The universal finite set
Joel David Hamkins, W. Hugh Woodin · arXiv (Cornell University) · 2017
We define a certain finite set in set theory $\{x\midφ(x)\}$ 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 $φ$ has complexity $Σ_2$, so that any affirmative instance of it $φ(x)$ is verified in any sufficiently large rank-initial segment of the universe $V_θ$; the set is empty in any transitive model and others; and if $φ$ defines the set $y$ in some countable model $M$ of ZFC and $y\of z$ for some finite set $z$ in $M$, then there is a top-extension of $M$ to a model $N$ in which $φ$ defines the new set $z$. Thus, the set shows that no model of set theory can realize a maximal $Σ_2$ 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.