# CONFLUENTES MATHEMATICI

Approximate isomorphism of randomization pairs
Confluentes Mathematici, Volume 14 (2022) no. 2, pp. 29-44.

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 preserved under various natural constructions, and formulate our guesswork for the general case.

Revised:
Accepted:
Published online:
DOI: 10.5802/cml.85
Classification: 03C66, 03C45, 03C35, 22F50
Keywords: continuous logic, randomization, approximate categoricity, $\aleph _0$-categoricity, $\aleph _0$-stability, beautiful pairs
James Hanson 1; Tomás Ibarlucía 2

1 Department of Mathematics, University of Maryland, 4176 Campus Dr., College Park, MD 20742-4015, USA.
2 Université Paris Cité, CNRS, IMJ-PRG, F-75006 Paris, France.
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James Hanson; Tomás Ibarlucía. Approximate isomorphism of randomization pairs. Confluentes Mathematici, Volume 14 (2022) no. 2, pp. 29-44. doi : 10.5802/cml.85. https://cml.centre-mersenne.org/articles/10.5802/cml.85/

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