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Axiomatic set theory is basically a first order logic theory, so I believe we must certainly begin with a treatment of first order logic. However, to define a first order logic vocabulary we need to define a set of constants, a set of variables, a set of relation symbols, etc, whereby we are already using set theory.

Is this avoidable? What would be the cleanest way to define these two subjects?

gdiazc
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    I was told this can be dealt with resorting to finitism. I've never seen it done and I would love to see it. – Git Gud Apr 25 '14 at 19:43
  • I remember having the same question on Logic course when I heard the notion of set. – Arash Apr 25 '14 at 20:06
  • If it makes you feel better, you could simply avoid the use of the word "set" by saying something like, these are the symbols are we can use: $\forall , \exists , \land , \lor$, etc. In any case, the word "set" is only ever used informally even in set theory. – Dan Christensen Apr 25 '14 at 20:10
  • @DanChristensen I'm not sure that solves the OP's problem, at least it doesn't solve mine. Of course I can interpret the term set as something which isn't an object in the universe of $\sf ZFC$, the real question is how to give meaning (not an intuitive one, but a formal as possible one) to that concept of set you allude to. – Git Gud Apr 25 '14 at 20:12
  • One needs (sub)string manipulations on finite strings over a finite alphabet. This is something we firmly believe to knmow how to handle, and is much less than the full power of set theory. – Hagen von Eitzen Apr 25 '14 at 20:12
  • @GitGud Formally stated in the notation of FOL with some extensions (e.g. '$\in$'), the axioms of ZFC do not use the word "set." – Dan Christensen Apr 25 '14 at 20:26
  • @DanChristensen You must have misunderstood me. I'm aware of that. But "to define a first order logic vocabulary we need to define a set of constants, a set of variables, a set of relation symbols". The question remains unanswered. – Git Gud Apr 25 '14 at 20:29
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    I may have written about this elsewhere on this site. One approach is to begin with a weak but reasonable metatheory that allows us some induction. Primitive Recursive Arithmetic is way more than needed. Using this you can develop first order logic for explicitly presented finite languages. Once you have this, you can formalize set theory, and use set theory to formalize first order theory in full generality within the set theoretic universe, where you can prove the formal counterparts of (generalizations of) the metatheorems you had previously established. – Andrés E. Caicedo Apr 25 '14 at 20:29
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    (Note in particular that the development is purely proof theoretic in the metatheory, and emphasizes syntax, and semantics only really comes into play once we have formalized logic within the set theoretic framework. Of course one can push a little what we can do in the metatheory in some cases, but this seems the reasonable course of action.) It is not quite turtles all the way down, but of course we have to assume something, be it $\mathsf{PRA}$ or some other weak system. – Andrés E. Caicedo Apr 25 '14 at 20:33
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    @AndresCaicedo Can you give a reference to see it being done? – Git Gud Apr 25 '14 at 20:35
  • @Git Gud: I think that this older answer of mine is relevant to your comments: http://math.stackexchange.com/a/202159/630 - let me know where to go from there, if I can write something in addition – Carl Mummert Apr 25 '14 at 20:35
  • @GitGud Any circularity is only in the informal narrative. I don't see that as a real problem. – Dan Christensen Apr 25 '14 at 20:37
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    @CarlMummert I read it a few minutes ago. The recent up vote was mine. I get the gist of it. What I'd like is a expository treatment of the process: "we start with this, we define this and that in a certain way, theorem, proof, theorem proof..." – Git Gud Apr 25 '14 at 20:37
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    @GitGud I would cite this, but perhaps you mean something more explicit. The answers to the question mentioned above are a good starting point. I do not know of a printed reference where this is done in any level of detail, but surely there should be some such treatment. (Perhaps looking at Kleene's Introduction to metamathematics may be a not entirely unreasonable first step.) – Andrés E. Caicedo Apr 25 '14 at 20:38
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    I also thought of Kleene's old book. Other options: if you look at something like Simpson's Subsystems of Second-order Arithmetic you will find a formal treatment of logic in second-order arithmetic in the style I think you're looking for (for example, you could follow that to study ZFC or any other countable first-order theory in second-order arithmetic). Smorynski's article in the Handbook of Mathematical Logic shows how to formalize the second incompleteness theorem, which is a suitable example to see how it can be done. – Carl Mummert Apr 25 '14 at 20:40
  • Thank you all. I will look into it. – Git Gud Apr 25 '14 at 20:41

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