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.

References

 [a1] G. Birkhoff, “On the structure of abstract algebras”, Proc. Cambridge Philos. Soc., 31 (1935), pp. 433–454. [a2] H.J. Keisler, “Ultraproducts and elementary models”, Indagationes Mathematicae, 23 (1961), pp. 477–495. [a3] H.J. Keisler, “Ultraproducts and saturated classes”, Indagationes Mathematicae, 26 (1964), pp. 178–186. [a4] J. Łos, “Quelques remarqes, théorèmes et problèmes sur les classes définissables d'algèbres”, Mathematical Interpretations of Formal Systems, North-Holland (1955), pp. 98–113. [a5] S. Shelah, “Every two elementarily equivalent models have isomorphic ultrapowers”, Israel J. Math., 10 (1971), pp. 224–233. [a6] A. Tarski, “Some notions and methods on the borderline of algebra and metamathematics”, Proc. Intern. Congress of Math. (Cambridge, MA, 1950), 1, Amer. Math. Soc. (1952), pp. 705–720. [a7] R.L. Vaught, “Model theory before 1945”, Proc. Tarski Symp., Amer. Math. Soc. (1974), pp. 153–172.
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Keisler–Shelah isomorphism theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Keisler%E2%80%93Shelah_isomorphism_theorem&oldid=39843