nLab
Keisler-Shelah isomorphism theorem

Contents

Idea

The Keisler-Shelah isomorphism theorem characterizes the partially syntactic concept of elementary equivalence in purely semantic form with the help of ultrapowers.

Statement of theorem

In practice, the theorem is usually stated as follows: let AA and BB be first-order \mathcal{L}-structures. Then AA and BB are elementarily equivalent, written ABA \equiv B, if and only if there is an ultrafilter 𝒰\mathcal{U} on some index set II such that there is an isomorphism of ultrapowers A 𝒰B 𝒰A^{\mathcal{U}} \simeq B^{\mathcal{U}}.

Keisler proved, assuming GCH, that when ATA \models T and BTB \models T have cardinality 2 |T|\leq 2^{|T|} then they have isomorphic |T||T|-indexed ultrapowers. Shelah removed the assumption of GCH at the cost of exhibiting the isomorphism for only 2 |T|2^{|T|}-indexed ultrapowers instead.

Examples

p p/𝒰 p𝔽 p((t))/𝒰 \displaystyle \prod_{p} \mathbb{Q}_p/\mathcal{U} \equiv \displaystyle \prod_{p} \mathbb{F}_p((t)) / \mathcal{U}

are elementarily equivalent. Assuming the continuum hypothesis (this is an example of where this technical distinction is vital), they are also isomorphic.

Remarks

ultraroot

ultrapower

ultraproduct

Los ultraproduct theorem?

References