2017/11/21 by Joel David Hamkins, Hamkins, Joel David, W. Hugh Woodin +1 · 2 citations
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge
paper · pdf · doi:10.48550/arxiv.1711.07952
openalex publication_date 2017/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define a certain finite set in set theory \x|φ(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 Σ2, so that any affirmative instance of it φ(x) is verified in any sufficiently large rank-initial segment of the universe Vθ; 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 Σ2 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.