Modal group theory

TL;DR

This paper introduces modal group theory, exploring groups through modal logic, demonstrating its greater expressiveness than first-order logic, and characterizing its validities within known modal systems.

Contribution

It develops modal group theory, shows its expressiveness surpasses first-order logic, and characterizes its propositional validities as exactly S4.2, answering an open question.

Findings

Modal language of groups is more expressive than first-order language.

The theory of true arithmetic is interpretable within modal group theory.

Propositional modal validities of groups under embeddings are exactly S4.2.

Abstract

I introduce modal group theory, in which we study the category of all groups, considering embeddability as providing a notion of modal possibility. Using HNN extensions and Britton's lemma, I demonstrate that the modal language of groups is more expressive than the first-order language of groups. I interpret the theory of true arithmetic in modal group theory, and show that, as sets of Goedel numbers, it is computably isomorphic to the modal theory of finitely presented groups. I answer an open question of Berger, Block, and Loewe by showing that the formulaic propositional modal validities of groups under embeddings are precisely S4.2. I also analyze sentential validities and worlds validating S5.