Keisler-Shelah isomorphism theorem

Interpretations for a first-order language $ L $ are said to be elementarily equivalent (in $ L $) provided that they make exactly the same sentences in $ L $ true (cf. also Interpretation). The Keisler–Shelah isomorphism theorem provides a characterization of elementary equivalence: Interpretations for $ L $ are elementarily equivalent if and only if they have isomorphic ultrapowers (cf. also Ultrafilter).

This theorem was formulated and proved by H.J. Keisler in 1961 [a2]. Keisler gave a second proof in 1964 using saturated ultrapowers [a3]. Both proofs used the generalized continuum hypothesis ($ \mathsf{GCH} $). In 1971, S. Shelah gave a third proof [a5] that avoided $ \mathsf{GCH} $.

Given a non-empty family $ (\mathfrak{A}_{i})_{i \in I} $ of interpretations for $ L $, and given an ultrafilter $ \mathcal{F} $ on $ I $, the ultraproduct $ \prod_{i \in I} \mathfrak{A}_{i} \big/ \mathcal{F} $ of the family is defined as the quotient system on the direct product of the family induced by $ \mathcal{F} $. When there is a fixed interpretation $ \mathfrak{A} $ for $ L $, so that each $ \mathfrak{A}_{i} $ coincides with $ \mathfrak{A} $, the ultraproduct $ \prod_{i \in I} \mathfrak{A}_{i} \big/ \mathcal{F} $ is denoted by $ \mathfrak{A}^{I} \big/ \mathcal{F} $ and is called an ultrapower of $ \mathfrak{A} $. It follows from results of J. Łos [a4] that $ \mathfrak{A} $ and any of its ultrapowers are elementarily equivalent (this is the Łos isomorphism theorem). Hence, interpretations with isomorphic ultrapowers are elementarily equivalent.

Let $ \lambda $ be an infinite cardinal no smaller than the cardinality of the set of sentences in $ L $, and let $ \mathfrak{A} $ and $ \mathfrak{B} $ be interpretations for $ L $ of cardinality less than or equal to $ 2^{\lambda} $. Let $ \lambda^{+} $ denote the successor cardinal of $ \lambda $. Keisler showed (assuming that $ 2^{\lambda} = \lambda^{+} $) that $ \mathfrak{A} $ and $ \mathfrak{B} $ are elementarily equivalent if and only if there are ultrafilters $ \mathcal{F} $ and $ \mathcal{G} $ on $ \lambda $ such that $ \mathfrak{A}^{\lambda} \big/ \mathcal{F} $ and $ \mathfrak{B}^{\lambda} \big/ \mathcal{G} $ are isomorphic.

Let $ \lambda $ be as above, and let $ \beta $ be the least cardinal such that $ \lambda^{\beta} > \lambda $. Shelah showed (without assuming that $ 2^{\lambda} = \lambda^{+} $) that there is an ultrafilter $ \mathcal{F} $ on $ \lambda $ such that given elementarily equivalent interpretations $ \mathfrak{A} $ and $ \mathfrak{B} $ for $ L $ of cardinality less than $ \beta $, the ultrapowers $ \mathfrak{A}^{\lambda} \big/ \mathcal{F} $ and $ \mathfrak{B}^{\lambda} \big/ \mathcal{F} $ are isomorphic.

The motivation for Keisler’s results can be found in a program propounded by A. Tarski [a6]: To provide characterizations of meta-mathematical notions in “purely mathematical terms”. A discussion of this program and its history can be found in [a7]. To appreciate what was intended here, recall G. Birkhoff’s 1935 characterization [a1] of the classes of models of sets of equations (the equational classes): A class of algebras is an equational class if and only if it is closed under sub-algebras, homomorphic images and direct products. This result characterizes equational classes without mentioning equations.

Among the consequences of the Keisler–Shelah isomorphism theorem is a comparable “mathematical” characterization of the classes of models of sentences in $ L $. Given a class $ \mathcal{T} $ of interpretations for $ L $, we say that $ \mathcal{T} $ is an elementary class provided that there is a sentence in $ L $ whose models are exactly the members of $ \mathcal{T} $; we say that $ \mathcal{T} $ is an elementary class in the wider sense provided that there is a set of sentences in $ L $ whose models are exactly the members of $ \mathcal{T} $. It follows from the compactness theorem that $ \mathcal{T} $ is an elementary class if and only if both $ \mathcal{T} $ and its complement (relative to the class of interpretations for $ L $) are elementary classes in the wider sense. Keisler [a2] showed (assuming $ \mathsf{GCH} $) that:

1) $ \mathcal{T} $ is an elementary class in the wider sense, provided that $ \mathcal{T} $ is closed under isomorphic images and ultraproducts and the complement of $ \mathcal{T} $ is closed under ultrapowers;

2) $ \mathcal{T} $ is an elementary class if and only if both $ \mathcal{T} $ and its complement are closed under isomorphic images and ultraproducts.

Whilst Keisler’s proof of this result used $ \mathsf{GCH} $, its application was restricted to establishing that elementarily equivalent interpretations have isomorphic ultrapowers. Hence, by eliminating $ \mathsf{GCH} $ from the proof of the latter result, Shelah also eliminated the use of $ \mathsf{GCH} $ from Keisler’s characterization of elementary classes.

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