The universal finite set

We define a certain finite set in set theory $\{x\mid \phi (x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition $\phi$ has complexity ${\mathrm{\Sigma }}_2$, so that any affirmative instance of it $\phi (x)$ is verified in any sufficiently large rank-initial segment of the universe $V_{\theta }$; the set is empty in any transitive model and others; and if $\phi$ defines the set $y$ in some countable model $M$ of ZFC and $y\text{of}z$ for some finite set $z$ in $M$, then there is a top-extension of $M$ to a model $N$ in which $\phi$ defines the new set $z$. Thus, the set shows that no model of set theory can realize a maximal ${\mathrm{\Sigma }}_2$ theory with its natural number parameters, although this is possible without parameters. Using the universal finite set, we prove that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle.