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STABLE ORDERED UNION ULTRAFILTERS AND cov $\left( \mathcal{M} \right) < \mathfrak{c}$

Published online by Cambridge University Press:  03 April 2019

DAVID JOSÉ FERNÁNDEZ-BRETÓN
DAVID JOSÉ FERNÁNDEZ-BRETÓN*
Affiliation:
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN 2074 EAST HALL, 530 CHURCH STREET ANN ARBOR, MICHIGAN, 48109, USA and KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNA WÄRINGER STRAßE 25, 1090WIEN, AUSTRIAE-mail: david.fernandez-breton@univie.ac.atURL: https://homepage.univie.ac.at/david.fernandez-breton/
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Abstract

A union ultrafilter is an ultrafilter over the finite subsets of ω that has a base of sets of the form ${\text{FU}}\left( X \right)$, where X is an infinite pairwise disjoint family and ${\text{FU}}(X) = \left\{ {\bigcup {F|F}\in [X]^{ < \omega }\setminus \{ \emptyset \} } \right\}$. The existence of these ultrafilters is not provable from the $ZFC$ axioms, but is known to follow from the assumption that ${\text{cov}}\left( \mathcal{M} \right) = \mathfrak{c}$. In this article we obtain various models of $ZFC$ that satisfy the existence of union ultrafilters while at the same time ${\text{cov}}\left( \mathcal{M} \right) = \mathfrak{c}$.

Keywords

Primary 03E35Secondary 03E1754D8022A15ultrafiltersHindman’s theoremiterated forcingproper forcingcardinal characteristics of the continuum

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 3 , September 2019 , pp. 1176 - 1193
Copyright
Copyright © The Association for Symbolic Logic 2019 

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