Saturation of reduced products

We study reduced products $M=\prod_n M_n/\mathrm{Fin}$ of countable structures in a countable language associated with the Fr\'echet ideal. We prove that such $M$ is $2^{\aleph_0}$-saturated if its theory is stable and not $\aleph_2$-saturated otherwise (regardless of whether the Continuum Hypothesis holds). This implies that $M$ is isomorphic to an ultrapower (associated with an ultrafilter on $\mathbb N$) if its theory is stable, even if the CH fails. We also improve a result of Farah and Shelah and prove that there is a forcing extension in which such reduced product $M$ is isomorphic to an ultrapower if and only if the theory of $M$ is stable. All of these conclusions apply for reduced products associated with $F_\sigma$ ideals or more general layered ideals. We also prove that a reduced product associated with the asymptotic density zero ideal $\mathcal Z_0$, or any other analytic P-ideal that is not $F_\sigma$, is not even $\aleph_1$-saturated if its theory is unstable.Comment: 32 page

Cardinal Arithmetic: From Silver’s Theorem to Shelah’s PCF Theory

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2019-2020, Tutor: Joan Bagaria PigrauThe main goal of this master’s thesis is to give a detailed description of the major ZFC advances in cardinal arithmetic from Silver’s Theorem to Shelah’s pcf theory and his bound on 2אω. In our attempt to make this thesis as self-contained as possible, we have devoted the first chapter to review the most elementary concepts of set theory, which include all the classical results from the first period of developement of cardinal arithmetic, from 1870 to 1930, due to Cantor, Hausdorff, König, and Tarski