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This question concerns Godel's Theorem on existence of classes in Set Theory of von Neumann–Bernays–Gödel.

This theorem implies that for any formula $\varphi(x)$ with one free variable $x$ whose quantifiers are of the form $\forall x\in\mathbf U$ or $\exists x\in\mathbf U$ the class $\{x:\varphi(x)\}$ exists.

Question. Is there a formula $\varphi(x)$ with one free variable in the language of set theory for which the class $\{x:\varphi(x)\}$ does not exist?

Added in Edit. Reading the comments I realized that such a formula indeed exists. So, I reformulate my question to

Question'. Find a (relatively) short formula $\varphi(x)$ for which the existence of the class $\{x:\varphi(x)\}$ cannot be proved in NBG.

I need such a formula to stress that the restricting quantifiers to run over sets in Godel's theorem on existence of classes is essential. Desirably to have such a formula as simple as possible.

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    $\begingroup$ Which theorem of Godel do you have in mind? What is $\mathbf{U}$? With "the class does not exist" do you mean it provably (in NBG) doesn't exist, or one for which it is consistent that it doesn't exist? When saying "the language of set theory" do you admit quantifiers over all classes or just ones over sets? $\endgroup$
    – Wojowu
    Commented May 6, 2020 at 21:36
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    $\begingroup$ If you admit quantifiers over all classes, then the answer is as follows: for provable nonexistence, the answer is probably no, as that would imply Morse-Kelley set theory is inconsistent. If you mean consistent nonexistence, the answer is positive, as it's consistent that the set of truths about $V$ (which is definable in the language of classes) doesn't exist, as its existence implies consistency of ZF, and hence NBG. $\endgroup$
    – Wojowu
    Commented May 6, 2020 at 21:41
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    $\begingroup$ @Wojowu: "as its existence implies consistency of ZF, and hence NBG", is it understood to mean "as its existence would imply consistency of ZF, and hence of NBG, is provable in ZF"? $\endgroup$
    – Qfwfq
    Commented May 6, 2020 at 22:35
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    $\begingroup$ @Qfwfq Existence of the truth class literally implies consistency of ZF, so if the existence was provable, so would be consistency. $\endgroup$
    – Wojowu
    Commented May 6, 2020 at 22:37
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    $\begingroup$ @LSpice "The existence of the truth class" is (formalizable as) a sentence in the language of NBG, and that sentence implies the (formalization of) the sentence "ZF is consistent". $\endgroup$
    – Andreas Blass
    Commented May 6, 2020 at 23:28

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