is a good visual representation of a general function:

Source: Wikipedia
Source: Wikipedia

The set X = {1,2,3} is the domain, the set Y = {A,B,C,D} is the co-domain, and the graph would be { (1,D), (2,C), (3,C) }. Is there some hidden logic or formula behind this mapping? Probably not, but it doesn't matter. It's still, for all intents and purposes, a function.

Notation and properties of a function:

When you define a function, you have to first say what your domain and co-domain are. Let's say we want to define the function f, with both the domain and the co-domain being the set of natural numbers. We write - f: ℕ→ℕ.

Next, we need to define what the function actually does (or just explicitly write the graph, but we rarely do that). For example, if we want f to map each element of ℕ to its successor, we would write - f(x)=x+1, with "x" being the element of that we want to map.

A function doesn't have to have a single formula for all elements in its domain. We can split a function to map different elements in different ways. For example, consider the famous function D: ℝ→ℤ defined by:

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This is a famous function, notated with a "D" after the man who first defined it - the accomplished German mathematician Johann Dirichlet, and is a classic example of a split function that maps different elements in the domain to different elements in the co-domain based of different conditions: each real number which is also rational is mapped to the integer 1, and each irrational real number is mapped to the integer 0. When defining a split function, you can split it to however many conditions you want, but usually no more than 2 are used.

Now, the most important property of a function, is that no single element in the domain can be mapped to more than one single element in the co-domain. In other, more formal words: Let A, B be non-empty sets and let f: A→B be a function. Let x∈A and y,z∈B. If f(x)=y and also f(x)=z, then y=z. Let's look at the example with the superheroes and superpowers again, but this time, we'll add in the element "Heat Vision" to the co-domain. If we want to map each of the superheroes to their superpower, we would have to map Superman to both "Flight" and "Heat Vision", meaning that it isn't a well-defined function unless we change its definition. Do note that if 2 different elements in the domain are mapped to the same element in the co-domain, that's completely fine. For example, if we added Deadpool to the domain, both Deadpool and Wolverine would have been mapped to the superpower "Healing". That's fine.

Let's consider some other properties that functions can have. For the following definitions, we will consider the sets A and B and the function f: A→B:

  • Total:f is total if every element in A is mapped to an element in B.
    • Formally: ∀x∈A(∃y∈B (f(x)=y) ).
  • Injective:f is injective if no two elements in A are mapped to the same element in B. An injective function is often called "one-to-one".
    • Formally: ∀y∈B( ∀x,z∈A ( (f(x)=y)^(f(z)=y)⇒(x=y) ) ) - in other words, if "x" and "z" are both mapped to the same element "y", then "x" and "z" are the same element.
  • Surjective:f is surjective if every element in B has an element in A that's mapped to it. A surjective function is often called "onto".
    • Formally: ∀y∈B( ∃x∈A (f(x)=y) ).
  • Bijective:f is bijective if it holds all three properties - total, injective and surjective. A bijective function is called a "bijection".

These properties are important in proving different claims and theorems regarding functions, and especially in proving the cardinality of the domain and the co-domain.

How do we prove that a function has these properties?

That is actually pretty easy.

Showing that a function is total is easiest and usually takes no more than a small sentence that explains it. It could be more than that sometimes, of course, when dealing with unorthodox and obscure domains and/or co-domains.

Showing that a function is injective is also not hard, if you know how t do it right. Basically as explained earlier, an injective function is a function where no two different elements in the domain are mapped to the same element of the co-domain. However, proving it could be a little tricky; luckily, we can use the contrapositive (which we already saw in the first entry). We will assume that two elements in the domain (say, x and y) are mapped to the same element in the co-domain, and prove that x and y have to be the same element. For example, let's prove that f: ℕ→ℕ defined by f(x)=x+1, which we saw earlier, is injective:

  1. Let x,y∈such that f(x)=f(y).
  2. That is, x+1=y+1, because that's how we defined f.
  3. Subtracting 1 from both sides leaves us with x=y, as required. Q.E.D.

See? That wasn't hard, right? There are other methods for proving that a function is injective. For example, showing that it is strictly monotone, but more on that perhaps in future entries - that's already stepping into the realms of calculus, aka satanism and dark sorcery.

Finally, let's see how we prove that a function is surjective: in order to do that, we basically need to consider a general, arbitrary element in the co-domain and show that there's an element in the domain that's mapped to it. For example, let's consider the function f:→ℤ, defined by f(x)=x+1 (try thinking why we couldn't have used the same function, but with the domain and co-domain being , like we did above):

  • Let y∈. We need to show that ∃x∈, such that f(x)=y.
  • Consider x=y-1. It's easy to see that (y-1)∈, given that y∈ℤ.
  • Therefore, f(y-1)=(y-1)+1=y.
  • Given that y is arbitrary, we deduce that ∀y∈( ∃x∈ (f(x)=y) ), as required. Q.E.D.

Proving that a function is surjective is also usually not too difficult.

The true challenge, most often, is trying to think of a function between 2 certain sets that satisfies some required property. Why is that necessary? Well, because that allows us to learn things about the cardinalities of these 2 sets.

Cardinality:

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"Cardinality" is the term we use when referring to the "size" of a set. For example, the cardinality of the set A = {1,2,3,4,5} is 5, because it has 5 elements in it. Likewise, the cardinality of the set B = {10,33,980,-12,0.6} is also 5. Cardinality is notated the same as absolute value, ergo: |A| = |B| = 5. Now, what is the cardinality of, say, ? More on that in the next entry.

Anywho, what does cardinality have to do with functions? More specifically, with the four aforementioned properties (total, injective, surjective, bijective)? Well, everything! Here's the thing: let's consider two non-empty sets A and B, and a function f: A→B.

  1. If f is total and injective, then we can deduce that |A| ≤ |B|.
  2. If f is surjective, then we can deduce that |A| ≥|B|.
  3. Finally, if f is bijective, then the only choice we're left with is |A| = |B|.

Now, this all makes sense and may sound pretty intuitive, but we gotta prove it. So, let's do it!

(Feel free to skip this following part if you want, you can just accept the above three facts).

1. f is total and injective. That means, for every x∈A there exists a y∈B, such that f(x) = y (since it is total). Moreover, there doesn't exist z∈A such that f(z) = y and z ≠ x (since it is injective). Now, let's assume by contradiction, that |A| > |B| (opposite of what we have to prove). Let's mark |A| = n, |B| = m, ergo, n > m. Now, let's look at the graph. f is total, and as such, each element in A is mapped to exactly one element in B. So, the graph has to have exactly n ordered pairs in it (one pair for each element in A). But, we have assumed that n > m, and as such, there aren't enough elements in B such that each ordered pair in the graph has a unique right component (ergo, there has to exist some y∈B such that y is the right component of at least 2 different ordered pairs in the graph). That means that there are 2 different elements in A, both of which mapped to y. So, f is not injective! We have reached a contradiction. Therefore, the assumption we made, by which |A| > |B|, cannot hold true, therefore |A| ≤ |B|. Q.E.D.

2. f is surjective. That means, for every y∈B, there exists x∈A such that f(x) = y. As before, let's assume by contradiction the opposite of what we have to prove - so |A| < |B|. Again we'll mark |A| = n, |B| = m, ergo, n < m. It's easy to see that the elements of A will be mapped to, at the very most, n different elements in B. Since we assumed that n < m, we are left with elements in B which have no element in A mapped to them. In other words, there exists y∈B, for which there does not exist any x∈A such that f(x) = y. So, f is not surjective! Again, we have reached a contradiction, and as such the assumption we made, by which |A| < |B|, cannot hold true, therefore |A| ≥|B|. Q.E.D.

3. That is the easiest part; since f is bijective, it's total, injective and surjective all at the same time, by the previous 2 proofs that we've now seen, the 2 conclusions must hold, ergo |A| ≤ |B| and |A| ≥|B|. That can hold true only when |A| = |B|. Q.E.D.

So now we know how functions are relevant to cardinalities.

In summary:

Awesome! Now we know what functions are. We have seen what makes a function and what different properties a function can have, which led us to seeing what cardinalities are, and how functions and cardinalities are relevant to one another.

That was quite a lot to take in, but now we have almost everything we need to discuss infinity. That would be a pretty tough subject to wrap our minds around, but it's super-interesting, trust me. Next entry coming soon (not half a year this time).

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The Red Viper's guide to mathematics - part 2: Ready, Set, Go!

So last time, we saw some basic symbolic logic and an introduction on formal proofs. Today we will expand on that, and also start talking about sets - which will lead us, in the future, to learn some pretty interesting stuff about Set Theory - which many consider to be the theoretical basis for all mathematics and is a personal favorite subject of mine.

Let's start with sets.

What is a set?

Well, a Set is any group of elements. A set of numbers, a set of words, a set of people, of objects, of ideas. Anything. Even a set of sets.

Generally, sets are denoted with curly brackets, i.e {...}. Inside the brackets, we put the elements of the set. For example, S = {1,2,3,4} is a set that has 4 elements, which are the numbers 1, 2, 3 and 4. The order of the elements is meaningless, and so are repetitions. For example, all the following sets are identical:

  • {1,2,3,4}
  • {3,2,4,1}
  • {1,1,2,4,2,1,3,3,1,4}

Almost any group of elements you can think of can be defined as a set. There are certain axioms that dictate what sets can or can't be defined, but more on that in the future. Sets can be elements of other sets, too; for example: {1,2,3,{4,5,6,7},8,9}. This is a set that has 6 elements: the numbers 1, 2, 3, 8 and 9, and the set {4,5,6,7}.

There exists one set which has no elements in it. It is called the "Empty Set", and is denoted: Ø. Sometimes it is also denoted {} (empty curly brackets), but the former notation is more common.

Notations and operations on sets:

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There are several notations we should mention before we keep talking about sets.

  • Belonging/membership: denoted "∈". x∈A means "x belongs to/is an element of A". Likewise, x∉A means "x doesn't belong to A". For example, if we define A to be the set of all Marvel heroes and B to be the set of all DC heroes, and we define x as Spider-Man, we can say that x∈A and x∉B.
  • Containment: denoted "⊆". Containment is very often confused with belonging as they are both very similar concepts. But whereas belonging is a relation between a set and an element, containment is a relation between a set and another set. We say that A⊆B (A is contained in B) if A is a subset of B, i.e every element of A is also an element of B. For example, {1,2,3}⊆{1,2,3,4}. As another example, if we define A to be the set of Marvel characters and B to be the set of Marvel villains, then we can say that B⊆A. Do note that if set A is an element of set B, for example, B = {1,2,3,A,4,5}, then it is NOT true that A⊆B, but it IS true that A∈B. Interesting note: the Empty Set is contained in every set. Since it has no elements to begin with, then we can, in fact, say that "every element of Ø is also an element of A", no matter what set "A" is. Also, every set is a subset of itself (A⊆A no matter what A is). Note that the notation "⊂" also exists and means "strict containment", i.e A⊂B means that A is a subset of B but is NOTequal to B.
  • Union: denoted "∪". The union of two sets A and B (A∪B) is the set that you get when you combine A and B. For example, if we define A = {1,2,3} and B = {4,5,6}, then A∪B = {1,2,3,4,5,6}. Or if A = {1,2,3} and B = {1,4,5} then A∪B = {1,2,3,4,5} (since repetitions are meaningless in sets, we drop the extra '1'). As another example, if A is the set of all decent MCU films and B is the set of all decent DCEU films, then A∪B = A (KIDDING!!! Don't kill me!!!)
  • Intersection: denoted "∩". The intersection of two sets A and B (A∩B) is the set of all elements that belong in A and in B both. For example, if we define A = {1,2,3} and B = {1,2,4}, then A∩B = {1,2}. If A = {1,2,3} and B = {4,5,6}, then A∩B = Ø. If A is the set of all things in the world, and B is the set of all things that Jon Snow knows, then A∩B = Ø.
  • Relative complement: denoted "\". The relative complement of two sets A and B (A\B) is the set of all elements of A that are not elements of B. For example, if we define A = {1,2,3,4} and B = {1,2,3,5}, then A\B = {4} (note: a set that only has one element is called a "Singleton"). If A⊆B, then A\B = Ø (since every element of A is necessarily an element of B).

There are several more of those, such as the Cartesian Product of sets which we will surely meet sooner or later, but these are the important ones for now. So, why do we need this to continue talking about true, false and formal proofs? Well, that's because without sets, we wouldn't be able to talk about quantifiers and predicates!

So let's talk about quantifiers:

Well, I lied; there's one more thing I wanna say about sets, but it's short, I promise!

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There are a few "special" sets that are commonly used in mathematics, and have their own special notations. I'll say a few words about each of them before we proceed:

  • Integers: denoted "ℤ" (fancy Z, from the German word "Zahlen" - "numbers". The Power of German Originality). An integer is a number whose fractional part equals zero (i.e has only 0 after the decimal point). Basically, ℤ = {0,1,-1,2,-2,3,-3,...}.
  • Natural numbers: denoted "ℕ" (fancy N). Natural numbers are the positive (or non-negative) integers. So basically, ℕ = {0,1,2,3,4,5,...}. Important note: sometimes 0 is considered a natural, sometimes it isn't, it mostly depends on which field of mathematics you're studying. In discrete mathematics and combinatorics for example, 0 is a natural number, and in calculus it isn't. For the purpose of this blog, we'll say that it is.
  • Rational numbers: denoted "ℚ" (fancy Q, from the Italian word "Quoziente" - "quotient"). A rational number is any number that can be expressed as the quotient of 2 integers. For example: ½ is a rational number since it is the quotient of the integers 1 and 2.
  • Real numbers: Everything else, basically. Denoted "ℝ" (fancy R). The set of real numbers is the set of all numbers, both rational and irrational. For example, π∈ℝ. √2∈ℝ (it can be proven that the square root of any prime number is irrational, maybe in one of the future entries). e∈ℝ (Euler's Constant, we will meet it again in future entries probably). Note that the set of irrationals has no special notation and is simply expressed as ℝ\ℚ (the relative complement of the reals and the rationals, basically all reals that aren't rationals).

Note that: ℕ⊆ℤ⊆ℚ⊆ℝ.

Set-builder notation: a way of defining our own subsets of certain sets which we already know, according to some condition. The set-builder notation is used as follows: A = {x∈[set] | [condition]}. For example, if we write: A = {x∈ℤ | x<0}, that would mean that we defined A to be the set of all negative integers.

Now we can continue so let's talk about quantifiers. There are two quantifiers which are commonly used in mathematics:

  • There exists: denoted "∃" (reversed E, as in "Exists").
  • For all/for each: denoted "∀" (reversed A, as in "All").

A quantifier comes before a certain variable which belongs to a certain set, followed by a certain proposition regarding the variable. For example, if we write: ∃x∈ℕ(x>5), it means "there exists a natural number "x" which is bigger than 5" (note that "there exists" doesn't necessarily mean just one). As another example, we can write: ∀x∈ℕ(x>-1), and that means "all natural numbers are bigger than -1".

Of course, if we want to negate a quantifier ("not for all" or "there does not exist"), we can use the negation symbol we saw in the previous entry: "¬". However, a more elegant way of negating a quantifier is using the other quantifier and reversing the statement. For example, if we want to say "there does not exist a natural number smaller than -5", we can either write: ¬(∃x∈ℕ(x<-5)), or we can write: x∈ℕ(x>-5). That is a common way of disproving/refuting mathematical statements. Most mathematical statements are in the form of "for all" (for example, "all differentiable functions are continuous"), and in order to refute such a statement we need to find a counterexample - or in other words, show that "there exists a differentiable function which isn't continuous" (there doesn't).

A statement can have more than one quantifier of course. For example: ∀x∈ℕ(∃y∈ℤ(x+y=0)), which means "for all natural numbers "x" there exists an integer "y" such that x+y=0". Also, these kinds of statements can (and usually do) use the symbolic logic we saw in the previous entry (¬ ⇒ ⇔ ^ ∨ ⊕). For example, consider Goldbach's Conjecture: Goldbach's Conjecture suggests that every even integer greater than 2 is the sum of 2 primes. How do we Write it? First, let's define:

  • A = {x∈ℕ | (x>2)^((x/2)∈ℤ}
    • In other words: A is the set of all naturals that are larger than 2, which yield no remainder when divided by 2.
  • B = {x∈ℕ |(x≥2)^(∀y∈ℕ(y≠1)^(y≠x)⇒(x/y)∉ℤ)}
    • In other words: B is the set of all naturals larger than or equal to 2, that if you divide them by any number other than 1 or themselves, you get a remainder.
  • We write: ∀x∈A(∃y,z∈B(y+z=x))
    • In other words: for each x in A, there exist y,z in B, such that y+z=x.
  • Side note: Goldbach's conjecture is yet to be proven or refuted to this day.
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∀∃ vs. ∃∀:

Yes, there's a difference. Consider the two following two suggestions:

  1. For each superhero, there exists a supervillain.
  2. There exists a supervillain for all superheroes.

These two sentences are a classic example for the stark difference between ∀∃ (for each... there exists...) and ∃∀ (there exists... for all...).

Consider the first suggestion: "For each superhero, there exists a supervillain." You name a superhero, and I name their villain. You say Superman, I say Lex Luthor. You say Thor, I say Loki. This suggestion implies that for each superhero there exists a villain, but nothing assures us that two heroes would have the same villain.

Now, consider the second: "There exists a supervillain for all superheroes." There's a specific villain you can name who is an enemy to all heroes. You name Thanos, I name the entire Marvel Universe. You name Darkseid, I name the entire DC Universe. You name Tom King, I name the entire comics industry.

Predicates:

Now, on to predicates. A predicate (usually marked as "P") is a proposition on a certain variable which can either be true or false. For example, if we define a predicate P, by P(x) = "x is even". The set A = {x∈ℕ | P(x)} would be the set of all natural numbers for which P is true (in this case, the set of all even naturals). A predicate can also be defined on more than one variable. For example, we can define P(x,y) = "x is smaller than y". This kind of predicate cannot be used for a set-builder notation, however. Let's see how we can express Goldbach's Conjecture which we saw earlier using predicates:

  • P(x) = "x is even"
  • Q(x) = "x is prime"
  • We write: ∀x∈ℕ((x>2)^(P(x)))⇒∃y,z∈ℕ((Q(y))^(Q(z))^(y+z=x))
    • In other words: For each natural number x, if x is larger than 2 and is even, it implies that there exist two naturals y,z such that y is prime and z is prime and y+z=x.

Examples for the use of quantifiers and predicates:

1. All math must die:

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  • A = the set of all men.
  • P(x) = "x must die".
  • We write: ∀x∈A(P(x)) - "for all men "x" there holds that x must die".

2. Lord of the Reals:

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  • A = the set of all living beings.
  • B = the set of all rings.
  • P(x,y) = "x rules y".
  • We write: ∃x∈B(∀y∈A(P(x,y))) - "There exists a ring "x" such that for each living being "y" there holds that x rules y".
    • Note that I used ∃∀ and not ∀∃.

3. Guns N' Quantifiers:

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  • A = the set of all people.
  • a = you.
  • P(x,y) = "x needs y".
  • We write: (∀x∈A(∃y∈A(P(x,y))))^(∃b∈A((b≠a)^(∃c∈A(P(b,c))))) - "For each person "x" there exists a person "y" such that x needs y, and there exists a person "b" that isn't you, for which exists a person "c" such that b needs c".

*Note: I'm using so many parentheses because that's the most "formal" way of writing such statements, even though it's kinda clustered. Usually, some parentheses are dropped where they aren't really necessary for understanding the statement.

In conclusion:

Well, this was a nice little intro into the fascinating world of mathematics. I wanted to start with those 2 entries, even though they might be a tiny bit dull, because that's really as basic as it gets, and in future entries we will again meet everything we've discussed so far. I promise, it's gonna get more interesting from now on. What we just discussed about sets prepares us for some Set-Theory, and discussions on some very interesting stuff like cardinalities of infinity, and now that we know how to use quantifiers and predicates we can use that to show and prove some pretty mind-blowing theorems. Trust me, there's more to come and it's gonna be fascinating.

Thanks for reading!

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The Red Viper's guide to mathematics - part 1: True or false?

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Why am I doing this?

So, I've decided to do this after making a rather shocking (and some would say disturbing) discovery: Math is awesome, and I love it. Last October, I started my computer science studies, and my freshman year involves several courses (6 actually, maybe 7 if you count Data Structures) in different fields of math. I came on my first day scared out of my senses; math was the bane of me back in high-school. I mean, yeah, I passed my finals with an OK-ish grade, but it was nerve-racking and each and every moment was complete and utter suffering. I knew I'm going to have tons of math studies in my freshman year, but I figured I should just grit my teeth and get it over with, since I really wanted to take CS. But after my first lecture of the semester... I found out it wasn't all that bad. By the end of the semester, I found out that, despite myself, I'm actually starting to enjoy it. So, I figured I might as well share some of that nerdy enthusiasm with you, and pass on some of what I learned in a more "friendly" way than they do in school or in the academy.

I took the idea for this from the late ImpurestCheese, who shared her knowledge and love for animals with this community for quite a while before losing her battle against cancer not 2 years ago. I'm not much of an animal expert, and I've never really interacted with her all that much, but I hope to honor her in some way by sharing my growing knowledge of mathematics with you. It'll help me memorize it better myself as well, and you guys just might enjoy the read, so everyone's a winner.

What am I going to do here?

I'll focus mostly on logic, discrete mathematics, and set theory, at least for now. Maybe I'll add some calculus (AKA "evil") or linear algebra too, but those'll be a tad harder to pass on through forum threads - so I make no promises. I'll assume you have some basic knowledge from school, but you're more than welcome to ask me anything you don't understand, and I'll try my best to answer. If you ask a good question and I can't answer it, maybe I'll even ask my professors!

Anyway, let's start with formal proofs - pretty much the very basics of the basics. It was the very first thing I learned in discrete mathematics, which turned out to be my favorite subject last semester, so I figured I might as well start here.

Why do we need it?

Why do we need proofs? Well, my friends and I have this joke that math studies is like law studies: all you do is learn how to lie elegantly and call it a "proof". I like to think of math as the mommy and daddy of all science. It's probably the most precise field of science there is, and if you want precision, you can't do anything without basing it off some tangible, concrete evidence. Which is why every theorem, every corollary, everything in math requires proof. If you can't prove you're right, then you're wrong. There is (usually) no middle ground.

How do we do it?

How do you prove something in math, though? Well, first you need to know what you want to prove, obviously; then you need to gather everything that you already know that could help you prove it. Sounds easy, right?

Let's start with a very simple example, that might not look like it has anything to do with math at first glance: a vending machine. Dibs on the "Hershey's"!

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Let's say that we have a vending machine and we want to know if it works properly. The only thing we know about vending machines is this: every time you insert a quarter into the vending machine, you should get a snack. Simple, right? Now, let's consider the following 4 possible situations:

  1. You insert a quarter, and the machine drops a delicious Snickers bar. Awesome! Everything worked properly!
  2. You insert a quarter, the machine does some machine-noises, but... nothing comes out. The machine must be broken!
  3. You don't insert a quarter, and the machine drops nothing. What did you expect?
  4. You don't insert a quarter, but the machine still drops a snack. Who doesn't love a free snack?

Now, in those 4 scenarios, it's quite obvious that in case 1 the machine worked properly, and in case 2 it was clearly broken. But... what about 3 and 4? Did the machine work, or did it not?

Now, if I were a vending machine technician, I might have given you a different answer. But, mathematically, the machine worked (or, at least, wasn't broken) in all cases other than case number 2. What we know is this: IF we insert a quarter, THEN the machine will drop a snack. Do we know what happens if we don't insert a quarter? Well, no, we don't, because we weren't told. In cases 3 and 4, we did not insert a quarter, and so the machine worked just fine as far as we care.

For another, more mathematical example, let's look at the following claim: let X be an integer. We say that X is odd if it yields a remainder when divided by 2. So, let's check whether X is odd, shall we? Again, 4 possible cases:

  1. We divide X by 2 and the division yields a remainder. X must be odd!
  2. We divide X by 2 and the division yields no remainder. Looks like X is even!
  3. We divide X by 3 and the division yields a remainder. Is X odd? Well, could be! We weren't told what should happen if we divide it by 3.
  4. We divide X by 3 and the division yields no remainder. Is X odd? Again, could be.

So, if we use the vending machine analogy, dividing by 2 is whether or not we insert the quarter, and getting the remainder or not is akin to getting our snack or not.

How do we write it?

Math is a universal language. Take a French mathematician and an Indian mathematician, I promise you they can communicate just fine if they keep it formal. Everything in math has its own unique marking, and this time we're talking about implications. Having a remainder when you divide X by 2 implies that X is odd, just like putting a quarter in the vending machine implies that you should get a snack.

We denote: P = our hypothesis (or assumption, or condition), and Q = our expected result (or conclusion). "P implies Q" is denoted: P⇒Q. Pretty intuitive, right? Another important symbol is "≡", which means "Is equivalent to", in the Boolean (true/false) sense. "True" and "False" are naturally denoted as just T and F respectively. Looking back at our vending machine example, we can properly denote it now:

Did we insert a quarter? (T if so, F otherwise)Implies...Did we get a snack? (T if so, F otherwise)Is equivalent to...Is the machine working? (T if so, F otherwise)Explanation
TTTWe put in the quarter, and got the expected result. Hooray!
TFFWe put in the quarter and got no snack. That is the opposite of the expected result.
FTTWe didn't put in a quarter and still got our snack. Since, as far as we know, getting the snack isn't solely dependent on inserting the quarter, the machine still works as far as we know.
FFTWe didn't put in a quarter and didn't get the snack. Nothing unexpected happened, so the machine works.

So, with this in mind, we can conclude that:

  • T⇒T≡T: The hypothesis held and we got the desired result - our assumption is true.
  • T⇒F≡F: The hypothesis held and we did not get the desired result - our assumption is false.
  • F⇒T≡T: The hypothesis did not hold but we still got the desired result - our assumption is true (but can still be refuted).
  • F⇒F≡T: The hypothesis did not hold and we did not get the desired result - our assumption is true (but can still be refuted).

There are, of course, more symbols that are important to familiarize with. Here are a few:

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  • Equivalence/"If and only if": denoted "⇔". P⇔Q means P⇒Q and Q⇒P, basically a double-sided implication. For example, if we say: P = "Walks like a duck and quacks like a duck" and Q = "It is a duck", and then we say: "It walks like a duck and quacks like a duck IF AND ONLY IF it is a duck", we can write it as: P⇔Q. If it walks like a duck and quacks like a duck, then it is a duck - and if it is a duck, then of course it walks like a duck and quacks like a duck.
  • Negation/"Not": denoted "¬". ¬P means "P is false". In general: (¬P≡T)⇔(P≡F). For example, if we say: P = "I'm happy", Q = "I cry", and also "If I'm NOT happy, then I cry", we can write it as: ¬P⇒Q. Do I cry if I am happy? Maybe. If I do, the claim still holds. Take note of double-negative: for example, if P = "I'm NOT happy", then ¬P = "I am happy", and naturally, ¬(¬P)≡P.
  • And: denoted "^". P^Q means "P is true AND Q is true". In general: [(P^Q)≡T]⇔[(P≡T)^(Q≡T)]. For example, if we say: P = "I'm hungry", Q = "I'm tired", and R = "I'm cranky", and then we say: "If I'm hungry AND tired then I'm cranky", we can write it as: (P^Q)⇒R. Am I cranky when I'm just hungry, or just tired? Maybe. If I do, the claim still holds. We can now also write our claim about the ducks in another, more elegant way: we say: P = "Walks like a duck", Q = "Quacks like a duck", and R = "It is a duck", and then we say: "It walks like a duck and quacks like a duck IF AND ONLY IF it is a duck", we can write it as: (P^Q)⇔R.
  • Or: denoted "∨". P∨Q means "P is true OR Q is true OR both of them are true". In general: [(P∨Q)≡T]⇔[(P≡T)∨(Q≡T)]. For example, if we say: P = "I'm hungry", Q = "I'm tired", and R = "I'm cranky", and then we say: "If I'm hungry OR tired then I'm cranky", we can write it as: (P∨Q)⇒R. Am I cranky when I'm just hungry, or just tired? Yes. Am I cranky in general? Also yes, but that's beside the point. An interesting note would be that standard implication, "P implies Q" is equivalent to "Not P, OR Q", i.e (P⇒Q)≡(¬P∨Q). Look at the table above if you don't believe me!
  • Exclusive or/"Xor": denoted "⊕". P⊕Q means "P is true OR Q is true but NOT both". In general: [(P⊕Q)≡T]⇔[((P≡T)^(Q≡F))∨((P≡F)^(Q≡T))]. For example, if we say: P = "Alice will ask me to dance", Q = "Susie will ask me to dance", and R = "I will dance", and then we say: "I will dance IF Alice asks me OR Susie asks me but NOT if they both ask me", we can write it as: (P⊕Q)⇒R. Calm down there, ladies. "Xor" conditioning is pretty rare in mathematics, but it exists and is still relevant sometimes.

A few more examples:

1. Implication Is Coming:

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  • P = "You kill me"
  • Q = "Your brother's a dead man"
  • We write: P⇒Q

2. The Discrete Knight:

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  • P = "You die a hero"
  • Q = "You live long enough to see yourself become the villain"
  • We write: ¬P⇒Q

3. Fleetwood Math:

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  1. P = "You love me now"
  2. Q = "You will love me again"
  3. We write: ¬P⇒¬Q

4. Bye Bye, Ms. American π:

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  • P = "I had my chance"
  • Q = "I could make those people dance"
  • R = "Maybe those people would be happy for a while"
  • We write: P⇒(Q^R)

Contrapositive:

Contrapositive is basically "proof by contradiction": switching and negating the hypothesis and conclusion of a conditional statement. With P being the hypothesis and Q being the conclusion, the contrapositive of P⇒Q is ¬Q⇒¬P. For example, if our statement is: "If you put a quarter in the vending machine, you'll get a snack", then its contrapositive will be: "If you did NOT get a snack, then you did NOT put a quarter in the vending machine". Our goal with using contrapositive is reaching a contradiction of out hypothesis: if we assumed the opposite of what we're trying to prove, and we see that it contradicts out hypothesis (or some other mathematical axiom), then we have our proof.

Many times, a theorem or a statement can be very hard to prove (or refute) directly, but contrapositive makes it a lot more easy - sometimes (rarely) trivial, even. Here's an example (mathematical and boring, there's a more entertaining one ahead):

Let X be a positive integer larger than 2. Assume X is even; therefore X is not prime (P = "X is an even integer greater than 2", Q = "X is not prime").

That example is actually easy to prove directly, but let's see how it can be done with contrapositive. We'll assume by contradiction that X is prime. Therefore, there does not exist any integer Y such that X/Y yields no remainder. Therefore, X/2 must yield a remainder. But we know X is even, so we reached a contradiction of our hypothesis that X is even, which means X cannot be prime!

Another example: Let's say I browsed the Battle forum the other day, and came upon an awesome CaV match, that had 4 great posts from each debater, and a very close competition on the votes. Therefore... Nick (@the_hajduk) was NOT one of the debaters. Proof:

Assume by contradiction that one of the debaters was @the_hajduk. We know that this debate ended after 4 posts from each debater and got as far as voting. But we also know Nick's track record at getting this far in a debate without quitting. That's a very obvious contradiction! Therefore Nick was most definitely not part of that debate, and our claim is proven.

So what does this whole implication thing have to do with proofs?

Well, everything! If I make a claim and I want to prove it, I need to know what my hypothesis is, and what the hypothesis implies. What we saw above is the basis for thousands of years in which all fields of math evolved. Every theorem we know started as P - a hypothesis, which people used to show the truth of some Q - a conclusion. And those theorems were used for finding other theorems, following the very same logic - and so on and so forth. It all started with a simple P, all those thousands of years ago, and is turned into P⇒Q⇒R⇒S⇒...⇒and now we have electric cars and space travel. Brilliant.

In conclusion:

Well, it isn't really a conclusion. This really is the very basics of it all, but there's still much and more to be said about proofs... but we will get there, this is just the first entry in this blog. Hopefully, there'll be enough feedback for me to make another, because I did enjoy making this one. I always enjoy sharing whatever knowledge I can with whoever I can. Don't worry, it'll get more and more interesting as we go on - but now the basics are out there for you to enjoy (or not). I'll again mention that I'm still on my freshman year, but even now I realize how much there is to it all that I really never knew back in high-school. I'm no expert on math - I don't even have a degree, heck I'm still on my freshman year. But I did gather knowledge of some really interesting stuff that I would love to share.

Anyway, that's it for now. Hope you liked it!

30 Comments

The_Red_Viper's song competitions

Current polls:

Future polls:

I have a poll-bank with a list of future polls I'm planning to make, but I'm willing to consider requests if you're interested in one of my polls to feature a certain artist or song.

2 Comments

The_Red_Viper's CaV's

Victories

Defeats:

Draws:

Ongoing:

Hope I didn't forget anything..

1 Comments

Why "You VS X" battle threads are stupid

A lot of times, when scouring the battle forums, I see people posting threads like "You VS Batman" and things like that (often those threads would involve some characters that are considered weak/incapable, of course, not Batman). Now, these threads, in my opinion, are really dumb. You just can't say you win and expect to be taken seriously. Here's why...

1. You're not really a trained fighter.

No, you're not. Even if you took karate in 8th grade, or took part in your high-school wrestling team, or practice fencing... unless you're Olympic/international level, you're still going to lose to 99% of fictional characters that can be justifiably used in any battle thread. Even the amounts of training you get in the army or law enforcement would literally put you at generic fodder level. Unless you can be objectively considered a professional fighter in a certain school of martial arts, then no, you're not more skilled than Curtis Holt, you're not a better fighter than Rorschach and you're not beating Theon Greyjoy.

2. You don't know how to fight a real enemy.

Let's say that you really are a professional in MMA, or kung fu, or fencing, or whatever. Cool. Have you actually ever fought anyone outside of the dojo, with no rules, a fight to the bitter end where everything goes? This makes a difference, you know. It's different when your opponent isn't restricted by this rule or another.

3. You're not a killer.

Let's say you are a professional fighter, AND that you were involved in some brawls and fights and hospitalized someone, maybe even scared off a mugger or a burglar or something. I doubt you ever killed anyone. And yes, it matters. It matters a lot. There's a reason so many works of fiction make a point about a certain character hesitating before killing an enemy. It's not easy. Unless you personally looked someone in the eye and killed the before, it's not going to come easily for you. You will most probably hesitate.

4. And finally...

Even if you're an international champion of ninjutsu, Muay Thai, aikido and fencing, AND you're a decorated veteran served 10 years in the special forces in Iraq, Afghanistan and Winterfell, AND you have over 300 confirmed kills to your name - more than half of which with your bare hands... who the hell do you think is going to believe you?

So stop with these threads. And if you see these kinds of threads, don't bother to comment on them.

12 Comments

Warrior Analysis #1: Spartacus

This is a new thing I wanna try here. My goal here is to determine, more or less, the fighting prowess of various characters, by analyzing their feats and abilities as objectively as possible.

If this first thread gets positive feedback, I will keep it up and make others in the future. I will start with human, live-action characters, and maybe in the future I will advance to other tiers and/or mediums as well.

What is going to be tested here?

The characters I will examine here will be judged by 3 main attributes: strength, speed and skill. In addition, other various factors will be put into consideration - such as endurance, stamina, gear, mindset and so on, depending on the character. In the end, a final verdict will be given, giving the character a numerical rating in strength, speed and skill, and detailing all its other noteworthy advantages and disadvantages.

How do we do this and keep it objective?

The three main attributes - strength, speed and skill, will be gauged by analyzing the feats of the character in question and comparing them to a certain bar, set by another, unrelated character. The bar set in each field will be higher than the capabilities of our test-subject, and in the end our test-subject will receive a numerical rating (X/10) based on how close they got to this bar. The character chosen for each category will be in pretty much the same tier as our test-subject, and will represent more or less the pinnacle of their category in this tier.

To make it clearer, here's an example: let's say we want to analyze the Punisher. In order to rate the Punisher's strength, we will compare it to the strength of Deathstroke - a character who's completely unrelated to the Punisher, coming from a whole other verse. Deathstroke's strength is clearly above that of the Punisher, and will represent a 10/10 in the field of strength. The Punisher's best strength feats will then be analyzed, using Deathstroke's strength as a measuring stick, and the Punisher will receive a rating according to how close his strength comes to that of Deathstroke. The closer the Punisher comes to matching Deathstroke's strength, the higher the rating (for example, if the Punisher comes very close, he will get a 8/10, maybe even a 9/10, but never higher than 10/10). The same will then be done with speed and skill (with different characters used to set the bar in each category most likely). After those three main fields are fully analyzed, we will look at some of our test-subject's other attributes - both good and bad. These different attributes may differ, depending on the character in question.

Hope that makes it clearer. If not, just read on and I'm sure you'll understand.

Who are we looking at here?

This first analysis will discuss everyone's favorite gladiator - Spartacus (unless you're more of a Russel Crowe guy, that is). I decided to make this about him since I think he suffers from a horrible overrating on Comic-Vine, and despite stating my opinion regarding him in detail many times over in different battle threads, and even though I know many people will disagree with me here - I am 100% certain of my opinion on the guy and I want to put it in detail once and for all. Even though it will feature quite a few misinterpreted feats that need to be put into proportion, don't get me wrong here, this isn't a "debunk" thread or a rant of any kind, I loved "Spartacus" and binged it like a madman. I think he is extremely formidable and will pose a challenge to most live-action melee fighters of his tier, I just don't like to see any character's capabilities blown out of proportion.

Before we start, let's set the mood. Spartacus is a gladiator, so this fine track (which personally I absolutely love) seems very appropriate here:

So, without further ado, let's get to it.

Spartacus:

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Overview:

Spartacus, as we all know, is a very skilled and experienced fighter. He was a battle-tested Thracian soldier before being captured and enslaved, then proceeded to become one of the most famous and successful gladiators in the Roman Empire. After leading a slave revolt and killing his master, Spartacus became the leader of an army comprised of former slaves, mainly gladiators, and dedicated his life to make the Roman Empire bleed. The Romans suffered great losses against Spartacus and his army, until eventually Spartacus was killed in an epic battle against a Roman force lead by Marcus Crassus - even though it was a pretty close call.

Spartacus boasts good physical stats and skill, mostly as a result of the grueling gladiator training that he went through. Let's start gauging just what this ruthless killer has to offer, beginning with a very important and pretty straightforward physical stat - strength.

Strength:

As explained above, Spratacus's strength will be analyzed and compared to the strength of another, completely unrelated character. That character would be Ser Gregor Clegane, AKA "The Mountain", from the TV series "Game of Thrones", a monster of a man notorious for his inhuman strength. To make things clearer, in case you did not quite understand the explanation I gave above, for the purpose of this thread - let's say Gregor's strength gets a 10/10 rating. Spartacus's own strength will be gauged by analyzing some of his best strength feats and receive a numerical rating out of 10 in comparison to Ser Gregor's strength.

Let's start by putting forth some of Gregor's strength feats, to use as a measuring stick:

  1. Easily lifts a man over his head.
  2. Pops a human skull like an overripe tomato with his bare hands - probably his best strength feat, which is absolutely superhuman.
  3. Throws Oberyn Martell over a considerable distance with one hand.
  4. Cuts a man's arm off and kills the man's horse with one swing of his sword (source video here).

Now, let's start going bit by bit, over some of Spartacus's most prominent strength feats. I'll start with a disclaimer - Spartacus has quite a bit PIS showings. He has some crazy super-soldier-level showings, but those are few and far between and every one of those can be countered with a comparable showing that's simply far inferior. I will start with those outliers, in order to get them off the table first.

Let's start with a very notorious feat, from season 1 episode 13:

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What we see here is Spartacus, with some help from Crixus who gave him boost with his shield, jumping straight to a 2nd-story balcony. That is definitely a pretty superhuman feat, no doubt about it. I can't think of many non-enhanced characters, if any at all, who could do such a thing. However, that is one of these outliers I was talking about. Spartacus has made many such jumps throughout the show, and has never achieved this kind of result again.

Let's look at this one for example, from season 1 episode 5:

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Same ally, same shield, exact same sprint before taking the jump - but Spartacus didn't achieve half of what he did in episode 13.

Here's another, from season 1 episode 10:

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A different ally this time, but exact same idea. The double beheading was a nice touch and will definitely be taken in mind here, but the jump itself, which is the point here, was nothing special. In fact, Spartacus literally asked for Varro to position his shield for that jump - which means Spartacus knew he couldn't accomplish that jump on his own. Can be seen here.

One last example, from season 3 episode 10, the very last episode of the show. Spartacus used a pile of dead bodies as leverage instead of an ally's shield in this one, but it doesn't make much of a difference so it doesn't matter:

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Again, not near as good as that superhuman jump.

Now, an argument could be made that Spartacus simply jumps as high as the moment requires him to. But that's just another way of saying "he is as strong as the author wants him to be", which means it's quite an illegitimate excuse. Think of all the times when that super-jump could have saved Spartacus. He could have jumped over high walls, clear entire enemy squadrons to reach their commander in the back, etc. That jump wasn't an evidence of Spartacus's real physical capabilities. However, his standard jumps, shown above, do show he is very athletic and in pretty peak physical condition. Credit is given where credit is due, but since I see so many people take that super-jump as if it was legitimate, I had to put this off the table first.

Edit [March 09th 2018]: Seeing as the question of super-jumping led to a discussion with some people in PM, who disagree with me about that super-jump being a complete outlier, I have decided to elaborate a little more.

Here are a few of the instances throughout the show where Spartacus COULD have used the super-jump to his advantage, but didn't, with no reasonable explanation other than genuinely not being able to jump like that:

  • Why didn't Spartacus just escape Batiatus's ludos in season one? He could have easily jumped over the walls. Why did he have to plan a daring, dangerous, complicated escape which he never even got to carry out? He had to wait for the gates to be voluntarily opened in order to make an escape. There were several instances in season one where Spartacus was practicing alone in the yard at night. He could have jumped over the walls and run away, and it would have been hours before anyone realized he was gone. He clearly wasn't afraid of being caught, since he strongly believes that it's better to die free than live as a slave - and he said as much to Agron in the show's last episode, moments before dying.
  • In season 3 episode 2, when the rebels capture Sinuessa, the city's governor locks himself in the city's food storage and threatens to burn it all to the ground, leaving the rebels with no provisions. Spartacus had to distract him while Gannicus and Crixus scaled the walls of the food storage from the other side of it in order to stop him from burning it all down. The walls weren't particularly high, why didn't Spartacus just jump over them and killed the governor himself?
  • When chasing Marcus Crassus and his party in season 3 episode 10, why did Spartacus have to fight the entire party, consisting of ~10 men or maybe even more, before engaging Crassus? Why didn't he just jump over them and got straight to Crassus?
  • In season 2 episode 5, when Spartacus and the rebels rescued Crixus and Oenomaus from the arena in Capua, Spartacus spotted Glaber watching him from the gallery. The gallery was definitely not higher than the balcony in Batiatus's villa, why didn't Spartacus jump there and killed Glaber? Instead he tried throwing a spear at him, why not jumping and going straight for the kill?
  • In season 3 episode 10, after knocking Crassus off his horse, a bunch of Romans formed a small shield-wall between Spartacus and Crassus, which prevented Spartacus from getting to him. Why didn't he just jump over them? He wouldn't have even put himself at a disadvantage, his army was all over the place.
  • In season 2 episode 1, Spartacus and a bunch of his men (Crixus and Agron included - his very best), infiltrated Capua to try and rescue Varro's wife. Sparacus spied Glaber standing a bare few meters away from him and ran forward, shoving past Illithyia and Lucretia to try and ge to him, only to be intercepted at the last moment by a Roman soldier. Why didn't he just jump at him? He had his men undercover all over the place so he wasn't really in danger of getting overwhelmed, and one small jump (they weren't even very far apart) could have solved all his problems right then and there. He could have jumped and killed Glaber while one of his men grabs Varro's wife and then make a run for it.
  • Season 2 episode 10. After Spartacus and his army infiltrate Glaber's stronghold near the end of the episode, why did Spartacus fight his way to Glaber instead of just jumping a couple of meters to get to him? Not a very long distance. Again, he had his men all around him, it wasn't much of a risk.
  • In season 3 episode 3, Tiberius Crassus attacks Spartacus and the pirates while they're having a trade deal outside of Sinuessa. Why didn't Spartacus jump over some soldiers and get to Tiberius? The plume on his helm easily marked him as the commander and Spartacus's army was all over the place, so, again, not really a risk.

Generally, if Spartacus has that much strength in his legs, how is it that he never kills, or at least incapacitates men with every single kick he throws? A kick from a leg that powerful would be like getting hit with a cannonball.

He kicked Glaber in season 2 episode 8:

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This was after Glaber betrayed Spartacus's trust and launched a surprise attack against him. A few seconds earlier in this very scene, Spartacus even told Glaber "someday soon I will have your life", and all that is beside the point that Glaber is the man responsible for Spartacus and his wife being sold into slavery in the first place and the show's main antagonist for the first 2 seasons. That kick should have at least sent Glaber flying outside the walls if not outright killed him, but it did no damage whatsoever other than making Glaber lose his footing.

Another one. Season 3 episode 5. Here Spartacus kicks Heraclio, seconds after the latter betrayed him and led a legion of Romans into Sinuessa:

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Same idea. Why was Heraclio still alive here? Why did he get up seconds after that and keep fighting like nothing happened? Why didn't this kick kill him? Because Spartacus is not superhuman.

These are just 2 examples off the top of my head. Spartacus never even knocked out anyone with his kicks. Conclusion - his legs aren't uber-powerful, and that super-jump was an outlier that does not correlate with everything else we have seen him doing (or not doing) throughout 3 whole seasons.

Another high-end feat that Spartacus has is this one, also taken from season 3 episode 10:

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What we see here is Spartacus sending two men literally flying, each with a single, almost casual strike of his sword. That is pretty high-end, to be sure, however it's inconsistent with Spartacus's standard striking strength, by far.

Let's compare with some other of the man's striking feats, both with and without weapons. For example, here's a showing taken from season 2 episode 7:

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As we can see here, Spartacus took a few meters sprint, jumped from an elevated position, and landed a Superman-punch flat on Nemetes's face. Now, you would expect that a man who can send people flying with casual strikes would have left Nemetes unconscious and with a shattered jaw, if not outright kill him with that kind of punch... but he didn't. He made him lose his footing for a moment and stumble backwards, but not much else. Now, of course, the man has better showings than this one. But nothing he's ever done comes close to this contemptuous ragdolling you saw above.

Now that we've cleared the PIS showings, let's get to real business. Spartacus has pretty good strength, which he gained by maintaining the grueling training routine of a gladiator. Other than sparring and weapon training, the gladiators went through many physical exercises as well:

Now let's take a look at a couple of Spartacus's better strength feats, that are still within consistency.

Here's a good striking feat, from season 3 episode 9, where you see Spartacus doing a real number on a Roman soldier with a powerful flying knee:

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Another fairly impressive feat is this, from season 1 episode 10. What we have here is Spartacus, who's having some serious emotional breakdown after being forced to kill his best friend Varro, punching the wall in his room. He punched hard enough to send chinks of stone flying, and even burrow a small hole in the wall, but he did kinda wreck his own wrists as well:

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Other than that, there's all the usual stuff. Breaking people's teeth, lifting heavy burdens, and chopping through limbs and necks is another day at the office for Spartacus here. Since there's nothing else I would deem especially remarkable to add, let's move on to the verdict.

Strength verdict:

Spartacus is a very strong man, no doubt about it. He may not be the superhuman that many people seem to think he is, but his feats show with a certainty that he has honed his body to peak human performance.

Comparing that to Gregor Clegane, Spartacus falls short. Burrowing a small hole in a stone wall with multiple punches is a great feat, but does not compare to popping a human skull. Sending a man flying with his knee is pretty damn good as well, but does not compare to tossing a man one-handed several meters away. However, as I said, Spartacus was never supposed to pass the bar set by the Mountain, our purpose here was to estimate how close he would come. And my verdict would be, that compared to Ser Gregor Clegane, Spartacus's strength earns a score of 7/10. Spartacus can't do anything that Gregor can, but he can at least come close.

Next up - let us gauge the man's skill.

Skill:

Same as strength, Spartacus's skill will be compared to a bar set by another character. This time, we will be comparing Spartacus's skill to that of CW Ra's al Ghul - a deadly melee fighter with centuries of experience. Skill is a difficult stat to put a number on; it is comprised of several factors, such as speed, agility, dexterity, precision, equilibrium and so on. Therefore, this segment would be more dependent on my own perspective than any of the other segments. The main things I will take in mind here will be martial skill, which means Spartacus's knowledge of fighting techniques and his ability to utilize them, and his ability to fight several opponents at once (though this will also be considered in the "speed" segment).

Here are a few of Ra's al Ghul's feats that we can use as a measuring stick:

  1. Fights and defeats 8 ninjas who have him surrounded in less than 15 seconds, without taking a hit.
  2. Despite starting the fight barehanded, Ra's easily beats Oliver Queen, who's already an adept fighter and uses 2 swords.
  3. Beats Thea Queen, also an adept fighter, with contemptuous ease.
  4. In their final fight, Ra's beats Oliver again, despite Oliver greatly honing his swordsmanship skills throughout the season and training to fight Ra's specifically. Oliver does kill him with a surprise-counter-attack in the end, but in my opinion Ra's could have killed him if he hadn't wasted his time blabbering. Worth noting that Ra's used a very similar move against Oliver in their first duel that you can see above.

In the above scenes, Ra's has shown a great degree of skill. He is very adept in swordsmanship techniques as well as various hand-to-hand and wrestling moves. Now let's see what Spartacus has to offer and see how close he comes to match the great skill of Ra's al Ghul.

Let's start with focusing more on martial skill - showings of specific fighting techniques and moves performed by Spartacus, both armed and unarmed. Let me start with a showing near the end of Spartacus's very last fight in the show (and in his life) - his duel with Marcus Crassus, as seen in season 3 episode 10:

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What we see here is Spartacus countering a counter-move by Crassus. I wanted to start here because it can be compared to Ra's al Ghul's failure to deal with a Oliver's counter-move at the end of their last fight (both moves were very similar, almost exactly the same really), and Spartacus really earns some extra credit for that. Although, as I said and you can see above, Ra's himself used a very similar move himself in the past too, against Oliver. Worth noting that Crassus used this very move in season 3 episode 1 to kill Hilarus - a gladiator who was said to be a famous champion. All in all, it is a fine showing. Not a very complicated or flashy maneuver, but shows good reflexes and adaptability.

Now, moving on to some of his flashier showings. This is a personal fave, from season 3 episode 2, where Spartacus pulls a total Ray Mysterio takedown on a Roman soldier, and killing another soldier with his sword while he's at it:

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A very high-level maneuver in my opinion. To be fair, we have to ask ourselves how applicable it really is in combat: perfect precision and timing are a must in order to pull this thing off and get away with it, and there are so many things that could put one in immediate mortal danger when pulling that sort of thing. An enemy behind Spartacus or at his flank in that moment would have killed him in the blink of an eye. A soldier with better strength and balance might have been able to withstand that jump (unlikely though), and a faster one would have been able to avoid it, even a small shift in his stance at the right moment could have spelled failure and death for Spartacus here. That said, it is the only time Spartacus pulled this sort of thing off in the entire show, so I would assume he is well aware of the dangers in trying a move like this and only went for it when he knew he could risk it. The fact he could do it nicely in combat earns him another nod here.

He did pull off something similar in season 2 episode 5:

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Now, that move was a lot less risky than the last, since Spartacus performed it against an enemy who was in no position to do anything about it. It was a little unnecessary, but it was flashy and nice nonetheless.

Moving on. Spartacus uses all sorts of martial arts moves in his fights pretty much all the time. For example here, in season 2 episode 6, where he countered a punch thrown by Gannicus, used his momentum and turned it into a throw:

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He has a lot of other showings of using certain moves and techniques in his fights, no point in showing them all, or this thread would last forever. Point is, the man's got moves.

Now, let's get to the other aspect of the "skill" segment - Spartacus's showings against multiple enemies simultaneously. Ra's al Ghul's bar of stomping 8 ninjas is a very high one for Spartacus to live up to, so let's see what we have.

Now, I'd like to start with some of his season 1 showings. The thing about season 1 is, that in it, Spartacus was still a gladiator - and most of his fights were in the arena. Fighting in the arena isn't the same as fighting on a real battlefield, for the reason that gladiators fighting in front of a huge crowd have it in top priority to prolong the fight as much as they can, in order to provide a good show. Because of that, gladiators often choose to toy with one another, and don't always fight as effectively and lethally as they would under other circumstances.

Let's start with Spartacus's first fight in the arena, from season 1 episode 1. In that fight, Spartacus wasn't even a gladiator yet - he was a prisoner on death row, given a sword and set against four trained gladiators by himself - seemingly impossible odds. While Spartacus wasn't a gladiator yet, and wasn't as skilled or as fit as he was throughout most of the show - he was still the veteran of many battles and knew how to fight. He was fatigued and malnourished though. Anyway, here it is:

(sorry for the Italian, good-quality Spartacus full scenes are borderline impossible to find online and the only one I had is this one, couldn't find an English one. The talking isn't all that important though).

Now, as the fight starts, Spartacus is surrounded by 4 gladiators. He manages to avoid some of their first swings at him, but they eventually overwhelm him and put him on the ground after laying some hard hits on him. The shield to the face that he takes in 0:27 and the axe-shaft to the head at 0:37 are an example of what I was describing earlier - either hit would have been a killshot in a regular battlefield, but in the gladiatorial arena, the gladiators chose to abuse their "victim" instead of killing him to amuse the crowd. Then they do the most typical thing for a gladiator - they disperse to flex their muscles for the crowd, with only one of them remaining to finish off the prisoner. However, Spartacus finds some source of inner strength after remembering something that his wife told him earlier in the episode and springs up with a fast attack before the gladiator can finish him off. Then comes the better part of this fight for Spartacus, where he fights the other three and does a lot better than he did before. While he did get hit once or twice more, he finishes all three of them off quite effectively. All in all - not perfect, but a good feat nevertheless.

Now, another one would be here, from season 1 episode 7 (warning - loud and crappy music added in the background. Told you good videos are hard to find):

What we have here is Spartacus, facing 6 men by himself (he wasn't supposed to face them alone, but chose to, because those were men from Thrace- Spartacus's homeland. Spartacus wanted to do this by himself because he didn't want anyone else killing men from his homeland). Anyway, as we can see in the video, he charges in (after doing that oh so idiotic thing with the spear) and starts fighting the six men. He does relatively well in the first ~15 seconds (hard to say for sure with all that slow-mo but it was somewhere between 15 and 20 seconds), tagging his enemies a few times but not avoiding getting hit himself - and then he takes a blow from a warhammer straight in the chest and goes hard to the ground. Again goes the same deal as above: his opponents lay some unnecessary abuse on him while also flexing their muscles for the crowd's amusement, one guy goes back to finish the job, Spartacus finds some inner strength after remembering some motivational cliche that his dead wife told him once, and the fight is on again. Except here, these Thracians do the most idiotic thing they could have possibly done (and I really have no idea why they did it) - they start coming at him literally one at a time. Then it becomes a cakewalk for Spartacus here, because fighting one half-trained dude six times in a row is far easier than fighting six half-trained dudes all at the same time. To be blatantly honest, this fight is really not much of a feat for Spartacus. Losing to 6 guys then killing them one by one isn't really a skill-ish showing. So, for all the people who love saying that "Spartacus solo'd 6 enemies" - he didn't.

Lastly, there's one more showing from season 1 that I would like to discuss - from episode 12. This time, the fight doesn't happen in the arena, but in Batiatus's villa, with the odds stacked heavily against Spartacus - he's forced to use two wooden practice swords against Roman soldiers wielding real swords (again sorry for the foreign language, but as before the talking isn't really important):

This fight was held for Glaber's amusement, as a "demonstration" of Spartacus's famous skill. Glaber had Batiatus give Spartacus 2 practice swords, then started sending his own soldiers against him. As we can see, first Spartacus is fighting one opponent, and takes him out pretty quickly. Then two more are put against him, and they too are soon out of the game. Then comes the real fight, where Spartacus faces 4 soldiers at the same time. First, only two charge at him, to which Spartacus reacts well. Then the third charges in, and as Spartacus dodges his charge, one of the first two kicks him in the back of the knee and drops him (2:29). If you look closely in the background, the 4th soldier - who is yet to join the party - actually punches the one that Spartacus just dodged. Whose side are you on, dude? Then he remembers that Spartacus is his enemy and joins in, and lays a beating on Spartacus along with the 2 other soldiers who are still standing. Then, check this, at 2:55 that friendly-fire soldier (or maybe it was another one, hard to say for sure) actually hits one of his comrades with his own sword - again, with the same savage blow that cut Spartacus's wooden sword in half. In other words, it's either that these guys' loyalty is very questionable, or they're just a lot less than competent. I'd bet on the latter. Now, after hitting his mate, that soldier kicks Spartacus down into the water. And then - when Spartacus is down flat on his back, with water in his face and his "weapon" broken in half - that soldier chooses to patiently wait for Spartacus to get up and give him an angry look before attacking. As we've seen before, letting this guy regain his footing and coming at him one by one is a very bad idea. Of course he makes short work of that soldier and the other one who wasn't already out, but it's still far from a clean showing - especially considering that one of these four Romans was actually taken out by his own comrade. We do have to take in mind the fact that Spartacus wasn't using real weapons here - he was all but unarmed. Sure, wooden swords can be used as blunt weapons and deliver a painful blow, but they're not designed for such fighting style like clubs or maces and using a wooden sword against enemies wielding real steel isn't something to be overlooked.

Now, let's talk about another scene of Spartacus facing multiple enemies - but this time only two. This one is from season 3 episode 9:

Earlier in the episode, Spartacus and the rebel army capture Tiberius, the son of Marcus Crassus, along with a handful of his men. This happens shortly after Naevia delivers Spartacus the news of Crixus's death, so Spartacus decides to hold "gladiatorial games" to honor Crixus. The "games" are held with Spartacus and his men facing Tiberius's men. The first fight has Spartacus fighting 2 men simultaneously - which is what we see above. Well, to put it simply - he humiliated them. Sadly, the videos that I posted originally - depicting the full fight - were removed from YouTube so we have to settle for this one. He started by letting them come at him, tagging them a little here and there, mostly to amuse the crowd - same as in real gladiatorial games. It goes on for a while until one of them tags Spartacus and puts a cut on his cheek. Shortly after, Spartacus ditches his swords (which is where this video starts) - but don't worry, this is the good part of the fight since this is where the real humiliation started. He takes on both soldiers, at the same time, barehanded - and wins. This is probably Spartacus's best showing against multiple enemies, because he was literally toying with them, and killed them both after taking only one cut himself.

Now, of course, he was outnumbered in almost all of his fights in the show, often relying on superior tactics and guile to fill in for numbers disadvantage, and many times he's come out none the worse for wear. But just as often, and perhaps more often, he gets tagged by his enemies and in some of the times, like you saw above - he's escaping death by the skin of his teeth and his enemies' idiocy.

For example, here, in season 3 episode 2, where he was disarmed, punched and nearly overwhelmed by a Roman guardsman:

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He did retaliate and regained control, but still not what you'd expect of the guy. Another example can be seen here, in season 2 episode 1, where he was tagged twice in a row when fighting 2 mercenaries:

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Again, of course it didn't stop him and he killed them both after a little while, but that cut to his back could have easily been a fatal one if it had gone just a little deeper, or was aimed a little higher. Spartacus was very lucky here. Another example is this one, from season 2 episode 9, where Spartacus gets momentarily overwhelmed by 2 Roman soldiers:

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It skips scenes right after that shot so we don't get to see the whole course of the encounter, but when it gets back to showing Spartacus and Gannicus, who were by there by themselves against a pretty big bunch of Romans, they seemed to be handling things pretty well.

One last showing for this bit is again from the show's last episode - season 3 episode 10. The original video that I posted here was removed from YouTube as well, and I couldn't find anything good enough online, so I went ahead and uploaded it myself (first time I ever uploaded anything to YouTube, it's a lot easier than I thought it'd be really):

What we have here is Spartacus, in his final battle, chasing Marcus Crassus's party for quite a long distance by his lonesome (you can see that the two armies fighting one another are pretty far off), and facing them in battle before engaging Crassus himself. Now, that was quite a lot of people there, and they didn't give Spartacus a very easy time. While they did not charge at him one by one, they didn't come at him all at the same time either. They came at him in pairs more or less, and in very quick succession, and he handled them pretty damn well. He was tagged a few times (0:38 for example, and 0:47), and it is worth noting that he didn't really kill them all (at 0:58 he throws one of the soldiers at three others without directly engaging them, and they didn't come back so I'm guessing they rolled down the hill or something, so that kinda counts for BFR). It's one of the man's better showings in my opinion, though, which also shows stamina in addition to skill - and a bit of strength with his killing blow against the spearman (1:03).

As for good showings against skilled opponents, the only two that I really feel have to be included here are his fight with Gannicus in season 2 episode 6 (relevant bit starts at 1:38), in which they were about equal until the fight was interrupted, and his fist-fight with Crixus in season 3 episode 7 (relevant bit starts at 1:35) which Spartacus won. Both Gannicus and Crixus are great warriors and matching/beating them are great feats on Spartacus's part. Not much to say on these, both were good fights and Spartacus was on the delivering end at least as often as he was on the receiving end in both cases.

Now, before moving on to the skill verdict, I'd like to point out one more thing that I always bring up on debates involving Spartacus: his main adversaries - the Romans - are awful fighters. I mean, regardless of how well trained Roman soldiers were (or weren't) in reality, in the show they're a disgrace. A few examples of that can be provided. One of those would be from season 2 episode 6, where three Roman soldiers get trashed by Ashur. Now, Ashur - by his own admission, and as seen throughout the prequel miniseries "Gods of the Arena" - was the lousiest gladiator in his brotherhood. He was very bad by gladiator standards, and what's worse is that this fight was about the first time he held a sword after a few years of recovering from a crippling injury. What makes this even worse still, is the fact that those three Romans were actually relatively high-ranked, meaning that they would have been better trained than common soldiers. In fact, one of them was Salvius - Glaber's own Tribune (second-in-command). Another example would be when Naevia defeated Tiberius Crassus in season 3 episode 9 (unfortunately there isn't one full, good-quality video of the fight so you'll have to settle for 4 parts - part 1, part 2, part 3, part 4). Now, Tiberius has shown several times throughout the season to be superior to regular Roman soldiers, and while Naevia wasn't an amateur or anything, she wasn't too good either. I mean, only a few episodes earlier - in season 3 episode 3 - she was almost killed in a fight with Attius, and Attius wasn't even a warrior - he was just a blacksmith, and in addition he was using an improvised weapon against Naevia's sword. One last example would be from season 3 episode 6, where Julius Caesar - one of the greatest Roman fighters in the show - almost loses to Donar in a fight. While it's true that Donar was a pretty beastly warrior, in his fight with Caesar he was suffering from pretty harsh injuries that he received before, and was also using a sword whereas his weapon of choice was always an axe. Caesar only gained an opening by hitting the deep stab wound that Donar already had in his belly from before this duel. If not for the preexisting wound, that punch would not have so much as annoyed Donar and Caesar would have died. With all this in mind, whenever Spartacus beats Romans and makes it look easy - it should be taken with a grain of salt. Beating Romans, in fact, IS pretty easy. Alright, NOW let's get to the verdict.

Skill verdict:

Spartacus is undoubtedly very skilled. He has shown knowledge on many fighting techniques and maneuvers, and has mastered several weapons as well (which I didn't really talk about here). While the moves he uses are sometimes flashier than those used by Ra's al Ghul, they're not more effective or anything. Spartacus's ability to counter the special move by Marcus Crassus earns him a big nod here as I said earlier too. When we consider Spartacus's showings against multiple opponents, he doesn't really come very close to Ra's al Ghul's feat of eating 8 ninjas who had him surrounded in only a few seconds. Spartacus has shown the ability to take on great numbers, but not so many at the same time, and he almost always gets tagged. His feat of toying with 2 Roman soldiers while unarmed was a very good feat, but I would say that Ra;s al Ghul beating Oliver Queen while unarmed tops that, since Ollie was already a trained warrior with great skill whereas the Romans... not so much. Spartacus has some good showings against skilled opponents such as Gannicus, Crixus and Marcus Crassus too. All in all, Spartacus is an extremely skilled individual. But compared to the very high bar set by Ra's al Ghul, I think that Spartacus's skill earns a score of 6.5/10. While he cannot take on as many people as Ra's al Ghul, certainly not with such contemptuous ease, he can handle himself fairly well while greatly outnumbered, in situations where even greatly skilled warriors will be hard-pressed to survive. He'd be able to handle himself very well against 2 or 3 enemies at the same time, he'll manage with 4, but will start having trouble with more than that (though he has shown to take on a lot more fairly well when they just don't come at him all at the exact same time). His knowledge of fighting techniques and especially unarmed moves that can come in handy even against armed opponents is pretty great, and he uses it often to great efficiency, even though sometimes he uses unnecessary moves for the sake of flashiness - but hey, you won't hear me complaining about that.

Now, for the last of the three big segments of this thread, before moving on to Spartacus's miscellaneous advantages and disadvantages, let's take a look at the man's speed.

Speed:

Here, in my opinion, Spartacus faces the highest bar yet. His speed will be gauged in comparison to that of one of my favorite TV badasses, and a ruthless killer with hundreds of kills to his name - Sunny from "Into the Badlands". When looking at speed, we will look mainly at combat speed - how fast can Spartacus fight, and reaction speed - how he deals with fast attacks coming his way. His ability to deal with multiple enemies at the same time, which we already saw in the "skill" segment will also be factored in here, in one way or the other. This segment will be relatively short since Spartacus's speed is pretty consistent, and it isn't complicated to gauge, like skill.

As usual, let's start with seeing some of Sunny's showings of speed for future reference. It's pretty hard pinpointing the man's exact speed feats since everything in this show, well... just happens so fast, but here is a fair taste:

  1. One of the first big fight scenes in the show, where Sunny faces a few dozen foes on his own. It's pretty standard for him really, but here are a few markers to note in regards to speed: 0:47-0:52 - Sunny, while flat on his back, removes the legs of quite a few people around him in about 4 seconds. 0:58-1:03 - Sunny kills 4 men who have him surrounded in as many seconds, then proceeds to kill some more in a pretty cramped space. 1:15 - deflects a thrown axe back at his enemies with contemptuous ease. 1:29-1:32 - kills 6 men who have him surrounded (pretty much off-screen so pay close attention) in about 3 seconds or so. Anyway, just watch the whole scene, it shows Sunny's standard combat speed pretty well in my opinion.
  2. Here Sunny deflects and dodges a multitude of crossbow bolts while running at the shooters through a corridor. Couldn't find it on YouTube and I couldn't fit the whole thing into a single GIF, but for reference, he only got hit twice in the entire run (including the one that appears in the GIF) which is pretty damn impressive.

Each of the 16 episodes of "Into the Badlands" features at least a few fight scenes, and I can't really post all of Sunny's impressive speed feats here - but these should give you the idea on how fast the man is.

Now, Spartacus has quite the bar to live up to. While he isn't as fast as Sunny he is no slouch whatsoever either, so let's see what we have here.

First of all, let's take a look at combat speed - the speed in which Spartacus can deliver effective blows.

One good example would be right here, in this scene from season 2 episode 1:

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What we see here is Spartacus killing 3 mercenaries in as many seconds (give or take, a bit hard to tell with all the slow-mo but I'd say it's 3 seconds tops). Now, I'm not really buying the whole "he's so fast that everything around him appears to move in slow-mo while he's maintaining normal speed" nonsense, it's just cool cinematography (which often makes people misjudge certain feats and characters). Spartacus isn't a speedster, he'd have to be like Spider-Man level at the very least to make everything look so slow compared to him, but that showing alone is safe to say that he is definitely quick. The first kill of the three here is a bit iffy, since Spartacus did it by throwing his sword at a distant opponent, but it's still nice. What I like most about this feat is how Spartacus reacted to the enemy coming from behind him a split-second after killing the horseman.

Here's another nice showing of combat speed, from season 2 episode 10:

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What we see here is Spartacus and his rebel army storming Glaber's stronghold (which was previously their own stronghold). Glaber, who's hiding in the back and lets his men get butchered while doing all the dirty work since he's a useless coward, has been the show's main antagonist in the first two seasons, so obviously Spartacus is going to be the one to see him killed in an epic fight. So, the man charges ahead of his army to get to Glaber, and what we see here is Spartacus tearing through 2 Roman soldiers in just about an instant, tossing Salvius around and then parrying and headbutting Glaber himself - all in pretty much a heartbeat (turned into 12 seconds by that goddamn slow-mo). Now, it skips scenes after that headbutt, and when it goes back to Spartacus we see him bashing Salvius's head against a pillar which kills him, and then he goes on to dueling Glaber (obviously winning in the end). Anyway, a nice showing of speed.

Now, Spartacus doesn't have anything else that I think is really worth mentioning in regards to combat speed. I can find other cool sequences, to be sure, but it'll just be more of the same. The guy is fast and his speed has been pretty consistent throughout the show and those two showings should give you the general idea. So, let's move to reaction speed.

First, let's talk about this showing from season 2 episode 6:

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What we have here is Spartacus (and Gannicus) aim-dodging an arrow fired by Mira. Now, I say "aim-dodging" because that's exactly what it was - you can see in the GIF that Mira warned Spartacus before she let the arrow loose so he wouldn't get hit. I think it's a good showing because Spartacus was locked in combat with Gannicus when Mira shouted her warning, and Gannicus isn't just some easily-disposed fodder, he is a very skilled fighter and in my (unconventional) opinion he's even a better fighter than Spartacus. Heeding Mira's warning and moving out of the way in time while fighting someone like Gannicus is a good feat of reaction speed in its own right, even though it doesn't compare to Sunny's suicide-run in that corridor while deflecting all those crossbow bolts.

Now, as for slower-moving projectiles, Spartacus has some very nice feats. Thrown weapons are usually of no concern to him. For example, there's this showing from season 1 episode 7, in his fight against the 6 Thracians:

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Not much to explain, what we see here is Spartacus dodging a thrown axe at relatively close range (looks like 3-4 meters away I believe). Pretty nice. Moving on to another feat of the same category which I personally like better, from season 3 episode 2:

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Here you see Spartacus putting a human shield between himself and a spear that was thrown at him. What makes this feat better than the last one in my opinion is that Spartacus only noticed it after it was thrown, so he had a pretty small time-frame to react, especially when you consider that the spear came from above him and probably traveled with greater velocity than it would have if it had just been a straight throw. He dodged a couple more spears in this scene, worth mentioning (one of these you can see in one of the GIFs featured in the "skill" segment). Now, it should be said that dodging an object thrown your way isn't in itself an overly impressive feat. I mean, there's a whole sport based around dodging things thrown at you, it's called "dodgeball". Even still, I think these showings are worth mentioning, dodging that spear for example was far from trivial in my opinion.

One more aspect of speed would be travel speed. While it's not something that is usually discussed and brought into question when discussing characters of this tier (it's usually more important for powerhouses and high-tiers like Superman or the Flash), there's one feat in particular that I'd like to bring up here. It's taken from season 2 episode 7 (relevant part starts at 3:10):

What we have here is Spartacus, reacting to a call from Mira and rushing to save Agron from Sedullus who was stupid-drunk and stupid-angry and trying to kill him. Now, many people seem to think that it was a split-second reaction and movement feat, since you see Sedullus already starting to bring his sword down (3:18). Now, what some people seem to miss is, that by the time Spartacus is already within arm's reach of Sedullus, the latter is clearly only bringing his sword upwards (3:22). The very simple explanation for this is that these two shots (3:15-3:20 and 3:21-3:22) happened simultaneously: Mira called for Spartacus when she saw Sedullus grabbing the sword, or maybe even just throwing Agron to the ground, and Spartacus reacted. In fact, somewhere between 3:21 and 3:22, there's a split second where you see Spartacus rushing forward, and at the left-end of the screen you see Sedullus only beginning to raise his sword. It's not easy to notice, but it's there. It's still a very nice speed feat, but nothing like that Deathstroke-level affair that many people seem to think it was, especially when it's evident that Spartacus was only 5 feet or so away from Sedullus when Mira called for him.

Lastly, one more thing I'd like to discuss in regards to speed is Spartacus's fights shown in the "skill" segment. Namely the ones that involve him fighting multiple opponents. Now, I don't have the means to measure his exact speed in such situations or anything, but the very prospect of fighting multiple enemies at the same time requires good amounts of speed as well as raw martial skill and experience. Spartacus has shown to be able to handle himself well enough against superior numbers, even if he gets tagged most of the time. Sunny, the man who set the bar here, usually gets tagged more than once in his fights as well - but, of course, he's usually facing far greater numbers, as shown in the scene I linked at the beginning of this segment. Another thing to consider here would be Spartacus's fight with Gannicus. Gannicus is, in my opinion, one of the best fighters in the show - even better than Spartacus himself. In the show, he has shown his speed and skill many times over and I believe that matching him in combat is worth mentioning here too.

Speed verdict:

Well, I'll just be blunt here - Spartacus doesn't come very close to Sunny. His speed, in all aspects, is pretty much the standard for peak-human level characters. He's quick enough to kill multiple foes in the blink of an eye, he's capable of avoiding weapons thrown at him from relatively close range and aim-dodging an arrow, but while it's very nice by "real-world" standards, it's nothing too unique in fiction. He gets credit for consistency, but his level of speed isn't something truly remarkable, and doesn't really live up to the standard set by someone such as Sunny - who, in my opinion, resembles pretty much the pinnacle of speed for live-action street level characters. High as the bar here is, I think Spartacus's speed earns him a score of 6/10. He's quick, he's definitely a peak-human and could undoubtedly match and maybe even surpass some world-class champion athletes, but not much more. He definitely can't do what Sunny does for breakfast, I don't think that needs too much explaining.

Alrighty then. Now that strength, speed and skill - the three major aspects that are being evaluated here - are done with, it's time to look over some of Spartacus's other qualities, as well as drawbacks.

Miscellaneous:

Now, this segment will be different from the previous three. Spartacus's various advantages and disadvantages that will be discussed here won't be compared to any certain bar and will not feature a thorough analysis like strength, speed and skill did. Each of Spartacus's miscellaneous advantages and disadvantages will get a short overview, with maybe some GIFs or a videos for reference.

Let's start with a more positive tone, and point out the man's advantages.

Advantages:

1. Stamina:

Spartacus's stamina is pretty great. One classic example for that appears in season 3 episode 10, right before his fight with Marcus Crassus (see scene above in the "skill" segment). Spartacus, after fighting Crassus's army for a while, runs a few hundred meters uphill (hard to gauge the distance but the video makes it pretty evident that it was a pretty long distance he had to run) and engages Crassus's party before fighting - and for all intents and purposes defeating - Crassus himself. Stamina is very important in fights that drag out over long periods of time. Spartacus's almost complete lack of armor lessens fatigue, and combined with the man's peak physical condition - he can go at it for hours probably if need be.

2. Endurance (pain tolerance):

While Spartacus's endurance isn't as freakish as that of other characters, such as Theokoles for example (who was a complete freak of nature in that regard), it's still very impressive. He can take multiple cuts and bruises and keep fighting like a champ, which happens pretty much every time he fights anyone anywhere throughout the show. He did show to get overwhelmed at times, just barely finding the willpower he needs to get up and keep fighting (for example in his first arena fight from season 1 episode 1, or in his fight against the 6 Thracians in season 1 episode 7), but he has fought through pretty bad injuries many times. For example, this cut that he took from Varro in season 1 episode 10:

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As you can see, Spartacus literally laughed it off after taking that cut, but in later episodes it was seen to be very deep and even made him collapse once or twice. Normally, I don't take the odd cuts and bruises as overly impressive showings of endurance, simply because it's only natural for the human body to ignore pain while in a life-and-death scenario and the adrenaline is pumping, but deep injuries like this one are not at all trivial to just shrug off. Worth mentioning that this fight with Varro was a "mock" battle. It took place at a birthday celebration for the governor's son, and was supposed to be just an exhibition for the entertainment of the birthday boy and the other guests. Neither combatant was supposed to die, and both Spartacus and Varro knew it, so I'd say it's very plausible that they weren't even in a "real" adrenaline rush as one would experience in a real fight to the death. Varro did end up dying there, but that's another story - it wasn't supposed to happen and came as a surprise for everybody present. Anyway, Spartacus has great endurance. The training routine of a gladiator, which Spartacus went through day after day for many months, involved getting hurt on pretty much a daily basis. He's been through pretty debilitating experience, such as fighting at the cruel underground fighting pits for several consecutive nights, and ended up living to tell the tale. I mean, the guy goes around headbutting Roman soldiers wearing helmets pretty often too, so yeah, he can definitely take some harsh punishment and keep fighting.

3. Agility:

Well, not much to say. Spartacus is pretty damn agile. He can avoid hits and dart around the battlefield with cool maneuvers such as this nice jump from season 2 episode 1:

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The man is extremely athletic. Not much needed to be said here, just look back at the GIFs and videos posted throughout the thread and you'll see. He's not Remy LeBeau or anything, but his agility is definitely impressive and he knows how to use it.

4. Mindset:

Not much to say on this one. The man is a complete savage. He was described in the show (season 1 episode 7) to "fight like a man possessed by the gods themselves" and has lived up to this description pretty nicely. Spartacus does not shy away from quick, efficient kills, he is a ruthless killer and has no remorse. Need proof, just look back through this thread.

5. Guile and leadership:

This one is different from all the other things that appear in this thread, as it doesn't have that much to do with personal fighting prowess. Spartacus has proven to be a great leader, not just because of his ability to inspire loyalty and motivation, but also because of his brains. He's a great strategist and has overcome seemingly impossible odds time and time again throughout the show with creative and brilliant plans and strategies. Be it long planning or quick improvising, the man has proven himself to be very, very smart and a great leader.

Now, moving on to the man's main drawbacks.

Disadvantages:

1. Armor:

This is Spartacus's main drawback in my opinion. His armor isn't what you'd call... good, really. He wears almost none most of the time, and when he does it never covers much of his body. He's used the odd lobstered arm-guards, vambraces, spaulders, greaves and even a breastplate every now and then, but it's a far cry from being consistent and even when he does use these things, they don't offer much protecion and still leave most of his body exposed to damage. He used to wear helmets in season 1 but that ended after episode 5 as well, when Spartacus started training in the "Dimachaerus" fighting style. Endurance and luck can only take you so far, and while it does help him with stamina and a little with mobility too, lack of armor is very, very dangerous.

2. Weapons:

Well, that one's pretty dependent on who he fights, but I think it's pretty fair to say that his weapons aren't ideal for Comic Vine's battle-board. First of all, the swords he usually favors are pretty short and offer very little reach. Second, if he has to fight someone from a more advanced time-period, his antique steel will put him in a pretty big disadvantage. Spartacus comes from around 100 BC, a time when metallurgy and technology weren't very advanced and his swords would easily nick and maybe even break if he has to face a fighter from an advanced period of time. Even in medieval times, metal was a far cry above what Spartacus has to offer. As I said, it depends on who he fights, and certain rules can be set in battle threads to even it out, but under default conditions - his weapons would put him at a disadvantage against most potential enemies.

Now, that's all she wrote folks. This is everything I think deserves mentioning when looking at our boy Spartacus here. Let's move over to one final summary to gather up everything we've had here so far.

Summary:

So, we started with gauging three basic attributes that are extremely crucial for every fighter - strength, skill and speed. We set an objective bar for each of those three attributes and analyzed them thoroughly, so let's see what we gathered.

As far as strength goes, Spartacus was assessed compared to a true behemoth - Ser Gregor Clegane, who in my opinion resembles a real 10/10 of strength in his tier. Spartacus's strength is often very overrated because of certain outlier showings which had to be dismissed in the start, but we did see some of his legitimate strength feats such as punching a small hole in a stone wall in a fit of red rage or kneeing a Roman soldier into oblivion. The man is not superhuman, but he does boast peak-human physicals, which earned him a rating of 7/10 in strength.

As for skill, Spartacus was compared to the deadly Ra's al Ghul from the CW verse. While he does not live up to the bar set by Ra's of killing 8 foes who have him surrounded in mere seconds, or utterly humiliating expert swordsmen with his bare hands, he does have some nice feats worth mentioning. He can take on superior numbers, albeit less than 8, and still end up winning, and he also has great knowledge of fighting and wrestling techniques with he masterfully employs into his fights. While his most common foes - the Romans - were pretty crappy fighters, his feats are still very nice and earned him a rating of 6.5/10 in skill. It may not appear as much at first glance, but you have to remember that bar of 10/10 was set by Ra's al Ghul - a killer of amazing skill that very few of this tier can match.

The last of the three main attributes, speed, had a very high bar set by Sunny, who's a speed-freak and is really the pinnacle of speed for this tier in my opinion. Heck, he may even push the limits into a higher tier with his feats sometimes. It wasn't a very close comparison, though; while Spartacus is definitely very fast in both combat and reaction speed, he just doesn't break past the limits of "peak-human" like Sunny does with ease. Set against another human who doesn't break the laws of physics, Spartacus will have the speed advantage more often than not. But not beyond that. All in all, Spartacus's fairly impressive feats earn him a rating of 6/10 in the speed department.

Lastly, we discussed all of Spartacus's other attributes as a warrior, both positive and negative. As for his advantages, they mostly have to do with his peak-human physical form. He is very agile and has great stamina, and he can take great amounts of punishment before going down. On top of that, he is a ruthless, savage killer who fights with brutal efficiency. He has fire in his eyes whenever he grabs a blade and that alone can overwhelm many opponents. In addition, he is an excellent strategist and tactician, which may be less relevant when gauging his personal combat prowess, but it's still one of Spartacus's dominant qualities and it's definitely worth mentioning here.

Spartacus's disadvantages revolve mainly around his gear. His very minimal, and often nonexistent armor makes him extremely susceptible to damage and could prove his downfall against another quick and skilled opponent. His twin swords, which are his most common weapon of choice, offer pretty small reach which would be a problem when fighting against someone with a bigger melee weapon, such as a longsword or a spear. In addition, the fact that Spartacus comes from an era when steel wasn't very advanced might be a problem when put against someone coming from a more advanced time period.

Bottom line, Spartacus has the stats of a peak human, straight-up great skill, zero remorse and a great tactical mind, all of which make him a very formidable opponent for just about anyone on his tier.

Well, that was a long thread and it took me the better part of a month to write down. I re-watched the whole series for it (admittedly I did skim through many parts that didn't really involve fighting and such), but I did have a lot of fun writing this. Hope you guys enjoyed the read, I would love to hear your feedback on this.

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The GoT spin-offs we would like to see

So it has been confirmed that we, the fans hungry for entertainment, will be getting us some spin-off for Game of Thrones! Woooooooo!!!!

But what will they be about? When and where will they be set? Will they have characters that we already know from the show? Will they have characters that we've heard of in the show but never got to see? When will they air on HBO? How many spin-offs are we even going to get?

Nobody knows, nobody even has any hint as of now. But, there are some obvious things that any fan of "Game of Thrones" or "A Song of Ice and Fire" would be thrilled to see on screen. Here I will discuss some of the more obvious ones, as well as a few ideas of my own that I would like to see.

I'll start with the "obvious" ones, which are sadly very unlikely to happen.

*Disclaimer: I didn't even mention the "Dance of Dragons" because it's so not going to happen that I'm not even gonna waste my time with it.*

1. The Tales of Dunk & Egg:

Let's start this off by saying - we're NOT going to get this one. It was confirmed by George R. R. Martin. Probably because he doesn't want the series to develop past the point where he stopped in his books, like "Game of Thrones" did.

Anyway, for those of you who haven't read GRRM's books and have no idea why would anybody want to dunk an egg, what the heck does it have to do with "Game of Thrones" and why would anyone want to see it on TV, here's a bit of background:

The Tales of Dunk & Egg are a series of stories by GRRM, set in Westeros a bit less than a century before the current timeline (the events of the first ASOIAF novel, or first season of the TV series, if you will). The main heroes are Dunk, formally known as Ser Duncan the Tall, and his young squire Egg, formally known as Prince Aegon Targaryen (who will one day become King Aegon V Targaryen, AKA "Aegon the Unlikely"). One novel on the tales of D&E has come out already, called "A Knight of the Seven Kingdoms", and it collects the first 3 stories of the saga: "The Hedge Knight", "The Sworn Sword" and "The Mystery Knight". There are also graphic novels, by the way!

Ser Duncan the Tall and the young Prince Aegon Targaryen
Ser Duncan the Tall and the young Prince Aegon Targaryen

The Tales of Dunk & Egg are fan-favorites and would make for a great spin-off series. But, as I said, it has been confirmed that sadly we won't get to see those two on our screens anytime soon. Perhaps that's for the best, and we're better off letting GRRM finish his books. Who knows, maybe we'll get to see those two sometime in the future. I sure hope so!

2. Robert's Rebellion:

Pretty sure it was confirmed that this isn't going to happen either, lol. We all know the events of the rebellion that brought an end (?) to the ancient Targaryen dynasty, but seeing it on-screen is something else entirely.

It's still possible that we will get to see choice moments of the rebellion in Bran's DMT trips flashbacks in season 8, but there's nothing to indicate that and it's merely speculative. We did see some key moments in the rebellion, such as the battle at the Tower of Joy, and Jaime killing the Mad King (sorta). Others we only heard of and can only imagine to ourselves, but come on, who wouldn't want to see Robert Baratheon in his prime, the Demon of the Trident, leading his army and smashing Prince Rhaegar Targaryen with his warhammer?

Robert Baratheon as he is about to land the killing blow on Prince Rhaegar Targaryen
Robert Baratheon as he is about to land the killing blow on Prince Rhaegar Targaryen

Robert's Rebellion had many events that were major turnpoints in the history of Westeros. The Tourney of Harrenhal that started the whole deal, the Sack of King's Landing, the Battle of the Trident, the Siege of Storm's End, and many more. Even seeing the battle at the Tower of Joy remastered could also be awesome.

This was one of the bloodiest, most important wars in the history of Westeros, and perhaps the whole World of Ice and Fire. Seeing it on-screen would break ratings records. But, unfortunately, it's unlikely to happen.

3. Aegon's Conquest:

The Conquest of Aegon the Conqueror, which brought the Seven Kingdoms under one king and started the glorious dynasty of Targaryen Kings in Westeros, is another one of the bloodiest and most important wars in the history of the World of Ice and Fire. Sadly, it's also unlikely to happen. It has too many dragons, lol.

Aegon's Conquest started around 300 years before the current timeline. One by one, the Seven Kingdoms fell to Aegon I Targaryen and his sisters, and their dragons Vhagar, Meraxes and Balerion the Black Dread. Iconic moments such as the Field of Fire and the destruction of Harrenhal would look amazing on-screen, but would require waaay too much CGI which costs waaay too much money. Sadly, we're unlikely to see this amazing conquest on-screen anytime soon.

The Targaryen siblings who unified the Seven Kingdoms with Fire and Blood. Left to right: Visenya, Aegon and Rhaenys
The Targaryen siblings who unified the Seven Kingdoms with Fire and Blood. Left to right: Visenya, Aegon and Rhaenys

Another reason why it won't happen is that we already know everything about the conquest. There's no element of suspension or surprise in this, much like Robert's Rebellion. The conquest and its major turnpoints and battles were referred to many times in the show and in the books. Guess we'll have to settle for our imagination. Oh, well.

4. The Long Night:

The events of the Long Night are also an "obvious" choice for an awesome spin-off series... and thus, very unlikely to happen, lol.

The Long Night occurred around 8,000 years before the current timeline. It was the name that was given to the murderously harsh winter that first brought the Others (AKA "White Walkers") and their army of the dead into Westeros. It only ended when the hero known as Azor Ahai, wielding the legendary sword Lightbringer, lead the living in the War for the Dawn against the dead and prevailed. Then the Wall was built and the Night's Watch was formed to guard the realms of men from the darkness that lurks in the north.

What a White Walker SHOULD look like
What a White Walker SHOULD look like

The main reason I would like to see the Long Night and the War for the Dawn is the chance to see the Others as GRRM desribed them:

"Strange, beautiful… think, oh… the Sidhe made of ice, something like that… a different sort of life… inhuman, elegant, dangerous."

In other words, the White Walkers should look like a creepy pale Legolas. Very different from the way they're portrayed in the show. In my opinion, the look that GRRM described is far more eerie and threatening (and here's why).

The problem with this, other than the tons of CGI (=money) that it would probably require, is again the "element of surprise". The long-forgotten secrets of Azor Ahai and how the hell did he defeat the White Walkers all this time ago are the mystery that Jon Snow is so determined to solve, both in the show and in the books. There resides the solution to his current war against the White Walkers. Releasing a spin-off series based on the Long Night after season 8 of "Game of Thrones" ends will spoil the suspense of seeing how Azor Ahai wins the War for the Dawn, and vice versa.

That's the last of the unlikely fan-favorites, now for the interesting part!

5. The First Blackfyre Rebellion:

Well, I lied. Kind of. That one is also a fan-favorite, but not as unlikely as the above. The First Blackfyre Rebellion, to the best of my memory, was never even mentioned in the show (it was mentioned in the "Histories and Lore" though). Here's a small background for those of us who haven't read the books:

King Aegon IV Targaryen, AKA "Aegon the Unworthy", sired many bastards over the years. On his deathbed, he legitimized all of them, and passed Blackfyre - the ancestral sword of House Targaryen - to the eldest of his bastards, Daemon Waters, even though it should have passed to his rightful heir - Prince Daeron. Having the sword passed to him, Daemon decided that he is the rightful king of the Seven Kingdoms, and what better way to seize the throne than a rebellion that splits the realm in half? Daemon founded House Blackfyre (named after the sword), and became the self-styled King Damon I Blackfyre. The war was a close thing from start to finish. It ended in a fierce battle known as the Battle of the Redgrass Field - named after the grass that was painted red by all the blood that was spilled that day, in which Daemon was killed.

The Coronation of Daemon I Blackfyre
The Coronation of Daemon I Blackfyre

The First Blackfyre Rebellion was a key point in Westerosi history and involved many intriguing characters - Daemon I Blackfyre, Aegor Rivers AKA "Bittersteel", Brynden Rivers AKA "Bloodraven" (AKA "The Three Eyed Crow"), Shiera Seastar and others. Seeing those characters interact and fight on-screen could be amazing. The cream on top is the fact that Daemon Blackfyre was one of the greatest warriors of all time in Westeros, and a spin-off based on the First Blackfyre Rebellion is sure to have some amazing fight scenes (the Battle of the Redgrass Field involved a prolonged duel that he had with Ser Gwayne Corbray of the Kingsguard).

Will it happen? Well, who knows? I sure hope so. Unlike the previous ones, I don't think anyone said that this is NOT going to happen, so there's still hope.

6. War of the Ninepenny Kings:

The name "First Blackfyre Rebellion" already hints that there were more of those. And indeed, four other attempts at seizing the Iron Throne were made by the heirs and allies of Daemon I Blackfyre, who were all banned to Essos after the first rebellion ended in their defeat. The fifth Blackfyre Rebellion, also called the War of the Ninepenny Kings, was the last of them (or is it?)

Again, a bit of background for those of you who are unfamiliar:

About 40 years before the current timeline, nine warlords in Essos struck an alliance, promising to fight together and win a kingdom for each of them, and calling themselves "The Band of Nine". One of those nine warlords was Maelys Blackfyre, AKA "Maelys the Monstrous", and the kingdom he was promised was- you guessed it - the Seven Kingdoms of Westeros. The Band of Nine made the Stepstones (a chain of small islands to the east of Dorne) their base of operations, and this is where the war took place, after King Jaehaerys II Targaryen sent a large Westerosi army to battle the Band of Nine. The war lasted nearly a year, and ended when Maelys the Monstrous fell to Ser Barristan Selmy in single combat, and with him fell House Blackfyre (the male line at least).

Ser Barristan Selmy fighting Maelys the Monstrous
Ser Barristan Selmy fighting Maelys the Monstrous

The War of the Ninepenny Kings could be awesome to see not only for the many great battles that it would feature, but also because we would get to see many characters from the main series in their heyday. Other than Ser Barristan Selmy, other prominent figures in the war were Brynden Tully AKA "The Blackfish", Tywin and Kevan Lannister, and even Lord Steffon Baratheon - the father of Robert, Stannis and Renly.

7. The Life of Aemon the Dragonknight:

This is an idea that I had, but I never heard anyone talking of it anywhere. Prince Aemon the Dragonknight was probably the most legendary knight who had ever lived in Westeros. He served in the Kingsguard under 5 different Targaryen kings: Aegon III AKA "Aegon the Dragonbane", Daeron I AKA "The Young Dragon", Baelor I AKA "Baelor the Blessed", his father Viserys II, and lastly his brother Aegon IV AKA "Aegon the Unworthy".

Aemon the Dragonknight had seen a lot, serving closely under 5 different kings, and has participated in many battles and exciting events. Seeing those events on-screen from Prince Aemon's point of view could be pretty awesome, in my opinion.

Aemon the Dragonknight saved from a pit of vipers by Baelor the Blessed
Aemon the Dragonknight saved from a pit of vipers by Baelor the Blessed

As for the private life of Aemon, very little is known. A spin-off series about his life could provide an interesting point of view on the man's life beyond the famous battles and deeds that every person in Westeros knows of. His loves, his feelings, all that.

8. Daario Naharis in Slaver's Bay:

At the end of season 6, Daenerys Targaryen left the city of Meereen and headed for Westeros, and left Slaver's Bay - now named "The Bay of Dragons" - in the hands of her thrusty ally, Daario Naharis. We know nothing on what became of the man and the cities that he now governs, and it's unlikely that we will know in season 8. Well maybe we will, but there's nothing to indicate that. A cool spin-off series could feature Daario as its main character, and will revolve around his governance of the Bay of Dragons, with a bit (or a lot) of pit-fighting here and there.

The man, the legend; Michiel Huisman as Daario Naharis in
The man, the legend; Michiel Huisman as Daario Naharis in "Game of Thrones"

Daario was always a cool character with many shades of mystery around him. We only saw the man as a supporting character to Daenerys Targaryen, and never really got to see his true motives, feelings and thoughts (though I'm willing to bet that his love for Daenerys was genuine, but that's another topic). Having his as a main character could also bring a lot of comedy into the fray. Ever the warrior, never the governor, Daario would probably have a heck of a hard time governing 3 huge and ancient cities that are in a pretty bad state as it is.

I don't know if we're going to see the man making a comeback in season 8, but if not, Daario's ventures in the east could definitely make for a really cool spin-off series.

9. Anything taking place in the far east:

The World that George R. R. Martin created is huge. Huge, I tell you. Unfortunately, we have mostly seen the far west, which is Westeros. We've seen very little of Essos, and the easternmost places we have witnessed on-screen (and on-page) are Vaes Dothrak and Qarth. There's so much more to see, so many stories to be told on the mysterious and exotic lands in Essos and Sothoryos: the Basilisk Isles, the Summer Isles, the Plains of the Jogos Nhai, the Golden Empire of Yi Ti, Asshai-by-the-Shadow, the Port of Ibben, and so many more places. A completely new and original story can be made in any of these places.

The port of Asshai-by-the-Shadow, a land shrouded in mystery
The port of Asshai-by-the-Shadow, a land shrouded in mystery

An example that could make all my wishes come true is a spin-off series revolving around an explorer, perhaps a maester of the Citadel, on his adventures through the known world, perhaps trying to map it, write a book about it, find some hidden treasure, or just enjoy the view. The adventures of an ordinary man through dangerous and exotic places would surely have me stuck to the screen, and probably most other fans. There's just something about all those places, the mystery about them, that makes me want to know EVERYTHING there is to know about them.

Map of the known world
Map of the known world

Honestly, any story taking place in even one of those places will be awesome to watch. It would be an interesting change from all the medieval-ish theme of knights in shining armor, tall castles and noble houses. Don't get me wrong, Westeros is awesome and there are many amazing stories to be told about it, but a change of scenery would be great and most welcome.

10. A background story an a popular character:

There are many characters in the show with little to nothing known about their past. A spin-off series featuring one of those characters as its main hero could be interesting and give some special view on things.

That way, we can see the background of people like Bronn, Melisandre, Tormund Giantsbane, Mance Rayder, Thoros of Myr and many other cool people we all know and love. Perhaps even a first-person point of view on stories we only vaguely know, such as the adventures of Euron Greyjoy or the Red Viper.

Euron Greyjoy sailing his ship - the infamous
Euron Greyjoy sailing his ship - the infamous "Silence"

The producers have pretty much 100% freedom in this, depending on the character(s) that they choose. Characters like Bronn, for example, don't really have any major background stories that would become major plot-twists at some point or anything like that, and a show about the adventures of everyone's favorite sellsword could be pretty awesome.

Another great option would be the story of Varys and Illyrio Mopatis in the Free Cities, before they became rich and famous. Varys is one of the most intriguing characters in the whole franchise and seeing where he started could be pretty dope. The options are almost endless, really.

11. Anything about The Hound and Tormund Giantsbane screwing things up:

C'mon. You know you want it just as much as I do. Assuming they survive season 8, that is.

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So, what are your thoughts? What's your favorite idea from the list? Any original idea of your own?

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