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I am studying set theory. In most of the references on set theory such as the Jech's book, they start with the axioms of ZFC and the first-order language of set theory and eventually they start to talk about certain 'objects' called sets. I see a certain logic jump. I want to understand how is formally defined the sentence '$x$ is a set' from the first-order language. I understand that this sentence means that we can find some well-formed formula $\varphi(u)$ with a free variable $u$ where the sentence $(\exists x)\varphi(x)$ is true in ZFC. Am I right?

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    $\begingroup$ It is not possible with with "standard" $\mathsf{ZFC}$ because we have no predicate $\text{Set}(x)$. Every element of the domain is a set. $\endgroup$ – Mauro ALLEGRANZA Mar 2 '18 at 8:37
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    $\begingroup$ There are versions of set-tehory with atoms (or urelements) i.e. objects that are not set. In this case, we may have a predicate $\text {Ur}$ such that $\text {Ur}(x)$ holds iff $x$ is not a set. $\endgroup$ – Mauro ALLEGRANZA Mar 2 '18 at 8:39
  • $\begingroup$ @MauroALLEGRANZA As far as I see, in Jech's book, it starts with the axioms but no domain is introduced, then eventually they talk about certain objects sets. It is always necessary to have a domain? $\endgroup$ – Littlefield Mar 2 '18 at 9:15
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    $\begingroup$ Every "formal" theory pressuposes implicitly that it "speak about" something. Every set theory "speaks about" sets. The "domain" is the technical term of the formal semantics relative to a theory used to formalize the informal notion of "speaking about something". $\endgroup$ – Mauro ALLEGRANZA Mar 2 '18 at 9:20
  • $\begingroup$ It might be useful to think of the analogous question for more concrete theories; e.g. how do we say "is a natural number" in the context of Peano arithmetic (where everything considered is a natural number)? $\endgroup$ – Noah Schweber Mar 2 '18 at 21:19
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When you work in standard set theory, everything is a set.

In other words, formulas in the language of set theory do cannot even be applied to things that are not sets.

So if you want a formula $\varphi(x)$ which is true when $x$ is a set and false if $x$ is not a set, this means that you want a $\varphi(x)$ that is always true, period. You can use any propositional tautology you want for that, such as $x\in x\to x\in x$, but that won't make you any wiser about sets.

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