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

TL;DR

This paper characterizes the modal logic of finite symmetry-preserving iterations as exactly S4, using ZF-provability, and proves both soundness and completeness through novel lemmas and model constructions.

Contribution

It establishes that the ZF-provable modal logic of finite symmetry-preserving iterations is precisely S4, providing new lemmas and model techniques for this class.

Findings

The logic of finite symmetry-preserving iterations is exactly S4.

A non-amalgamation lemma shows axiom (.2) fails for these iterations.

Model constructions demonstrate the completeness of S4 for this modality.

Abstract

We determine the ZF-provable modal logic of the modality $\Box_{\mathrm{sym}}$, where $\Box_{\mathrm{sym}}\varphi$ means '$\varphi$ 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.