Countable ultraproducts without CH
The author studies ultraproducts of countable structures (mainly \(<\omega,<>\) and \(<\omega,+,*>)\) over countable index sets. If one assumes CH then the theory is very simple: all ultrapowers of a given structure are isomorphic. Without CH things become basically untractable: CH can fail in too many ways. The author (wisely) restricts his attention to models of \(\neg CH\), obtained by adding Cohen and/or random reals to a model of CH. Especially Cohen reals allow a variety of ultrapowers: the author's section 4.12 gives a good summary. The author cleverly uses the fact that one Cohen real actually produces a perfect set of Cohen reals, to show that ground-model ultrafilers can be extended, in the extension, to ultrafilters for which the corresponding ultrapowers have prescribed saturation, cofinality and coinitiality. Random reals do not have al the flexibility of Cohen reals in this respect; the main problem is that every new function is dominated by an old one. This puts restrictions on the cofinalities and saturation one can expect. After adding random reals to a model of CH all ultrapowers will have cofinality \(\omega_ 1\). As to saturation, one gets it for types that are consistent not too high in the ultrapower. This paper is recommended to anyone interested in Cohen/random reals and ultrafiltes. Unfortunately the quality of the paper is impoverished by many misprints.
scientific article; zbMATH DE number 4053581
scientific article