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  1. A relative of the approachability ideal, diamond and non-saturation.Assaf Rinot - 2010 - Journal of Symbolic Logic 75 (3):1035-1065.
    Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square _{\lambda}^{\ast}$ together with 2 λ = λ⁺ implies $\lozenge _{S}$ for every S ⊆ λ⁺ that reflects stationarily often. In this paper, for a set S ⊆ λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved: 1. if I[S; λ] is fat, (...)
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  • Forcing axioms, supercompact cardinals, singular cardinal combinatorics.Matteo Viale - 2008 - Bulletin of Symbolic Logic 14 (1):99-113.
    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 (...)
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  • The Proper Forcing Axiom and the Singular Cardinal Hypothesis.Matteo Viale - 2006 - Journal of Symbolic Logic 71 (2):473 - 479.
    We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].
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