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Vive la différence. I: Nonisomorphism of ultrapowers of countable models. (English) Zbl 0789.03035

Set theory of the continuum, Pap. Math. Sci. Res. Inst. Workshop, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 26, 357-405 + Erratum (1992).
[For the entire collection see Zbl 0758.00014.]
It was shown by the author that any two elementarily equivalent structures of cardinality \(\lambda\) have isomorphic ultrapowers with respect to an ultrafilter on \(2^ \lambda\) [the author, Isr. J. Math. 10, 224-233 (1971; Zbl 0224.02045)]. Further on it is known that CH implies that for any two elementarily equivalent countable structures and any ultrafilter \(\mathcal F\) on \(\omega\), the ultrapowers with respect to \(\mathcal F\) are isomorphic.
Here the author shows that it is not provable in ZFC that any two countable elementarily equivalent structures have isomorphic ultrapowers relative to some ultrafilter on \(\omega\). Starting with a model \(V\) of CH the author constructs a generic extension via an iterated proper forcing of length \(\omega_ 2\) in which are elementarily equivalent countable graphs \(\Gamma\), \(\Delta\) such that for every ultrafilter \(\mathcal F\) on \(\omega\), \(\Gamma^ \omega/{\mathcal F}\not\cong \Delta^ \omega/{\mathcal F}\).
He shows that the graphs \(\Gamma\) and \(\Delta\) can be replaced by countable sequences of finite graphs. Further on he shows that adding \(\aleph_ 3\) Cohen reals to a model of ZFC gives a structure with elementarily equivalent graphs \(\Gamma\) and \(\Delta\) such that for some ultrafilter \({\mathcal F},\Gamma^ \omega/{\mathcal F}\) and \(\Delta^ \omega/{\mathcal F}\) are not isomorphic.
Reviewer: M.Weese (Berlin)

MSC:

03C55 Set-theoretic model theory
03C15 Model theory of denumerable and separable structures
03C20 Ultraproducts and related constructions
03E50 Continuum hypothesis and Martin’s axiom
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