In the aftermath of the intense, mathematical mind-bendery (yes, I made that word up) I subjected you to in part 2, I thought it would be fun to look at a somewhat abstract, but thoroughly interesting consequence of what we just discussed.

Alice and Bob play a game

One fine morning, Alice and Bob decide to play a game. The game works like this. Alice and Bob will be given two balls at a time. Each ball with have a number on it (the first two balls will be 1 and 2, the second pair will have 3 and 4, etc). On each toss, Alice must throw the ball with the smallest label, and Bob must throw the even numbered ball. Let’s take a look at how the first three throws would play out:

1. Alice throws away the ball labeled 1, Bob throws away the ball labeled 2.

2. Alice and Bob both receive balls labeled 3 and 4. Alice, who still has a ball labeled 2 which she hasn’t thrown away, throws away 2. Bob whose only even ball is labeled 4, throws away 4. After this toss, Alice has balls 3 and 4 and Bob has balls 1 and 3.

3. They receive balls 5 and 6. Alice throws away 3 and Bob throws away 6. After this toss, Alice has balls 4, 5 and 6 and Bob has balls 1, 3 and 5.

We can begin to notice a pattern here. After twenty throws, say, Alice will have thrown away balls 1-20 and Bob will have thrown balls 2, 4, 6, … , 40. Each has 20 balls left. Extending our thinking slightly, we realize that after any (finite) number of throws, Alice and Bob retain the same number of balls. Now suppose that they threw an infinite number of tosses (I know it’s hard to imagine but bear with me here). As our pattern has shown, they should have the same number of balls left, right? Wrong, and this is where it gets a little crazy.

Think about the set of balls Alice is holding after an infinite number of throws. Does she still have ball 1,000? I claim she doesn’t because she got rid of it on toss 1,000. Once again extending our line of reasoning, we realize that for any number we can think of, Alice must have thrown the ball with that label away at some point “before infinity!” if I ask you for the number of any ball that Alice is still holding, you will not be able to come up with one! She has no balls left! Bob, on the other hand, never threw away an odd numbered ball, so he still has an infinite number of balls, labeled 1, 3, 5, 7, … This conclusion completely defies our intuition, but it is the direct consequence of some simple rules sprinkled with the madness of infinity.

Conclusion

For those of you who have made it this far, I hope you’ve enjoyed. In my opinion, understanding infinity represents a most distinctly human ability: the capacity to think in the abstract, to envision and reason about things we do not physically encounter. Infinity is as elusive as it is grand, unfathomably huge and, on the surface, deeply paradoxical. To me, our development of its rigorous treatment over the last few centuries to where we can actually harness its enormity and make real use of it (all over calculus and physics, for instance) marks one of humanity’s greatest intellectual achievements.