Infinity Blows my Mind

Don’t let anyone ever tell you that math is boring.

Anyone who says that simply hasn’t spent enough time with math to have their socks seriously knocked off. Because trust me, sooner or later math will and then you’ll become one of those math nerds going around telling everyone how cool math is while they roll their eyes…

Here’s the thing, infinity.

It’s a flipping MIND BLOWING IDEA!

Sure most people nowadays are comfortable with the idea that things can go on forever. Theoretically you could keep counting on and on. The universe is ever expanding. Buzz Lightyear goes to infinity and beyond. Even childhood debates tend to end in some theoretical set theory debate involving multitudes of infinity and infinity plus ones.

So what’s so crazy about infinity? Seems like we’ve all got it down pat.

But what if I came up to you and said, “hey friend, which is bigger? The set of all counting numbers, you know 1, 2, 3 and so forth, or the set of all even numbers, 2, 4, 6, and so on?”

What do you think?

The natural inclination would be to say that the set of all numbers is larger than the set of even numbers. I mean even numbers are a subset of the natural numbers. Isn’t a subset smaller than a set??

Well, not always.

Sure it is if we’re talking about FINITE sets, but with infinite sets it a whole new game. And that game is full of twists and turns, surprise endings and conundrums that will leave you silent and scratching your head.

So which is bigger?

Normally if we want to know which of two sets is bigger, we’d simply count them. Of course we can’t manually count sets that are infinite. So we’ve got to come up with a more generalized way of counting.

Okay, so tell me this how would you count without numbers? No, I’m serious how do you count without numbers?

That may sound like a silly question because we’ve always just counted with numbers and so you never have had to come up with a different system, but it’s not difficult to do.

Way, waaaaaaaay back at the beginning of this blog I talked about the origin of numbers and counting. I told you about this shepherd who needed to keep track of his sheep, but hadn’t any notion of numbers. So in the morning as his sheep left the pastor he’d place a single stone for each sheep in a pile. And in the evening when he gathered his sheep for the night he’d remove one stone from the pile for each sheep. So long as there were no extra sheep and no extra stones, he knew all of his sheep were accounted for.

Our dear shepherd was creating a one-to-one correspondence (or more formally a bijection) between his sheep and the stones.

And when you think about it, that’s really all counting is. You see, we humans came up with some symbols and gave them names and said that the first stone in the piles was to be called “1”, the next stone “2”, and so on. Once we memorized this system of named symbols we needn’t carry around a bag of stones, we could simply draw from our memory and voila! count things.

So to count without numbers is simply to make a pairing between one set of things and another. If at the end one group has things left when the other doesn’t, you know that that is the bigger group.

The Solution

We want to compare two sets:

  • Set A is the set of all numbers. It looks like 1, 2, 3, 4, …
  • Set B is the set of all even numbers. It looks like 2, 4, 6, 8, …

To begin our general method of counting we need to make a bijection between Set A and Set B.

To do this I’m just going to write the start of each set in order and draw a line between the ordered elements of each set, one by one.

A simple bijection

Now it appears that we could just go on in this way forever and ever, matching up each pair one by one. And if I can then that means that the two sets are equal in size.

We have one thing left to do to secure our argument.

Since we can’t actually write out the entirety of each set we need to describe a formulaic relationship between the two sets to ensure I could keep matching up the elements in this manner forever.

Well, that turns out to be rather simple. Each element in Set B is twice the value of it’s paired element in Set A.

So our relationship is 2 times n where n is the set of natural numbers.

And because we found a bijection between the two sets with a bulletproof relationship, we have confirmed that contrary to our intuition the set of all natural numbers is equal in size (formally: the same cardinality) to the set of all even numbers.

Boom.


Friends, that’s just the tip of the iceberg with infinities. There are countable infinities and uncountable infinities. There are orders of infinity. There are infinities that are the same size that seem impossible to be true, yet are easily provable. And yes, paradoxes galore.

Truly,

honestly

mind

blowing

stuff.