Omitting types for infinitary [0, 1]-valued logic

TL;DR

This paper introduces an infinitary logic for metric structures, capable of expressing complex analytical concepts, and proves an omitting types theorem that leads to new results in Banach space theory.

Contribution

It develops a new infinitary logic for metric structures and establishes an omitting types theorem with applications to Banach space separability results.

Findings

Established an omitting types theorem for countable fragments.

Proved a two-cardinal theorem strengthening previous Banach space results.

Demonstrated the expressive power of the new infinitary logic in analysis.

Abstract

We describe an infinitary logic for metric structures which is analogous to $L_{\omega_1, \omega}$. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.