Modal group theory: homomorphisms

Abstract

I investigate modal group theory for arbitrary homomorphisms. Possibility is interpreted by the existence of a group homomorphism out of the given group, so the semantics is governed by the possibility of collapse: elements may be identified, parameters may be killed, and new relations may hold in the target. I show that the modal language nevertheless expresses cyclic subgroup membership, subgroup generation by a fixed finite tuple, cyclicity, finite generation by a fixed number of elements, and torsion. I use these definability results to interpret arithmetic, and prove that, as sets of Goedel numbers, the homomorphic modal theory of finitely presented groups is computably isomorphic to true arithmetic. I also analyze propositional modal validities: sentential validities are exactly S5, the trivial group has exact parameter-validities S5, and uniformly prime-indivisible groups have exact parameter-validities S4.2.