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ON ${\omega _1}$-STRONGLY COMPACT CARDINALS

Published online by Cambridge University Press:  17 April 2014

JOAN BAGARIA  and
MENACHEM MAGIDOR
JOAN BAGARIA
Affiliation:
ICREA (INSTITUCIÓ CATALANA DE RECERCA I ESTUDIS AVANÇATS) AND DEPARTAMENT DE LÒGICA, HISTÒRIA I FILOSOFIA DE LA CIÈNCIA UNIVERSITAT DE BARCELONA, MONTALEGRE 6 8001 BARCELONA, CATALONIA, SPAIN E-mail: [email protected]
MENACHEM MAGIDOR
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS, THE HEBREW UNIVERSITY OF JERUSALEM EDMOND J. SAFRA CAMPUS, GIVAT RAM JERUSALEM 91904, ISRAEL E-mail: [email protected]
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Abstract

An uncountable cardinal κ is called ${\omega _1}$-strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$-complete ultrafilter on I. We show that the first ${\omega _1}$-strongly compact cardinal, ${\kappa _0}$, cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$. We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$. Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$.

Keywords

Primary: 03E5503E75. Secondary: 54D1554D2054E35ω1-strongly compact cardinalSingular Cardinal HypothesisLindelöf spacesecond-order reflectioncountably chromatic graphmetrizable spacecompletely regular space
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 266 - 278
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

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