Learn set theory. Then, notice than the only definition of "thirteen" you can formulate is "the thirteenth number". Then, define "thirteenth" by induction (those are the ordinals, who have a precise meaning in set theory).
Now, you have a meaning for "the thirteenth something". Numbers are not significant after all.

I know a fair bit of set theory, as it happens, and I understand that it's sometimes convenient to identify numbers with sets, but I deny that it's correct (or even intelligible) to do it in the way that, say, the von Neumann ordinals tell us that 0 = {}. There's actually quite a sizeable literature on this problem in the philosophy of maths.

Set theory is one of the options open to you, is it, OP? You have research set theorists in your dept as well as research algebraists?

Ugh, Set Theory didn't seem that interesting to me (what little I learned of it). I think the math class I enjoyed the most was Graph Theory, with Number Fields a close second.

take whatever rick kenyon is teaching

Sets are building blocks in mathematics. Everyone agrees on some basic operations/properties on sets, in an axiomatic sense. So everything you build is in fact a set.

As for numbers, ordinals and numbers are a quite different thing. You can take any well-ordered set isomorphic to omega (the first infinite ordinal) and call it the set of natural numbers. That's reminiscent of peano axiomatic, all those sets will behave the same way algebrically. You are thus not forced to say that zero "is equal to" the empty set.
When you construct relative numbers, you usually take equivalence classes of ordered pairs of integers. You identify N with some subset of Z, and you don't talk any longer about your 'original' N. Your "embedded" N is as good as the old one.

Real numbers are traditionnaly defined as 'equivalence classes of sequences of equivalence classes of ordered pairs of equivalence classes of pairs of finite ordinals' if you take N to be the first infinite ordinal. And zero is not the empty set.

Well, for example, surely nobody would argue that the elements of the ring of integers are not numbers?

By "linear" I assume you mean "linear algebra", which gets boring and technical when you try to learn too much of it (at least finite dimensionally).

You may then want to learn algebra. I used to really like algebra when I was in my first two university years (I'm french, scolarity is difficult to compare though) but it did bore me at some point. I then fell in love with functional analysis and partial differential equations, and everything that I'm interested in now as a phd student is related to "applied" mathematics in some sense : calculus of variations, optimization, numerics, PDEs, probability theory for instance.

Conclusion : jump first and try to fly.
My advice : complex analysis. It's really cool, there are wonderful concepts and really deep stuff. It motivated half of the cool math of the 20th century.

I'm sure DrunkHamster understands all this very well. But just because you can base mathematics on set theory, doesn't mean you have to, and it doesn't mean you should. The fact that you can is merely a demonstration that set theory is powerful enough to express other mathematics. But it is hardly unique in this respect.

Set theory is easy to express with formulas, is widespread, has been thoroughly studied and is convenient for people to agree on (at least it is precise enough to isolate the points one should disagree with). That's the standard point of view since 100 years in mainstream mathematics, and even if you don't study it, you know it exists and you can sleep.

When you want to conceptualize "the first", "the second", and so on, ordinals come in handy, and you should use {}, {{}}, ... because of the very natural ordering induced by the relation "is an element of". But finite ordinals are not natural numbers. Numbers can be "numbered" by finite ordinals (by definition) and you can construct them the way you want, you'll get the same theorems. It was the original point of DrunkHamster imo, and I only wanted to point out that you have so many multiple copies of the same object at the same time that you really shouldn't begin to think about how they are constructed.

But feel free to number the natural numbers with finite ordinals.

Does the reduction of stuff to set theory really help people sleep? I mean, it doesn't seem out of the realm of possibility to me that one day people find that, say, the Axiom of Replacement allows you to derive a rather nasty contradiction from the rest of ZF. Would this result cause number theorists or probabilists or group theorists or topologists to have a crisis of faith in the coherence of their subjects? I doubt that it would, somehow (OK, maybe topologists because they actually use set theory), because at the end of the day the whole program of set theoretic foundations is something which is done in the first chapter of textbooks and then quickly forgotten or ignored. The idea that number theory somehow rests on secure foundations because we can find an interpretation of the integers in ZFC is somewhat ridiculous, to me at least; I have much more certainty in the consistency of PA than I do in ZFC.

Well in the interests of full disclosure, I'm partial to a structuralist philosophy of maths which says that all there is to numbers is their structural properties and relations; there is no truth of the matter whether 2 = {{{}}} or 2 = {{{}}, {}} or any of the infinitely many other possibilities, because to even ask the question which set is 2 is meaningless given that '2' only makes sense in the context of a natural number progression.

Consistency of ZF is undecidable, isn't it ? So what you describe is unlikely to appear.
You're right, PA is a much better (and simpler and more natural) way to deal with numbers, but you can build simple enough models of PA in ZF to be happy with ZF as a whole. Nonetheless, I have been taught ZF way after I was introduced to N with Peano axioms. And it didn't cause me insomnia.

Ordinals for instance are important in some fields of analysis though, set theory arose from studying topology and measure theory. So familiarity with ZF(C) is good for a lot of mathematicians.

That's the point of any axiomatic theory imo, nothing new under the sun. You use axioms to prove things, not models. You use natural numbers ? Use PA to construct addition and multiplication and do all the stuff you want with the semi-ring structure. Constructing an explicit model in ZF is only to show you there exists a constructive way to have numbers, and this is important for some people (constructivists, intuitionnists, computer scientists, young people who need some faith in mathematics, ...).

+1

Number theory was probably my favorite, but it was mostly a result of having a kick ass professor.

I really enjoyed functional analysis, it's mind bending.

Combinatorics was my favorite by a lot. It was pretty easy too.

My favorite college math class was Analytic Number Theory. I was blown away by the proof of Dirichlet's theorem (If a and b are relatively prime there exists infinitely many primes of the form a + mb). It was also cool to see the proof of the prime number theorem, but to me, Dirichlet's proof was much prettier.