An approximate Herbrand's theorem and definable functions in metric structures

TL;DR

This paper extends Herbrand's theorem to continuous logic and demonstrates that definable functions in infinite-dimensional Hilbert spaces can be approximated by affine functions, with applications to structures expanded by groups or subspaces.

Contribution

It introduces an approximate Herbrand's theorem for continuous logic and applies it to characterize definable functions in various Hilbert space structures.

Findings

Definable functions in Hilbert spaces are piecewise approximable by affine functions.

Herbrand's theorem can characterize definable functions in certain classical logic structures.

Applications include structures expanded by groups of unitary operators or subspaces.

Abstract

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in some absolutely ubiquitous structures from classical logic.