The Keisler-Shelah Isomorphism Theorems

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Abstract

In this chapter we use Hekin-Keisler models to prove various formulations of the Keisler-Shelah Isomorphism Theorem: two interpretations are elementarily equivalent iff they have isomorphic ultrapowers. This results is proved in Keisler [1961] assuming the Generalized Continuum Hypothesis (GCH). Keisler [1964] contains a second proof using saturated ultrapowers. This proof also assumes GCH. Shelah [1972] contains a third proof. This proof avaoids assuming GCH. In the following we give two proofs of Keisler’s original version of the theorem. The first assumes GCH, the second does not. We then indicate how the second proof can be extended to yield a stronger version of the theorem. In Chapter 9 we obatain yet another proof. This proof, like that of Keisler [1964], uses saturated ultrapowers and assumes GCH.

Since U and any of its ultrapowers are elementarily equivalent and isomorphic interpretations are elementarily equivalent, it is immediate that U and B are elementarily equivalent if they have isomorphic ultrapowers.