We say that a complete theory T has the Schröder-Bernstein property, or simply, the SB-property, if for any two models M and N of T that are elementary bi-embeddable then they are isomorphic. The purpose of this talk is to study the SB-property in the continuous context for Randomizations. Informally, a randomization of a first order (discrete) model, M, is a two sorted metric structure which consist of a sort of events and a sort of functions with values on the model, usually understood as random variables. More generally, given a complete first order theory T, there is a complete continuous theory known as randomized theory, T^R, which is the common theory of all randomizations of models of T. Randomizations were introduced first by Keisler in [5] and then axiomatized in the continuous setting by Ben Yaacov and Keisler in [4]. Since randomizations where introduced, many authors focused on examining which model theoretic properties of T are preserved on T^R, for example, in [4] and [3] it was shown that properties like ω-categoricity, stability and dependence are preserved. Similarly in [1] it is proved that the existence of prime models is preserved by randomization but notions like minimal models are not preserved. Following these ideas, we will prove that a first order theory T with ≤ ω countable models has the SB-property for countable models if and only if T^R has the SB-property for separable randomizations. This is a joint work with Alexander Berenstein, presented in [2].

References:

[1] U. Andrews and H. J. Keisler. Separable models of randomizations. The Journal of Symbolic Logic, pages 1149–1181, 2015.

[2] Argoty, C., Berenstein, A. & Cuervo Ovalle, N. The SB-property on metric structures. Archive for Mathematical Logic (2025). https://doi.org/10.1007/s00153-024-00949-y

[3] I. Ben Yaacov. Continuous and random vapnik-chervonenkis classes. Israel Journal of Mathematics, 1(173):309–333, 2009.

[4] I. Ben Yaacov and H. Jerome Keisler. Randomizations of models as metric structures. Confluentes Mathematici, 1(02):197–223, 2009. [5] H. J. Keisler. Randomizing a model. Advances in Mathematics, 143(1):124–158, 1999

The partition relation A→(P)^μ_λ, despite its brevity, is remarkably expressive. This fundamental combinatorial principle asserts that every λ-coloring of μ-sized subsets of A is constant on a subset in the class P. By adjusting the parameters A, μ, λ, and P, one can express a wide variety of large cardinal principles, including weak compactness, Ramsey-ness, and even supercompactness. In this talk, we focus on the case where A is an uncountable cardinal κ and P is the class of stationary subsets of κ. By work of Baumgartner 1977, it turns out that this corresponds to certain ineffability properties of κ. We also describe how this fits into a different hierarchy of ineffability properties described in current joint work with Matthew Foreman and Menachem Magidor.

The partition relation A→(P)^μ_λ, despite its brevity, is remarkably expressive. This fundamental combinatorial principle asserts that every λ-coloring of μ-sized subsets of A is constant on a subset in the class P. By adjusting the parameters A, μ, λ, and P, one can express a wide variety of large cardinal principles, including weak compactness, Ramsey-ness, and even supercompactness. In this talk, we focus on the case where A is an uncountable cardinal κ and P is the class of stationary subsets of κ. By work of Baumgartner 1977, it turns out that this corresponds to certain ineffability properties of κ. We also describe how this fits into a different hierarchy of ineffability properties described in current joint work with Matthew Foreman and Menachem Magidor.

This is the second in a series of lectures going through notes entitled "Naive Descriptive Set Theory" that are available on ArXiV.  

In the last 20 years the field has had many applications to areas in Analysis and Dynamical Systems The lectures are intended to be an opportunity to learn the subject matter, and will be interspersed with research lectures during the quarter.

No background beyond basic elements of the 210 sequence are required.