This talk analyses when different generics for a given poset yield the same extension gives rise to countable Borel equivalence relations. We characterize when these relations are smooth. We also explore Prikry and Cohen forcing. This is joint work with Filippo Calderoni. 

Quasiminimal classes form an abstract analogue of strongly minimal theories.  Following what can be done in the strongly minimal case, we consider two expansions of quasiminimal classes with a unary predicate:  beautiful pairs and H-structures.  We show each of these expansions can be axiomatized with a single Lω1ω (Q)-sentence and that both expansions are ω-stable. We will explain why these expansions are natural in the strongly minimal context and how to extrapolate some results to the new setting.  Conversely, we show how to produce new examples of quasiminimal classes using beautiful pairs.

This is joint work with E. Vassiliev.

Historically, proofs of the Boolean Prime Ideal Theorem (BPI) in choiceless models of ZF have taken considerable effort. We present a new conceptual framework with which to prove BPI in choiceless models, based on Harrington's proof of the Halpern-Läuchli theorem. This allows for new proofs of several results from the literature, including the fact that ZF+BPI cannot prove the axiom of choice. We describe these results and the connection to the Halpern-Läuchli theorem that is implicit in this approach.

We state the dense-set Halpern-Lauchli theorem and sketch Harrington’s proof of the theorem. Harrington’s proof is then adapted to give a new proof of the Boolean Prime Ideal theorem in the first Cohen model. This method is flexible and is shown to admit two natural generalizations. First, it is applied to give a simpler proof of Pincus’ result that ZF+BPI+DC_\kappa does not imply choice. Second, it is applied to make progress towards the goal of proving BPI in iterations of symmetric extensions. Lastly, we show that this proof generalizes Stefanovic's theorem that the Halpern-Lauchli theorem can be derived from BPI in the Cohen model.

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