Expressive power of infinitary [0, 1]-valued logics

TL;DR

This paper investigates the expressive capabilities of three infinitary logics for metric structures, demonstrating their relative strengths, invariance properties, and the existence of Scott sentences in this context.

Contribution

It introduces a new example distinguishing the expressive power of these logics and establishes automorphism invariance characterizations for definable functions.

Findings

One logic is strictly more expressive than the others.

All three logics share the same elementary equivalence relation.

Automorphism invariant functions are exactly those definable in the more expressive logic.

Abstract

We consider model-theoretic properties related to the expressive power of three analogues of $L_{\omega_1, \omega}$ for metric structures. We give an example showing that one of these infinitary logics is strictly more expressive than the other two, but also show that all three have the same elementary equivalence relation for complete separable metric structures. We then prove that a continuous function on a complete separable metric structure is automorphism invariant if and only if it is definable in the more expressive logic. Several of our results are related to the existence of Scott sentences for complete separable metric structures.