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I'm reading a book on mathematical logic by Ebbinghaus, Flum and Thomas. It turns out that set theory is used to build formal languages. I mean, one begins with the definition of an alphabet, which is just a non-empty set $\mathcal{A}$. So actually, I imagine that one starts with two primitive notions: set and member.

My confusion is the following: if we are trying to construct all the logical symbols ($\forall, \exists, \wedge$, etc.) as just elements of an alphabet, we therefore can not use logic to build logic; if we are working with sets, the ZF axioms use logic ('there is a set $z$ such that $x\notin z$ for all sets $x$'); so how does one begin? Maybe I'm confusing myself. I'll try harder:

If I want to build languages and logic and start by the primitive notions of set and member, then for example I need the axiom of empty set which uses logic (?), because otherwise what are symbols? what are strings (which are in fact sequences and therefore we need $\mathbb{N}$, so we need the empty set, so we need the axiom and we need logic)? Well maybe we don't need $\mathbb{N}$ since we are concern with finite sequences, so we just need the definition of tuple. Still, we need the empty set.

Maybe it is not possible to construct mathematical logic, I don't know. It seems that there are always terms that have not been defined, or perhaps cannot be defined.

If I have not made myself clear, please let me know.

NoetherNerd
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    There are always undefined terms and there are always assumptions made. At some point you have to decide if you believe it exists or not but it is always going to be a leap of faith. – CyclotomicField Dec 06 '21 at 15:20
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    The question here isn't exactly the same, but I think Henning Makholm's answer will resolve your question: Why can we use induction when studying metamathematics? – MJD Dec 06 '21 at 15:23
  • You use the basic language of set theory to develop the mathematical theory of logic, in the same way you use the basic language of set theory to develop e.g. probability theory. – Mauro ALLEGRANZA Dec 06 '21 at 15:23
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    This is an important question, and if I recall correctly Stephen Cole Kleene, brought it up in his textbook Mathematical Logic. The basic idea lies in separating out logic from meta-logic used to talk about logic. In other words, we have a sort of informal reasoning used to construct how formal reasoning will work in terms of it's basic rules and assumptions. – Doug Spoonwood Dec 06 '21 at 15:24
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    You do not need set theory to understand the notions of symbol and expression: we have already because we know hoe to read and write. We simply formalize the intuitive notions of natural language with mathematical jargon. – Mauro ALLEGRANZA Dec 06 '21 at 15:25
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    See the same question in this post – Mauro ALLEGRANZA Dec 06 '21 at 15:32
  • It's not the same question, but Define a model for $\mathbb N$ without set theory comes from a similar place. – Mark S. Dec 07 '21 at 12:42
  • Attempt without separation of logic and meta-logic: Starting with the conclusion, that we do have the ZF system, you might define algorithms that yield all possible concepts of logic according to an invented quality criterion; among others it might yield the 2nd Order Logic. That criterion binds the class logic. Possibly you can prove, that the ZF system is the simplest creator of the assumed class "logic" and so the "mother of logic". – Sam Ginrich Dec 07 '21 at 19:39
  • "Set" in not actually defined in the ZFC axioms so you are free use it as defined in any good dictionary, even in mathematics AFAIK. Likewise "element" and "member." – Dan Christensen Dec 08 '21 at 01:28
  • Also aside from all of the comments above, and the distinction between logic and meta-logic, I think it may be helpful to take a look at the first few sections of Devlin's "Constructibility" book, where they "simulate" first-order logic in the language of set theory inside a fragment of ZF! Back when I had similar questions, reading it was helpful for me to understand the difference between logic as interpreted in ZF and logic in the sense of meta-logic. – Shervin Sorouri Dec 08 '21 at 09:13

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