Undecidabiliity for the additive AIA fragment of the theory of normed spaces

Abstract

An AIA formula is one of the form 'A implies B' where A and B are purely universal. Up to a simple reduction AIA formula are both EA and AE. In an earlier paper Solovay, Harrison and I proved the undecidability of validity for the AIA fragment of a two-sorted first-order language for normed vector spaces. In this note we find that validity remains undecidable for AIA sentences in the additive sublanguage, i.e., when multiplication is disallowed.