View source for Keisler-Shelah isomorphism theorem

Interpretations for a first-order language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100601.png" /> are said to be elementarily equivalent (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100602.png" />) provided that they make exactly the same sentences in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100603.png" /> true (cf. also [[Interpretation|Interpretation]]). The Keisler–Shelah isomorphism theorem provides a characterization of elementary equivalence: interpretations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100604.png" /> are elementarily equivalent if and only if they have isomorphic ultrapowers (cf. also [[Ultrafilter|Ultrafilter]]).

This theorem was formulated and proved by H.J. Keisler in 1961 [[#References|[a2]]]. Keisler gave a second proof in 1964 using saturated ultrapowers [[#References|[a3]]]. Both proofs use the generalized [[Continuum hypothesis|continuum hypothesis]] (GCH). In 1971 S. Shelah gave a third proof [[#References|[a5]]]. This proof avoids the generalized continuum hypothesis.

Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100605.png" />, a non-empty family of interpretations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100606.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100607.png" /> an [[Ultrafilter|ultrafilter]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100608.png" />, the ultraproduct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k1100609.png" /> of the family is the quotient system on the direct product of the family induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006010.png" />. When there is a fixed interpretation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006011.png" />, such that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006012.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006014.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006015.png" /> and is called an ultrapower of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006016.png" />. It follows from results of J. Łos [[#References|[a4]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006017.png" /> and any of its ultrapowers are elementarily equivalent (the Łos isomorphism theorem). Hence, interpretations with isomorphic ultrapowers are elementarily equivalent.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006018.png" /> be an infinite cardinal no smaller than the cardinality of the set of sentences in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006019.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006021.png" /> be interpretations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006022.png" /> of cardinality less than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006023.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006024.png" /> denote the cardinal successor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006025.png" />. Keisler showed (assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006026.png" />) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006028.png" /> are elementarily equivalent if and only if there are ultrafilters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006032.png" /> are isomorphic.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006033.png" /> be as above and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006034.png" /> be the least cardinal such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006035.png" />. Shelah showed (without assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006036.png" />) that there is an ultrafilter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006038.png" /> such that, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006040.png" />, elementarily equivalent interpretations of cardinality less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006043.png" /> are isomorphic.

The motivation for Keisler's results can be found in a programme propounded by A. Tarski [[#References|[a6]]]: to provide characterizations of meta-mathematical notions in  "purely mathematical terms" . A discussion of this programme and its history can be found in [[#References|[a7]]]. To appreciate what was intended here, recall G. Birkhoff's 1935 characterization [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006044.png" />. Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006045.png" />, a class of interpretations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006047.png" /> is an elementary class provided that there is a sentence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006048.png" /> whose models are exactly the members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006049.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006050.png" /> is an elementary class in the wider sense provided that there is a set of sentences in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006051.png" /> whose models are exactly the members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006052.png" />. It follows from the compactness theorem that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006053.png" /> is an elementary class if and only if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006054.png" /> and its complement (relative to the class of interpretations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006055.png" />) are elementary classes in the wider sense. Keisler [[#References|[a2]]] showed (assuming GCH) that:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006056.png" /> is an elementary class in the wider sense, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006057.png" /> is closed under isomorphic images and ultraproducts and the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006058.png" /> is closed under ultrapowers;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006059.png" /> is an elementary class if and only if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110060/k11006060.png" /> 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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Birkhoff,   "On the structure of abstract algebras"  ''Proc. Cambridge Philos. Soc.'' , '''31'''  (1935)  pp. 433–454</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.J. Keisler,   "Ultraproducts and elementary models"  ''Indagationes Mathematicae'' , '''23'''  (1961)  pp. 477–495</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.J. Keisler,   "Ultraproducts and saturated classes"  ''Indagationes Mathematicae'' , '''26'''  (1964)  pp. 178–186</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Łos,   "Quelques remarqes, théorèmes et problèmes sur les classes définissables d'algèbres" , ''Mathematical Interpretations of Formal Systems'' , North-Holland  (1955)  pp. 98–113</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Shelah,   "Every two elementarily equivalent models have isomorphic ultrapowers"  ''Israel J. Math.'' , '''10'''  (1971)  pp. 224–233</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.L. Vaught,   "Model theory before 1945" , ''Proc. Tarski Symp.'' , Amer. Math. Soc.  (1974)  pp. 153–172</TD></TR></table>