The Modal Logic of Finitely Symmetry-Preserving Iterated Extensions is Exactly S4

Abstract

We determine the ZF-provable modal logic of the modality $\Box_{\mathrm{sym}}$, where $\Box_{\mathrm{sym}}φ$ means '$φ$ holds in every finite symmetry-preserving iteration' of the symmetric method. We prove that the exact logic is S4. Soundness (axioms T and 4) follows from reflexivity and transitivity of the underlying accessibility relation. Exactness is obtained by (i) a non-amalgamation lemma showing that axiom (.2) fails for finite symmetry-preserving iterations (no common finite symmetry-preserving iteration above the parent), and (ii) a $p$-morphism/finite-frame realization producing, within ZF, models whose $\Box_{\mathrm{sym}}$-theory matches any finite reflexive-transitive frame.