The decision problem for normed spaces over any class of ordered fields

TL;DR

This paper explores the logical decidability of theories of normed spaces over various ordered fields, showing that such theories are generally undecidable when they include all spaces of a fixed dimension.

Contribution

It extends known undecidability results from real normed spaces to those over arbitrary ordered fields, demonstrating that the theory remains undecidable in this broader context.

Findings

Theories of normed spaces over any ordered field include all spaces of a fixed dimension.

Such theories admit a relative interpretation of Robinson's theory Q.

These theories are undecidable in general.

Abstract

It is known that the theory of any class of normed spaces over the reals that includes all spaces of a given dimension d > 1 is undecidable, and indeed, admits a relative interpretation of second-order arithmetic. The notion of a normed space makes sense over any ordered field of scalars, but such a strong undecidability result cannot hold in the more general case. Nonetheless, we find that the theory of any class of normed spaces in the more general sense that includes all spaces of a given dimension d > 1 over some ordered field admits a relative interpretation of Robinson's theory Q and hence is undecidable.