Is it possible to map mathematics without advanced set theory?

Question

I wish to make a large digraph (network) linking various proofs together in mathematics from, say, the definition of a group to Galois theory. I got in my head that I wanted to do this after reading about the "truth mines" in Greg Egan's book, diaspora, which I would wholeheartedly recommend.

The issue is, at the moment I could only start this graph from naive set theory, starting with the "definition" of a set, of the natural numbers, and so on (though definition stretches things as I could not rigorously define a set as it stands). It seems like this potentially invites danger as there's a reason such set theory is called naive, it leads to paradoxes.

If I were to start this graph from naive set theory, could I then substitute in more advanced material later on and amend it, so for instance eventually linking the ZFC axioms to where I had previously started my graph? Or is naive set theory so far from the truth that I could not possible hope to untangle it?

I understand that this is a strange question, but don't you think it would be amazing to be able to see a map of the mathematical landscape?

Answer 1

That depends what you wish to include in this graph.

If you wish to limit yourself to results that were proved before 1878, which include some of the seminal results of Galois theory mentioned on this page, and many results in basic analysis, then by all means, naive set theory is more than enough. You could probably extend the range to 1900, or even slightly later without having to worry about encountering anything advance in set theory.

If your graph will include, from "group" a "Whitehead group" and then the question "Is every Whitehead group free?", then you cannot proceed further without axiomatic set theory, because to prove or disprove whether or not every Whitehead group is free, you will need to assume axioms which extend from $\sf ZFC$ and certainly go well beyond naive set theory.

Perhaps you'd like to prove that every Abelian group with a discrete norm is free. But the only proof I am aware of, for the general case, uses heavy set theoretic tools (transfinite induction on the cardinals, and Shelah's compactness theorem for singular cardinals).

Mmmm, okay, maybe you'd rather avoid these two questions. But then you can ask yourself, suppose that you have $\aleph_1$ sets of reals, each of Lebesgue measure $0$. Is their union a null set also? Again, you'd have to defer the question until additional set theoretic axioms have been stated. What about meager sets? Well, same thing.

And so on and so forth. Can you map mathematics without appealing to advanced set theory? Sure, but depends on what you call mathematics, if your mathematics is solely based on finite objects, or the natural numbers, or pre-20th century... sure. You can probably avoid it. But if you don't want to live in the past, then you will have, at some point, to run into axiomatic set theory in one way or another.

Because set theory is the tool through which we understand infinite sets, and when you ask something about "every ring" or "every $R$-module" or "every group", you ask a question about pretty large infinite sets, and then some.

Answer 2

Well, your graph is never going to be a complete description of all of mathematics! So a simple answer is that you shouldn't worry about it too much, and just focus on what's useful to you.

In other words, if what you're really interested in is Galois Theory, then maybe you want to have a node that represents $P$, where $P$ is the statement "$S_5$ is not solvable." There'd probably be an arrow to $P$ from the node that represents the statement "$A_5$ is simple," as well as an arrow to $P$ from the statement "$A_5$ is not abelian", and others as well. Each of these nodes would have arrows pointing to them from other nodes, too, but the point is that it's up to you how far back you want to continue tracing these chains before you're not interested anymore.

If it were me, I believe I'd quit long before any propositions of pure set theory, naive or otherwise, really showed up. In fact, what you could do is just declare that "set theory" is going on in the background of your entire graph. (This is really what mathematicians do most of the time in practice; rarely do we explicitly state all of our axioms and methods of inference unless we are specifically interested in that type of mathematics.)

This project could definitely be illuminating and certainly could help you to understand an area of mathematics better, but probably not if you're extremely worried about getting every detail correct and including absolutely everything. Just try it and see what you get, and then add in more and more details/nodes/edges as you go. Enjoy!