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I want to axiomatize "the concept of set" in my head, but every time I face some circular definition or intuition.

In predicate logic, we quantify over some "Universe of Discourse". Intuitively Universe of Discourse is a set, a collection. But then, in naive set theory or axiomatic set theory, we define "set" by using quantifiers (over Domain of Discourse)... Isn't it circular?

I mean, we assume that "Universe of Discourse" is some kind of "set", then by using first order predicate logic we define set again?

Barış Akalın
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  • The universe of discourse is a proper class in the axioms of set theory. – Dustan Levenstein Dec 02 '13 at 23:48
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    You don't define sets. They are primitive notion. – azarel Dec 02 '13 at 23:48
  • (I suggested one thread where this topic has been discussed, but I am sure that there is at least one more, and I encourage people to go out there and find those as well.) – Asaf Karagila Dec 02 '13 at 23:52
  • Thanks. I'll examine your suggestions until I have a crystal clear concept... – Barış Akalın Dec 03 '13 at 00:01
  • For our axioms about sets to be true we just need there to be sets, and for them to be extensional, etc. The idea of a universe in the model theoretic sense belongs to metalinguistic talk, it isn't a thing that our first order theory can talk about to "define things in terms of". It's a device to talk about theories. – Malice Vidrine Dec 03 '13 at 00:10
  • Circular intuition is (usually) good -- it means your understanding of various aspects are all coherent and reinforce each other. –  Dec 03 '13 at 00:24

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You're right to think that there is a seeming tension between the model theory of first-order logic and the interpretation of set theory. For instance, we know that first-order logic is sound for set-sized models - if $\phi$ is provable from $\psi$ and $\psi$ is true in a set-sized model $M$, then $\phi$ is also true in $M$. But that doesn't tell us whether first-order logic is sound with respect to non-set-sized interpretations, say the intended interpretation of set theory which has as its domain the collection of absolutely all sets.

This raises two questions:

(i) What are these non-set-sized interpretations, if not sets?

(ii) How do we know that the account of logical consequence in terms of set-sized models doesn't lead us astray? That is, how do we know that any formula true in all set-sized models is true in all interpretations and vice versa?

There are a few viable ways of answering (i) and an argument due to Georg Kreisel that answers (ii). A good exposition of these matters, plus an introduction to other issues in the area, can be found here.