2011 seminar talk: The Definable Failure of the Singular Cardinal Hypothesis (SCH)

Talk held by Radek Honzik (Charles University, Prague) at the KGRC seminar on 2011-03-10.

Abstract

We show first that it is consistent that Κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(Κ⁺). Then with further forcing we show that it is consistent that GCH fails at ℵ_ω, ℵ_ω strong limit, while there is a lightface definable wellorder of H(ℵ_ω+1) ("definable failure" of the singular cardinal hypothesis at ℵ_ω). The large cardinal hypothesis used is the existence of a Κ⁺⁺-strong cardinal, where Κ is Κ⁺⁺-strong if there is an embedding j:V → M with critical point Κ such that H(Κ⁺⁺) is included in M (this is almost optimal). The fine structure of the canonical inner model L[E] for a Κ⁺⁺-strong cardinal is used throughout. This is joint work with Sy D. Friedman.