John Baldwin : Henkin models in the continuum

Posted by Filippo Calderoni , part of the Logic Seminar.

At

March 10, 2020, 3 p.m.

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427 SEO

Abstract

We describe Shelah’s construction of atomic models in the continuum, as reformulated with Laskowski as a Henkin construction [2]. Then we discuss briefly the connection with the Ackerman-Freer-Patel [1] proof that if $M$ is a countable structure for a relational language $L$ with trivial definable closure then there is an invariant probability measures on the countable $L$-structures that concentrates on $M$. We explain while the sufficient conditions for the model in the continuum include those with trivial definable closure, our theorem applies more generally to obtaining a atomic model in the continuum of the first order theory of a countable atomic extendible structure admitting a formula-based geometry. [1] Ackerman, N. and Freer, C. and Patel, R., Invariant measures concentrated on countable structures, Forum of Mathematics, Sigmas, vol. 4 (2016), no. X, pp. 59. [2] Baldwin, J. T. and Laskowski, M.C., Henkin Constructions of Models in the Continuum, Bulletin of Symbolic Logic, vol. 24(2019), no. 1, pp. 1–34.