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Consider the minimal normal modal logic $K$ (axioms = classical propositional logic + $(\Box(p\land q)\leftrightarrow\Box p\land\Box q)$ + $(\Box\top)$, nothing else).

Its canonical model with no variables is a descriptive frame $(W,R)$. Among several possible descriptions of this frame is the following. Let $\mathbf V$ denote the Vietoris endofunctor on the category of Stone spaces. Then $W$ is a final $\mathbf V$-coalgebra; in particular $R:W\to\mathbf V(W)$ is a homeomorphism.

Reading this $R$ as $"\ni"$ one obtains certain model of set theory. The above fact about the final coalgebra means that for any closed subset $C\in\mathbf V(W)$ there is a unique $c\in W$ with $C=\{x\mid x"\in"c\}$.

Note that powersets exist for those $c\in W$ corresponding to clopen $C$. Indeed for the latter, $\Box C:=\{x\in W\mid R(x)\subseteq C\}$ is again clopen. (An aside - is $\Box C$ closed for $C$ closed?) Then there is (as a particular case of the above) a unique $pc\in W$ with $\Box C=\{x\mid x"\in"pc\}$, so that $x"\in"pc$ holds iff $R(x)\subseteq C$, i. e. iff $\forall y\ (y"\in"x)\Rightarrow(y"\in"c)$.

This theory seems to be rather weird, e. g. there is the universal set $w$ with $x"\in"w$ for all $x$ (so necessarily $w"\in"w$; foundation is violently violated), many such things.

Is axiomatic description of this set theory and its properties worked out anywhere?

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There is a 'research report(?)' from the Institute for Logic, Language, and Computation (a rersearch institute of the University of Amsterdam) by Goivanna D'Agostino, Angelo Montanari, and Alberto Policriti titled "Modal Logic and Set Theory: a Set-Theoretic Interpretation of Modal Logic" (look under title on the Web). This report discusses the very topic (the set-theoretic interpretation of modal system $K$) of your question. I hope it is helpful to you.

You might also want to take a look at "Modal Deduction in Second-Order Logic and Set Theory--$I$" by the aforementioned authors and Johan van Benthem. It also discusses the set-theoretic interpretation of system $K$.

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