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Mendelson, in Introduction to Mathematical Logic, 4th ed, 1997, had a more elegant approach to comprehension than predecessors, in my opinion.

With $x\in\mathbf{V}$ short for $\exists y(x\in y)$, and $\alpha$ any formula in the language of set theory (possibly without =), use the axiom schemas:

SE: $\exists y(y=\{x|\alpha\})$ and CA: $\forall x(x\in\{x|\alpha\}\leftrightarrow x\in \mathbf{V}\wedge \alpha)$

A class is a set just if it is a member of $\mathbf{V}$. Mendelson goes on and develops NBG set theory on the basis of these, and further assumptions.

Question: Have others explored other set theories, with an approach as this?

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    $\begingroup$ @JoelDavidHamkins The "$\in{\bf V}$" part of CA gets around this unless I'm missing something: letting $\rho=\{x\vert x\not\in x\}$, we get from $\mathsf{CA}$ that $\rho\in \rho\leftrightarrow \rho\in {\bf V}\wedge\rho\not\in\rho$, which of course isn't a contradiction. $\endgroup$
    – Noah Schweber
    Commented Jul 14, 2021 at 21:42
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    $\begingroup$ Note that any theory $T$ whatsoever has an accompanying extension $\hat{T}$, which is to $T$ as $\mathsf{NBG}$ is to $\mathsf{ZFC}$. Roughly, a model of $\hat{T}$ is a pair $(M,S)$ where $M\models T$ and $S$ is a collection of subsets of $M$ which satisfy the obvious comprehension scheme. As with $\mathsf{NBG}$ this can be phrased in a couple different ways (e.g. two-sorted or not), and for each $M\models T$ the structure $(M,\mathcal{P}_{def}(M))$ is the minimal model of $\hat{T}$ with first-order part $M$. Really, this is just second-order logic with Henkin semantics built on top of $T$. $\endgroup$
    – Noah Schweber
    Commented Jul 14, 2021 at 21:45
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    $\begingroup$ I won't be able to sleep because of the mistmatched parenthesis. $\endgroup$
    – Andrej Bauer
    Commented Jul 14, 2021 at 21:48
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    $\begingroup$ @AsafKaragila "Language of axiomatic set theory" I think - I've seen this abbreviation in some old textbooks. $\endgroup$
    – Noah Schweber
    Commented Jul 14, 2021 at 22:20
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    $\begingroup$ @NoahSchweber Your second comment seems to answer the question (more than completely). $\endgroup$
    – Andreas Blass
    Commented Jul 14, 2021 at 23:31

1 Answer 1

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Yes! There are other set theories explored along this way generally speaking.

(1) Quine's Mathematical Logic $\sf ML$ adopted a similar approach on top of his $\sf NF$, and easily one can get a similar treatment on top of $\sf NFU$. See: Quine, Willard Van Orman (1951), Mathematical logic (Revised ed.), Cambridge, Mass.: Harvard University Press, ISBN 0-674-55451-5, MR 0045661

(2) Randall Holmes in his theory about symmetric extensions also used a similar approach. https://randall-holmes.github.io/Drafts/symmetryrevisited.pdf

I also used this approach in a closely related article on: https://arxiv.org/abs/2012.08299

(3) Vopenka's alternative set theory

(4) Holmes Pocket set theory

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  • $\begingroup$ Thank you for the reminder of (2-4)! I believe other interesting approaches may be explored, but, obviously, and also as evidenced from the comments, the class jargon may be modified. It may nonetheless be useful when thinking about many matters. $\endgroup$
    – Frode Alfson Bjørdal
    Commented Jul 17, 2021 at 17:49

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