How do we avoid circularity when we build a structure for ZFC? [closed]

Question

when investigating ZFC as a formal language a structure is a set, are we not engaging in circular logic here? Or is 'set' thought of in a more primitive sense?

Answer 1

From the point of view of mathematics, we are not in the circle. We are using the language of set theory for expressing our definitions, theorems and proofs, because we know how good and suitable this language is (although probably there exist also other possibilities). In order to be mathematically correct, we have formalized and axiomatized this language in the form of first order logic and ZFC axiom system. So our proofs are proofs in ZFC and our theorems are statements provable in ZFC. We have built many theories (collections of definitions, theorems and proofs), one of which is also model theory. And we are also studying, no surprise, models of ZFC. If we prove a theorem about models of ZFC, we just prove a statement in no way different from other theorems we have proved. As we are making all our proofs in ZFC, we see that ZFC is occurring here twice, first as an axiomatic system in which we are making our thoughts, and second as an axiomatic system we are studying. These are two distinct things, some people would say, "in two different levels".

Now, to the question. What is a set? Depends in which level we are. In the first level (the one in which we are doing all our mathematics), this is the just an ordinary set that we all know very well. But if we are studying models of ZFC, we have also sets in the second level. Here the notions started to mix. In model theory (we developed in our common first level), structures are first-level sets. If this is a structure for ZFC then the elements of the structure are also called "sets", now these are second-level sets. The word "set" has two different meanings and we should distinguish between them. If we do it, there is no circularity in saying that second-level sets are (first-level-)elements of an first-level set, and therefore are first-level sets, too.

To make things more complicated, in ZFC model theory it is common to consider models in which there are sets which are again models of ZFC. I would say that now we are somewhere in the third level. Word "set" has three different meanings and we have to be careful, which use of this word means which level. But again, there is no logical circle in doing that.

Psychologically, this question was struggling me (and some my friends) for a long time. I now think that problem was caused by the the fact that I was too limited by the idea that ZFC is a universal framework for doing all mathematics, that I could not imagine it in a different position as an object of study.

Answer 2

When set theorists investigate ZFC, they use sets (or classes) as models, and their existence is of course given by the axioms of ZFC again. This indeed seems at first sight circular, but in fact it is not. The salient point here is $where$ we let our investigation happen (as it is a mathematical one we need some axiomatic frame for this task). And ZFC is easily capable of expressing all the things that are needed to examine models of ZFC. So whenever one talks about models of set theory, ZFC is tacitly assumed to be the axiomatic frame where the whole discourse takes place. So there is no vicious circle in the end. You start with assuming the axioms of ZFC and then use models, whose existence is now guarateed, to investigate the axioms of ZFC again, i.e. their gödelized equivalent.