Forcing axioms, supercompact cardinals, singular cardinal combinatorics

Bulletin of Symbolic Logic 14 (1):99-113 (2008)
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Abstract

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 ↠ 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.

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References found in this work

Reflecting stationary sets.Menachem Magidor - 1982 - Journal of Symbolic Logic 47 (4):755-771.
A very weak square principle.Matthew Foreman & Menachem Magidor - 1997 - Journal of Symbolic Logic 62 (1):175-196.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
The strenght of the failure of the singular cardinal hypothesis.Moti Gitik - 1991 - Annals of Pure and Applied Logic 51 (3):215-240.
Fragments of Martin's Maximum in generic extensions.Y. Yoshinobu & B. Konig - 2004 - Mathematical Logic Quarterly 50 (3):297.

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