Ralf Schindler

Semi-proper forcing, remarkable cardinals, and Bounded Martin's Maximum

We show that L(ℝ) absoluteness for semi-proper forcings is equiconsistent with the existence of a remarkable cardinal, and hence by [6] with L(ℝ) absoluteness for proper forcings. By [7], L(ℝ) absoluteness for stationary set preserving forcings gives an inner model with a strong cardinal. By [3], the Bounded Semi-Proper Forcing Axiom (BSPFA) is equiconsistent with the Bounded Proper Forcing Axiom (BPFA), which in turn is equiconsistent with a reflecting cardinal. We show that Bounded Martins Maximum (BMM) is much stronger than BSPFA in that if BMM holds, then for every X ∈ V , X# exists. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The strength of choiceless patterns of singular and weakly compact cardinals

Abstract We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that AD , the Axiom of Determinacy, holds in the L ( R ) of a generic extension of HOD : (a) ZF + every uncountable cardinal is singular, and (b) ZF + every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.

Bounded Martin’s Maximum and Strong Cardinals

We show that if Bounded Martin’s Maximum (BMM) holds then for every X ∈ V there is an inner model with a strong cardinal containing X. We also discuss various open questions which are related to BMM.

A new condensation principle

Abstract.We generalize ∇(A), which was introduced in [Sch∞], to larger cardinals. For a regular cardinal κ>ℵ0 we denote by ∇κ(A) the statement that and for all regular θ>κ, is stationary in It was shown in [Sch∞] that can hold in a set-generic extension of L. We here prove that can hold in a set-generic extension of L as well. In both cases we in fact get equiconsistency theorems. This strengthens results of [Rä00] and [Rä01]. is equivalent with the existence of 0#.

Generic Vopĕnka's Principle, remarkable cardinals, and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures

A criterion for coarse iterability

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Homogeneously Souslin sets in small inner models

C of the same type, there exist