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I know this question is a cliché but this is something I just cannot understand. This is my context: We define logic so we can define a formal language and then set theory, but to define logic we need set theory. So, which came first, the chicken or the egg?

This is my attempt to understand this dilemma.

We take a system of axioms (must probably ZF) that we might call meta-set theory and using the basic rules of logic and meta-mathematics (natural numbers, recursion, etc.) we develop a formal language and then, under such rules of language we construct the theory of logic, then we define set theory formally (again ZF but formally) and finally all the mathematical structures. (In this case I'd say ZF is both a system of axioms(a meta-theory) and also a formal theory).

This would be something tricky that avoids answering the question because we take as given both meta-logic and meta-set theory at the same time (or at least the question of asking which was first loses all sense and interest).

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I think questions very close to this one have been asked here before, so you may want to browse for a while, and may find additional information. One approach to this issue is to start with a weak meta-theory (how weak is an interesting question. You seem to need some basic induction). There you develop logic (for finite or explicitly presented countable vocabularies). Using this, you can develop formal set theory, and inside set theory you can re-develop logic, and go on to model theory, etc, now without any size or definability restrictions. –  Andres Caicedo Jul 9 '13 at 3:10
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Aside, it is curious how variants of this question seem so pressing when one is beginning, and how little they seem to matter after a while, because the answers end up not affecting the actual mathematics we do. –  Andres Caicedo Jul 9 '13 at 3:12
    
@AndresCaicedo Thanks for your comments Andres. I did this question some time ago: math.stackexchange.com/questions/307231/foundations-of-logic and it's basically the same. But the answer is not the one I'm looking for. –  Daniela Diaz Jul 9 '13 at 3:37
    
@AndresCaicedo Do you know of some book that explains this approach you say? It would be of great help and I'd be very grateful. –  Daniela Diaz Jul 9 '13 at 3:39
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Try reading/browsing these in the given order: Introduction to metamathematics by Kleene, Mathematical logic by Shoenfield, Set theory and continuum hypothesis by Cohen. –  hot_queen Jul 9 '13 at 8:38

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Logic dates back to Aristotle (at least). Set theory was not formalized in any sense (so far as I know) until centuries (millennia?) later. (Also, you can't actually tag Asaf in a question.)

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I'm not sure what kind of logic Aristotle did but first order logic is a recent development in mathematical logic. –  hot_queen Jul 9 '13 at 8:58
    
Here you go: Aristotelian logic. A lot of it should look familiar. –  Cameron Buie Jul 9 '13 at 12:07

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