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Saturation of reduced products

Ben De Bondt, Ilijas Farah, Alessandro Vignati

Abstract

We study reduced products $M=\prod_n M_n/\mathrm{Fin}$ of countable structures in a countable language associated with the Fréchet 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 $F_σ$ 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 $F_σ$, is not even $\aleph_1$-saturated if its theory is unstable.

Saturation of reduced products

Abstract

We study reduced products of countable structures in a countable language associated with the Fréchet ideal. We prove that such is -saturated if its theory is stable and not -saturated otherwise (regardless of whether the Continuum Hypothesis holds). This implies that is isomorphic to an ultrapower (associated with an ultrafilter on ) 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 is isomorphic to an ultrapower if and only if the theory of is stable. All of these conclusions apply for reduced products associated with ideals or more general layered ideals. We also prove that a reduced product associated with the asymptotic density zero ideal , or any other analytic P-ideal that is not , is not even -saturated if its theory is unstable.
Paper Structure (19 sections, 22 theorems, 54 equations)

This paper contains 19 sections, 22 theorems, 54 equations.

Table of Contents

  1. Preliminaries
  2. Notational conventions
  3. Language
  4. Types, saturation, compactness
  5. Arity
  6. Ideals, reduced products
  7. Gaps
  8. Simple Cover Property of Reduced Products
  9. Simple cover property
  10. Stability and order property in models with simple cover property
  11. Failure of saturation
  12. The proof of Theorem \ref{['T.non-saturated++']}
  13. Ultrapowers
  14. Saturation and layered ideals
  15. Reduced products with stable theory
  16. ...and 4 more sections

Key Result

Theorem 1

If $M_n$, for $n\in \mathbb{N}$, are structures in a countable language and the theory of $M=\prod_n M_n/\mathop{\mathrm{Fin}}\nolimits$ is stable, then $M$ is $\mathfrak c$-saturated. In particular, if $|M_n|\leq \mathfrak c$ for all $n$, then $M$ is saturated.

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 2.1: $h$-formulas
  • Theorem 2.2: palyutin2
  • proof
  • Definition 2.3
  • ...and 41 more