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Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

Published online by Cambridge University Press:  15 January 2014

Matteo Viale
Matteo Viale*
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Strasse 25, A-1090, Wien, AustriaE-mail: matteo@logic.univie.ac.at
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Extract

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.

The first example I will treat is the proof that the proper forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA.

The second example aims to show that the transfer principle (ℵω+1, ℵω) ↠ (ℵ2, ℵ1) fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory.

Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 1 , March 2008 , pp. 99 - 113
Copyright
Copyright © Association for Symbolic Logic 2008

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