Approximate isomorphism of randomization pairs

TL;DR

This paper investigates the approximate categoricity of theories of beautiful pairs in continuous logic, disproves a conjecture by providing a counterexample, and identifies cases where approximate categoricity holds.

Contribution

It demonstrates that certain theories of beautiful pairs are not approximately -categorical, countering previous conjectures, and establishes conditions under which approximate categoricity is preserved.

Findings

The theory of beautiful pairs of randomized infinite sets is approximately -categorical.

Counterexample provided with the theory of randomized infinite vector spaces over a finite field.

Approximate -categoricity is stable under natural constructions in specific cases.

Abstract

We study approximate $\aleph_0$-categoricity of theories of beautiful pairs of randomizations, in the sense of continuous logic. This leads us to disprove a conjecture of Ben Yaacov, Berenstein and Henson, by exhibiting $\aleph_0$-categorical, $\aleph_0$-stable metric theories $Q$ for which the corresponding theory $Q_P$ of beautiful pairs is not approximately $\aleph_0$-categorical, i.e., has separable models that are not isomorphic even up to small perturbations of the smaller model of the pair. The theory $Q$ of randomized infinite vector spaces over a finite field is such an example. On the positive side, we show that the theory of beautiful pairs of randomized infinite sets is approximately $\aleph_0$-categorical. We also prove that a related stronger property, which holds in that case, is stable under various natural constructions, and formulate our guesswork for the general case.