An Infinitely Unbounded Stack of Innumerable Infinities

This is the first post in a new series of mine, “Before I Forget”. This series is an exposition of all the awesome (in the British sense), marvelous, haunting mathematics I learned while in college. I’m not sure how often I’ll add to it or when it will end, but I know that I’ve been itching to write about some of this stuff for a while. I’m writing it to be somewhat accessible. I know I have several readers who skip over any math stuff, but I encourage those people to read this anyway if they have the faintest interest. Some of the notation might be confusing, but, if I’ve accomplished my goal, you should be able to understand it anyway because I’ve been redundant enough that you should be able to understand it. :-p Oh, and of course questions and comments are welcome. Without further ado, here is the first article, about Infinity and the Ordinal Numbers (which will form a basis for the rest of the series).

The Real Infinity

While I was a tutor in Calculus 2 (in 2005 I think) I noticed that the students were quite comfortable with the concept of infinity. But their comfort was not well-founded, as it turns out. The first sign of trouble was when they would write constraints like “4 < x < ∞” instead of “4 < x”; not a big deal, but an indication that they were missing the details somehow. Given a problem like: “find limx -> ∞((x2+x)/(x2+1))”, their reasoning would often proceed “the limit of x2+x is infinity, and the limit of x2+1 is infinity, so the result is ∞/∞ which is… (they look to the little table in the book)… undefined! That’s the answer, undefined!” (Exercise for the reader; why is that wrong?). The main problem is that they were using infinity as a number—a mathematical object—that you could do arithmetic on with the results as defined in the book (much like the IEEE floating point standard or Nullity… shudder).

The problem is that infinity is not a number. It is just a notation which stands for something more precise, depending on context. Saying “limx -> 2 f(x) = ∞” doesn’t mean that the value of the “limit function” is “equal to infinity” there, it means that I can make f(x) get as big (but still finite!) as I want by getting x close enough to 2. You want f(x) to be 1,000,000,000,000,000? If I set x to 2+10-400 I can do that. That’s what that expression means. More simply, if you say x ε (-∞,6], you’re just saying x ≤ 6.

So infinity is not an object… or rather, it wasn’t, until we started talking about sets.

Numbers, sets of numbers, sets of sets, it’s all the same to me.

A very long time ago, only some 2,400 years after Pythagoras, mathematicians wished to rebuild mathematics from the very foundation, in order to make sure (and prove, as we will see later in the series, in vain) that their field of study was consistent. You’d think that they would want to do this starting with numbers and building everything else from there. But no, it turns out mathematicians don’t like numbers that much: they’re spooky, even the Greeks weren’t sure if some of them existed (√2 for example).

They decided to build everything out of this concept of a set: an unordered collection of things, at most one of each. But in order to have sets of objects, you have to have objects, right? Well, actually, no. And that’s what we will start with. The empty set, the unique object which contains no other objects, is the basis for our universe. Every object in the universe will come down to this; set theory is made up of sets of sets of sets of sets of the empty set; that’s it, there is nothing else.

There are a bunch of axioms for how sets should behave, “intuitively”, and I certainly won’t talk about them here. To me, that’s the boring part of set theory (at least at this level; they get interesting later on). I want to cut to the chase. We want numbers (as mathematicians), but we have only sets. So we have to come up with a way that sets can represent numbers.

Put yourself in the shoes of a 19th century mathematician. How would you do this? We should start simple, right, let’s not worry about fractions or negatives (or certainly not square roots or imaginaries!). Let’s just try to get the natural numbers: whole numbers greater than or equal to zero. Everything starts at zero, let’s say that the empty set represents zero. Okay, okay. Good. Half way there. Only infinitely many left to go. What’s the simplest thing we can make which isn’t the empty set? How about the set containing only the empty set, {{}}. Let’s call that one. Okay, okay. Good. Almost there. Only infinitely many left to go.

The next step is when things can diverge. You might “follow the pattern” and call {{{}}} 2. Of course, there are a lot of patterns that are consistent with those first two elements, and it turns out that the one where the next element is {{{}}, {}} is more useful for representing 2. Note that this is the mathematician’s choice. We choose to define 2 as the set containing {{}} and {}, or the set containing 1 and 0 by our previous definitions. Something many people don’t realize is that mathematicians make choices like this; we make up objects and then ask questions about them. Numbers are not absolute, and if we had defined {{{}}} to be 2 instead, we would get different answers to some of the questions we asked about 2 (but we haven’t defined what it means to add yet or anything; one hopes that we’d define those things appropriately for each of our definitons of 2 so that the answers to any important questions, i.e. independent of representation, would come out the same; much of math, in fact, is giving a new definition and then proving that our definitions are sane in this way).

Can you guess the next definition? There’s a hint hiding in the previous paragraph. It’s 3 = {{{{}},{}},{{}},{}}, which is getting awfully hard to read, so let’s write it as 3 = {2,1,0}. Okay, now it’s clear what we’re doing. Each number is just the set of all smaller numbers. Weird, huh? This has the nice property that the size (or “cardinality” in math 13375π33|≤) of the set representing a number is the number itself. 3 has 3 elements. Clever.

So we go on forever defining these numbers, which are sets of numbers, which are sets of numbers, …, and ultimately it’s just sets of sets until you get to the empty set. Building a world on nothing. Notice that we didn’t stop somewhere, say, at infinity (as the Calc 2 students would believe). We couldn’t have. Say we did, and call the place we stopped ω (“omega”). We’d just define the next number to be ω ∪ { ω } (ω and everything smaller than it, which happens to already be collected in ω by our definition of a number being the set of all numbers smaller than it), so obviously we actually didn’t stop there. QED.

Welcome His Highness, The Axiom Of Infinity

Mathematicians realized that they can only get so far with that. We’re talking about sets of things, so the set of all natural numbers is something we’d like to talk about. But we don’t have that. It’s kind of pedantic why we don’t have that, really. It’s because the logical language in which the axioms of set theory are written is finite; you can’t have infinite sentences. So we can only ever construct finitely many numbers this way. We can describe how to construct them, and then using our definition, prove that 6 is a number, 10 is a number, and (with a really long sentence) 500,000,000 is a number. But we can never get all of them. It’s okay if you don’t understand why. Suffice to say that we were not satisfied by that, so we added another axiom which says, essentially, “the set of all natural numbers exists” (where number is defined by our construction previously). What a cheat! Anywho, let’s call it, oh, ω.

But then something mystical happens. ω is the set of all natural numbers, so it is a set of numbers. This qualifies for a number by our definition; specifically it is the next number bigger than all the naturals. Yeah, you’re right, infinity! Well, sortof. Let’s just call it ω, and not associate with that tricksy infinity concept. It’s the set of all naturals, and it is a number bigger than all of them. To be a little precise, it’s a number bigger than any number you can write like 1+1+1+1+1+…+1 some finite amount of times. Yeah, that’s kind of circular. Mathematics is about precision, and I spent a good deal of my first year at college precisely, non-circularly defining all of this; trust me, you’d be bored. The only reason I wasn’t is because I’m a freak.

To Infinity, and Beyond!

So then let’s look at this set, which we have actually mentioned earlier in a proof by contradiction: ω ∪ { ω } = { 0,1,2,3,4,…,ω }. We just said that ω qualifies as a number, so this is contiguous set of numbers, so it qualifies as a number too. But it’s greater than ω, because it contains ω. That’s why we didn’t want to think of ω as infinity, because there’s nothing greater than infinity, right? Well, this, in a grade-school punchline, is infinity plus one: ω + 1. It’s greater than ω, and there’s nothing between it and ω (in our current “whole-number” formulation).

We can keep going in a similar manner, defining ω ∪ { ω, ω + 1 } to get ω + 2, ω + 3, …. Then what about the set ω ∪ { ω + n | n ε ω } (the naturals together with the set of things that look like ω + n, for every natural n). This is greater than every ω + n, as long as n itself is less than ω. We call this ω + ω, or ω × 2 (two times infinity!). We keep going to get ω × 2 + 1, ω × 2 + 2, …, ω × 3, ω × 4, … even ω × ω (called ω2).

Yes, you guessed it, keep going! ω3, ω4, …, ωω. And more: ωωω, ωωωω. And now we come across something which our petty notation cannot represent: the union of { ω, ωω, ωωω, … } (the ellipsis is a powerful tool in set theory!). This is essentially ωω an infinite number of times. They call that ε0, but of course, it’s just a name.

And now, out of nowhere, you continue! ε0+1, ε0+2, ε0+ω, ε0ε0, … forever until you throw up your hands and become an English major.

These numbers are called the Ordinals, a gigantic stack of infinities, so you can always say “infinity plus one” in an infinite variety of ways. There are way more ordinals than there are numbers, obviously. But we will see next time that the ordinals we have seen so far do not actually outnumber the regular, finite numbers, despite appearences. All these—this gigantic stack—are really only a way of arranging our usual supply of finite numbers, and we could move things around so that they’re all just plain old ω again. The situation gets interesting again as we prove the existence of a new infinity that we can not do that to (which is larger than everything we have seen so far!). And, it turns out, we can prove that we will not ever be able to know how large that actually is!

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