Some new results on decidability for elementary algebra and geometry

TL;DR

This paper systematically investigates the decidability of various theories in elementary algebra and geometry, revealing decidability in some cases and undecidability in others, with implications for understanding the logical complexity of these mathematical structures.

Contribution

It provides a comprehensive classification of the decidability status of theories of vector spaces, inner product spaces, Hilbert spaces, normed spaces, Banach spaces, and metric spaces, including sharp results and complexity bounds.

Findings

Theories of real vector spaces, inner product spaces, and Hilbert spaces are decidable.

Theories of normed spaces, Banach spaces, and metric spaces are not arithmetical, with some being undecidable.

Certain fragments like the universal and existential fragments of normed spaces are decidable.

Abstract

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.