Doubts on sets in logic

Question

I am a university student and this is the first year I stumbled upon logic.

During our lectures we usually define sets (like the set of formulae, the set of proofs...) and prove theorems on them, sometimes using results strongly related to set theory, such as Zorn's lemma, but I don't really understand why we can.

How can one talk about a set of symbols, for example? I can't think of any way to build one from the axioms. How come that we can even use Zorn's lemma on an object that we "arbitrarily" (again, I'm a newbie, take it easy on me) consider a set? Is any collection of objects considered a set? I doubt it, it would lead to paradoxes like Russell's one; in this case, why are we allowed to use Zorn's lemma on it (take for example the compactness theorem)?

Moreover, aren't L-structures supposed to "come before" ZF? They sure should, since for the enunciation of the axiom of separation both languages and variables should be a known argument, yet, interpretations are functions... and functions are particular sets.

What I'm asking is: what's the most efficient (and logically rigorous) order everything should be introduced in? I know for a fact my professor didn't present it to us because she just needed to give us some elemental piece of knowledge in preparation for future courses and she adopted another order for didactical purposes.

Hopefully I've been sufficiently clear with my doubts.

Does this answer your question? How to avoid perceived circularity when defining a formal language?, or also this answer on MO — Vsotvep, Nov 20 '21 at 22:03
Here's a popular related post Does mathematics become circular at the bottom? What is at the bottom of mathematics? — Joe, Nov 20 '21 at 22:04
You have to distinguish between sets as a mathematically defined thing, and sets as defined through logical axioms. When we use sets to build up logic systems (sets of symbols, sets of formulas, etc), we use the plain old mathematical definition of sets…. the one that you presumably have been using in other math courses before you got to this course. But once you have a formal language like logic in place, you can start defining sets as the actual things that satisfy certain logic axioms, so that comes later. — Bram28, Nov 20 '21 at 22:16
A post on math overflow about your first question: the sets in mathematical logic — Joe, Nov 21 '21 at 00:31
I was expecting the answer could have been of this kind. So we don't care about the paradoxes which such objects could lead us to? What are functions and relations in this context then, just ways of mapping sets wothout any deeper link to sets thenselves (in set theory we think of those concept as particular sets)? Final question: how are we allowed to use the axiom of choice for sets which are not axiomatically built? Thanks for the answer, by the way. — R4ff0, Nov 21 '21 at 08:27
Do you also happen to know how to answer to the other questions? Up. — R4ff0, Nov 23 '21 at 01:00
If you are asking @Bram28, then you should tag to generate a notification. — Joe, Nov 23 '21 at 03:55
Yes, that user will (or did already) receive a notification from my comment. I phrased my comment the way that I did so that you are aware of that feature for future comments. — Joe, Nov 23 '21 at 14:22

Doubts on sets in logic

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