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I am confused about some definitions in logic/ axiomatic set theory:

We stated in our logic lecture the ZFC axioms and called the members of a ZFC-model "sets". But to define formulas and structures, we needed sets as in "A structure is a non-empty set with functions and relations" and also for formulas, we needed, e. g. a variable set.

Could you help me solving my problem? For me it currently seems circle-reasoning.

Best regards

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    I believe that there are a few threads like this one on this site already. – Asaf Karagila Oct 09 '14 at 13:46
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    Here is one: http://math.stackexchange.com/questions/121128/when-does-the-set-enter-set-theory – Asaf Karagila Oct 09 '14 at 13:46
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    You are confusing the word "set" with the word "set". –  Oct 09 '14 at 13:49
  • I really recommend checking out Asaf's link. Further, I'd say that one can eliminate the notion of a set from the syntax of a first-order language by adopting a weaker meta-theory, say primitive recursive arithmetic. – Nagase Oct 12 '14 at 02:24

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There is no circularity here: just an usual confusion between metalanguage and object language, or, more precisely, between the English-language word 'set' and the object-theoretic word '$set$' in ZFC. What you have observed is that in order "to define the formulas and structures" in ZFC we need to dispose of sets in our mathematical universe of the meta-language (to which extent this assumption is innocent is a question for the philosopher of mathematics).

Tourkalis makes an illuminating analogy in his Lectures in Logic and Set Theory I (2003) distinguishing those two levels of abstraction:

This is analogous to building a "model airplane", a replica of the real thing, with a view of studying through the replica the properties, power, and limitations of the real thing. (p.3)

Bruno Bentzen
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