×

On ultrapowers of Boolean algebras. (English) Zbl 0598.03040

The main result of the paper is the following theorem: For each regular cardinal \(\kappa\) with \(\omega_ 1\leq \kappa \leq 2^{\aleph_ 0}\) there exists a uniform ultrafilter p on \(\omega_ 0\) such that there is an \((\omega_ 0,\kappa)\)-gap in the ultrapower \(^{\omega_ 0}\omega_ 0/p\) with the lower part of the form \(\{\) ň,n\(\in \omega_ 0\}\). The main tool used in the proof is a family of large oscillation modulo a filter. As a consequence, if \(2^{\aleph_ 0}>\aleph_ 1\) one obtains non-isomorphic H-fields of cardinality \(2^{\aleph_ 0}\). Some generalizations of this result for uncountable cardinals are also proved.
Reviewer: L.Bukovský

MSC:

03E05 Other combinatorial set theory
03C20 Ultraproducts and related constructions
03E50 Continuum hypothesis and Martin’s axiom
PDFBibTeX XMLCite