III: Undecidability of the Elementary Theory of Groups

Abstract

Publisher Summary This chapter presents an axiomatic theory G with standard formalization, called the elementary theory of groups. The purpose of the chapter is to show that the elementary theory of groups is undecidable. It provides several theorems and proofs of each theorem. Theory J +(/) is weakly interpretable in some inessential extension of Theory G. Let G′ be the axiomatic theory obtained from G by including the individual constant c in the system of non-logical constants, but without changing the system of non-logical axioms. G′ is an inessential extension of G. In order to show that J +(/) is weakly interpretable in G′, it suffices to construct a theory T with the properties such as: T is consistent; T is a common extension of G′ and J +(/) ; and for each of the non-logical constants of J +(/) there is a valid sentence of T that is a possible definition of this constant in G′.