Keisler-Shelah isomorphism theorem

Interpretations for a first-order language are said to be elementarily equivalent (in ) provided that they make exactly the same sentences in true (cf. also Interpretation). The Keisler–Shelah isomorphism theorem provides a characterization of elementary equivalence: interpretations for 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 use the generalized continuum hypothesis (GCH). In 1971 S. Shelah gave a third proof [a5]. This proof avoids the generalized continuum hypothesis.

Given , a non-empty family of interpretations for , and an ultrafilter on , the ultraproduct of the family is the quotient system on the direct product of the family induced by . When there is a fixed interpretation, , such that each coincides with , is denoted by and is called an ultrapower of . It follows from results of J. Łos [a4] that and any of its ultrapowers are elementarily equivalent (the Łos isomorphism theorem). Hence, interpretations with isomorphic ultrapowers are elementarily equivalent.

Let be an infinite cardinal no smaller than the cardinality of the set of sentences in , and let and be interpretations for of cardinality less than or equal to . Let denote the cardinal successor of . Keisler showed (assuming that ) that and are elementarily equivalent if and only if there are ultrafilters on such that and are isomorphic.

Let be as above and let be the least cardinal such that . Shelah showed (without assuming that ) that there is an ultrafilter on such that, given and , elementarily equivalent interpretations of cardinality less than , and are isomorphic.

The motivation for Keisler's results can be found in a programme propounded by A. Tarski [a6]: to provide characterizations of meta-mathematical notions in "purely mathematical terms" . A discussion of this programme 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 subalgebras, homomorphic images and direct products. This result characterizes equational classes without mentioning equations.

Amongst the consequences of the Keisler–Shelah isomorphism theorem is a comparable "mathematical" characterization of the classes of models of sentences in . Given , a class of interpretations for , is an elementary class provided that there is a sentence in whose models are exactly the members of ; is an elementary class in the wider sense provided that there is a set of sentences in whose models are exactly the members of . It follows from the compactness theorem that is an elementary class if and only if both and its complement (relative to the class of interpretations for ) are elementary classes in the wider sense. Keisler [a2] showed (assuming GCH) that:

1) is an elementary class in the wider sense, provided that is closed under isomorphic images and ultraproducts and the complement of is closed under ultrapowers;

2) is an elementary class if and only if both and its complement are closed under isomorphic images and ultraproducts.

Whilst Keisler's proof of this result used the generalized continuum hypothesis, its use was restricted to establishing that elementarily equivalent interpretations have isomorphic ultrapowers. Hence, by eliminating GCH in the proof of the latter result, Shelah also eliminated the use of 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