# Atomic saturation of reduced powers

@article{Shelah2021AtomicSO, title={Atomic saturation of reduced powers}, author={Saharon Shelah}, journal={Mathematical Logic Quarterly}, year={2021}, volume={67} }

Our aim was to try to generalize some theorems about the saturation of ultrapowers to reduced powers. Naturally, we deal with saturation for types consisting of atomic formulas. We succeed to generalize “the theory of dense linear order (or T with the strict order property) is maximal and so is any pair(T,Δ) which is SOP3”, (where Δ consists of atomic or conjunction of atomic formulas). However, the theorem on “it is enough to deal with symmetric pre‐cuts” (so the p=t theorem) cannot be…

## 4 Citations

Model-theoretic applications of cofinality spectrum problems

- Mathematics
- 2015

We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is λ-saturated iff it has…

Universal graphs between a strong limit singular and its power

- Mathematics
- 2022

The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of GCH κ is supercompact and the…

Keisler's order via Boolean ultrapowers

- MathematicsArch. Math. Log.
- 2021

It is shown that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler's order, answering a question posed by Benda in 1974.

LIST OF PUBLICATIONS

- Environmental Science
- 2022

1. Sh:a Saharon Shelah. Classification theory and the number of nonisomorphic models, volume 92 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New…

## References

SHOWING 1-10 OF 27 REFERENCES

Ultraproducts which are not Saturated

- MathematicsJ. Symb. Log.
- 1967

This paper proves that, assuming the generalized continuum hypothesis (GCH), for each cardinal α there is an ultrafilter D over a set of power α such that for all structures, D-prod is α + -saturated.

More on regular reduced products

- MathematicsJournal of Symbolic Logic
- 2004

The consistency of the failure, relative to the consistency of supercompact cardinals, of the following is shown: for all regular filters D on a cardinal λ: if Mi and Ni are elementarily equivalent models of a language of size ≤ λ, then the second player has a winning strategy in the Ehrenfeucht-Fraïssé game.

Model-theoretic applications of cofinality spectrum problems

- Mathematics
- 2015

We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is λ-saturated iff it has…

Model Theory for a Compact Cardinal

- Mathematics
- 2013

We would like to develop model theory for T, a complete theory in L_{theta,theta}(tau) when theta is a compact cardinal. We already have bare bones stability theory and it seemed we can go no…

Cofinality spectrum theorems in model theory, set theory and general topology

- Mathematics
- 2012

We connect and solve two longstanding open problems in quite different areas: the modeltheoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set…

Saturated models of Peano arithmetic

- MathematicsJournal of Symbolic Logic
- 1982

Abstract We study reducts of Peano arithmetic for which conditions of saturation imply the corresponding conditions for the whole model. It is shown that very weak reducts (like pure order) have such…

Limit ultraproducts

- MathematicsJournal of Symbolic Logic
- 1965

This paper is a sequel to our earlier paper, “Limit Ultrapowers”, [6]. In that paper we introduced the limit ultrapower construction and proved that is isomorphic to a limit ultrapower of if and only…

On regular reduced products*

- MathematicsJournal of Symbolic Logic
- 2002

For M as above and D a regular ultrafilter over λ, Mλ/D is λ++-universal, and it is shown that the second player has a winning strategy in the Ehrenfeucht-Fraïssé game of length λ+ on ΠiMi/D andΠiNi/D.