The many worlds interpretation of mathematics

This is the first in a series of posts about model theory.

I’ve talked about this stuff before when writing about why I do math, but I wanted to write a standalone piece to emphasize the ideas I’ll discuss here.

Prerequisites

Basic familiarity with sets, logic. Familiarity with “different sizes of infinity” is also helpful (for instance, that $\mathbb N$ is smaller than $\mathbb R$ since no function from $\mathbb N$ to $\mathbb R$ covers all the elements of $\mathbb R$).

What is independence?

As alluded to in the prerequisites, you may know from Cantor’s Diagonalization Theorem that $|\mathbb N| < |\mathbb R|$. But is there some set $S$ whose size is strictly in between, i.e. $|\mathbb N| < |S| < \mathbb R|$? This question is known as the Continuum Hypothesis.

You may have heard that the Continuum Hypothesis is “independent of ZFC”. So what does that actually mean?

What is ZFC?

It turns out mathematics is governed by a set of rules which we call “axioms”, and the language of mathematics is really the language of sets. For instance, we can encode the natural numbers as sets. And ZFC does two things:

1. It gives us certain sets to work with, such as the empty set and the set of natural numbers¹.
2. Given a collection of sets, we can perform operations on them to get new sets. For instance, given two sets $A$ and $B$, we may find their union $A \cup B$.

This is a very imprecise picture, but it will do good enough.

At the same time, mathematics may be considered the study of whether things are true or false. And formally, a proof is a syntactic object which starts from a list of assumptions and derive a conclusion. For instance, if we start with $a \implies b$ and $b \implies c$, we may conclude $a \implies c$. And so on. The point is, we are given a list of rules which tell us how we may combine assumptions to reach conclusions. As another example, we are allowed to combine $a \land b$ with $a \land \lnot b$ to derive $a$. These syntactic considerations are what we will refer to as “first-order logic”.

The point is, ZFC (along with first-order logic) describes how mathematics ought to behave.

Many worlds

I’ll give you the punchline up front: there are multiple worlds of mathematics satisfying the rules of ZFC, and they are genuinely distinct.

We’ve just established that “mathematics” is “anything satisfying the rules of ZFC.” Let me give you an analogy: When we study linear algebra, we study “vector spaces”, which is “anything satisfying the rules of a vector space”. Importantly, we are not studying just one particular vector space: when we create definitions such as “linearly independent”, or prove theorems such as “all linearly independent spanning sets of a vector space have the same size” (this is the definition of dimension), we are doing this for every vector space, because we are only using the language and axioms of vector spaces.

The same holds for mathematics. Who says that ZFC only describes one world of mathematics?

Common misconceptions

If you’re anything like me, you may have watched certain YouTube videos that say “we cannot prove whether the Continuum Hypothesis is true or false.” This is technically true, but it is some very misleading phrasing.

In particular, if you were to explicitly unpack your assumptions about what this means, they might look something like this:

1. There is one world of mathematics.
2. There is no proof that the Continuum Hypothesis is true. There is also no proof that it is false.
3. Furthermore, we somehow know that we can never produce such a proof.
4. So I guess we’ll just never know whether the Continuum Hypothesis is true or false, and we can never know.

This is completely wrong. Here’s what’s actually going on: assuming there is a world of mathematics satisfying the rules of ZFC, there is some world of mathematics where the Continuum Hypothesis is true, and another world of mathematics where the Continuum Hypothesis is false.

Godel’s Completeness Theorem

From now on we abbreviate the Continuum Hypothesis as CH.

So why does “CH is independent of ZFC” imply that there is a world of ZFC mathematics where CH is true, and another world where CH is false? On the surface, “there is no proof of CH or (not CH)” is a statement about proof manipulations (syntactics) while the statement about multiple worlds is about actual worlds of mathematics obeying the rules of ZFC (semantics). But as we will soon see, our system of syntactics and semantics are carefully designed to be tightly connected.

This shows itself in Godel’s Completeness Theorem:

Suppose a set of rules does not lead to any contradictions; we call such a system consistent. Then there is an object satisfying those rules. (The converse also holds.)

A few other related facts:

1. If I can prove something given a set of rules, then it is true for every object satisfying those rules. (For instance, if I prove “every basis in a vector space has the same dimension”, then it is actually true of every vector space.)
2. And importantly, if I can’t prove a statement given a set of rules, then there is an object that doesn’t satisfy that statement. (For instance, since I cannot prove “every vector space is finite-dimensional”, then there exists an infinite-dimensional vector space.)

Given these rules, here is how you prove that CH is independent of ZFC. Start with some universe of ZFC. Use it to construct a universe of ZFC that satisfies CH and a universe of ZFC that does not satisfy CH. Now we can’t prove CH or not CH, since there is a universe of ZFC that breaks each of these statements. Voila! We have independence.

Godel’s Incompleteness Theorem

But this isn’t actually the end of the story. So far, the impression I’ve been giving is that “there is a universe where CH is true and one where CH is false.” But this isn’t really the correct picture.

Instead, it would be more accurate to say the following:

There either exists no universe of mathematics satisfying ZFC, OR there exists a universe where CH holds and a universe where CH does not hold.

Furthermore, we have no idea whether there exists a universe of mathematics satisfying the rules of ZFC. Now, why would we bother studying a system of rules if we can’t even show there’s a universe of mathematics satisfying these rules?

The short answer: we have to. Godel’s Second Incompleteness Theorem² says

No system of mathematics can prove its own consistency.

In short, we cannot prove any system of mathematics is consistent without working in a stronger system of mathematics, which we then need an even stronger system of mathematics to show the consistency of, and so forth…

This means that, at some point, we just need to take the consistency of our system for granted. We can rely on heuristic arguments such as “we’ve tried really hard and haven’t found a contradiction” and “the rules of ZFC seem simple enough.” But we can never know for sure… unless we find a contradiction. (For example, naive set theory has a contradiction in Russell’s Paradox, so we know for sure it is not consistent and need to toss it out.)

Finally, we should talk about why inconsistency results must exist, regardless of which system of mathematics we work in. This is a consequence of Godel’s First Incompleteness Theorem, which says

Given any system of mathematics, there exist statements that can neither be proven nor disproven.

This necessarily implies that there exists a world where the statement is true and another where the statement is false. Why? Because if there were no world where the statement were true, then it would be false in every world, which means we can disprove it by Godel’s Completeness Theorem. Similarly, if there were no world where the statement is false, then it would be true in every world, which means we can prove it.

What does all of this mean? If we have a system of mathematical rules, we don’t know and can’t know how many worlds satisfying these rules exist (unless we derive a contradiction in these rules). However, we know that there cannot be exactly one such world.

Independence is a good thing, actually

We know that no matter how many rules we tack on, we will never get a collection of rules that carves out exactly one universe of mathematics. But we can at least try to patch up our theory. For instance, if we know CH is independent of ZFC, why not just assume that CH holds? That could make proving certain things simpler. After all, we literally have more assumptions to prove theorems with. So why don’t we?

The core of mathematics is to prove things in as much generality as (reasonably possible). We could, for instance, study linear algebra only in the vector space $\mathbb R^3$ and prove theorems such as “every basis of $\mathbb R^3$ has the same size”. That certainly would make things easier, but we would be losing a lot of insight. And sometimes we want to focus on specific vector spaces, such as finite-dimensional vector spaces. Then it makes sense to tack on an extra assumption. But if we don’t need it, we really shouldn’t use it.

Studying ZFC mathematics is much the same. For instance, all of real analysis still holds whether CH holds or not. So when we are proving, for example, that “$\mathbb Q$ is dense in $\mathbb R$”, we aren’t just pointing to a specific world of mathematics and saying $\mathbb Q$ is dense there. Rather, we are making a statement about the entire multiverse of mathematics.

Isn’t that neat?