And Beyond?

You know nothing about infinity. You probably think you can add it to your argument in order to win, or add one to it to win even more. Nope, nothing about infinity, but I know everything about it (ha!) so allow me to enlighten you. There's this statement you might have heard before, that if an infinite number of monkeys on an infinite number of typewriters type for an infinite amount of time, then at some point they will produce the complete works of Shakespeare. This is an idea that comes from probability theory, and what infinity does to it. Because when you let the time line approach infinity then the probability that anything will happen approaches 1 (i.e. 100%). Therefore we could actually reduce the goofiness down to one monkey on one typewriter. Thus if Bubbles (yes I'm naming it Bubbles) types for an infinite amount of time then at some point it'll produce Hamlet. So on a long enough time line any damn thing can happen, anything.

Let's break this down a bit and consider an easier probability problem. Suppose you decide to flip a coin, even if you don't know anything about probability, you know that there's a 50-50 chance of landing heads or tails, whichever one you want. So the probability of getting a heads we say is 0.5, but this isn't entirely true. Why? Because there's a chance (however small) that the coin will land on its side, there's also a chance that it will hit the ground and shatter into a million pieces. The probability that a flipped coin will blow up your house  is extremely small, so small that we don't ever count it (i.e. we safely round it down to 0). So in truth the probability of getting a heads is actually 0.49999999... and a lot more 9s, ditto for getting a tails. Now if you flip a coin long enough eventually it will land on its side. It might take a billion years for it to happen, but assuming you can live forever, and you decide to spend eternity flipping a coin, eventually it will. So, again, letting the time line approach infinity the probability that anything will happen approaches 1. Thus if you flip a coin for an infinite amount of time at some point it will turn into a metal bird and fly away. What's the reason we don't see coins transforming into all sorts of metal creatures and fleeing from their owners? Because no one can flip a coin for an infinite amount of time. Have you ever just sat around flipping a coin? Seriously, ten minutes is pushing it, but what you should take away from this discussion is that when infinity is involved, and you aren't careful, anything can happen, and I mean anything.

Of course probability isn't my field of expertise, far from it in fact, so if any probabilists out there have a beef with what I've just said then I welcome your rants. Instead here's another fun problem that can show you how infinity misbehaves when you aren't looking closely. Consider the following infinite sum: suppose we take 1, subtract 1 from it then add 1 then subtract 1 then add 1 so and and so forth forever and ever. i.e.
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ...

Now let's group everything in the following way:

(1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + ...

Which, as you can probably see, is an infinite sum of zeros,

(0) + (0) + (0) + (0) + ...

Which is zero. Now let's regroup the first guy again in this new way:

1 + (- 1 + 1) + (- 1 + 1) + (- 1 + 1) + (-1 +1) + ...
 Which is now 1 plus an infinite number of zeros,
1 + (0) + (0) + (0) + (0) + ...
Which equals 1. So we have the same sum, but it equals 2 different things, and nothing we did here was wrong. But wait there's more, act now and I can show you that this sum is equal to any number you want. That's right, pick your favorite number, let's call it z. z can be anything you like: positive, negative, real, or complex. So let's say it's complex just to stay as general as possible. Now here comes the fun part. Those of you who aren't good at math may want to skip down to the bottom. Those of you who are at least familiar with sigma notation and infinite sums note that the infinite adding and subtracting of ones can be written as

Why is this do you think? Every time k is even we get a +1 and every time k is odd we get a -1. Ok now noting that 1 = z - (z -1) we can multiply anything we want by [z - (z-1)] since multiplying by 1 doesn't mess up our equations right? Right, thus our sum becomes

So there we go, we got the above infinite sum to equal our favorite number. Thus adding and subtracting by 1 an infinite number of times gives us any number we want, and it's the infinity that causes this to happen. If we stopped adding and subtracting ones at, say, the one millionth term we would get either 1 or 0 and only that result. But when we drag it out to infinity anything can happen, and I do mean anything. Does this always happen with infinity? Actually no, when you begin studying infinite sums (which we call series) we learn of a criteria to keep this from happening, but to hell with that. What's the fun of infinity if you can't play with it and let it go wild? Some stallions you just can't tame, and sometimes infinity is a kicking biting steed that breathes fire. Let it go free and on a long enough time line it'll grow into a snake with a firetruck face that'll blow up your house with a coin ... on a long enough time line that is. Who ever said math wasn't fun?

(Also later I do plan to explain my above steps, but I just don't have the energy to do so right now. Those of you who are interested be on the lookout for it)