We analyze scales in LpG Ω (R, Ω↾HC), the stack of sound, projecting, Θ-g-organized Ω-mice over Ω↾HC, where Ω is either an iteration strategy or an operator, Ω has appropriate condensation properties, and Ω↾HC is self-scaled. This builds on Steel's analysis of scales in L(R) and Lp(R) (also denoted K(R)). As in Steel's analysis, we work from optimal determinacy hypotheses. One of the main applications of the work is in the core model induction.
Abstract. In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees implies that in some homogenous generic extension of V there is a transitive model M containing Ord ∪ R such that M AD + + Θ > θ 0 . In particular, this implies the existence (in V ) of a non-tame mouse. The results of this paper significantly extend J. R. Steel's earlier results for tame trees.
This paper explores the consistency strength of The Proper Forcing Axiom (PFA) and the theory (T) which involves a variation of the Viale-Weiß guessing hull principle. We show that (T) is consistent relative to a supercompact cardinal. The main result of the paper is Theorem 0.2, which implies that the theory "AD R + Θ is regular" is consistent relative to (T) and to PFA. This improves significantly the previous known best lower-bound for consistency strength for (T) and PFA, which is roughly "AD R + DC".satisfying the hypothesis of the previous sentence is called κ-approximated by X. So a hull X is κ-guessing if whenever a ∈ X and whenever b ⊆ a is κ-approximated by X, then b is X-guessed.In this paper, we study the strength of the following theories • The Proper Forcing Axiom (PFA);• (T): there is a cardinal λ ≥ 2 ℵ 2 such that the set {X ≺ (H λ ++ , ∈) | |X| = ℵ 2 , X ω ⊆ X, ω 2 ⊂ X, and X is ω 2 -guessing } is stationary.Guessing models in [26] are ω 1 -guessing in the above notations. It's not clear that the theory (T) is consistent with PFA (in contrast to Viale-Weiß principle ISP(ω 2 ), which asserts the existence of stationary many ω 1 -guessing models of size ℵ 1 of H λ for all sufficiently large λ). However, it's possible that (T) is a consequence of or at least consistent with a higher analog of PFA.The outline of the paper is as follows. In section 1, we review some AD + facts that we'll be using in this paper. In section 2, using a Mitchell-style forcing, we prove Theorem 0.1. Con(ZFC + there is a supercompact cardinal) ⇒ Con(T).Of course, it is well-known that PFA is consistent relative to the existence of a supercompact cardinal. Theorem 0.4 suggests that it's reasonable to expect PFA and (T) are equiconsistent.Recall, for an infinite cardinal λ, the principle λ asserts the existence of a sequence C α | α < λ + such that for each α < λ + ,
Assume [Formula: see text]. If [Formula: see text] is an ordinal and [Formula: see text] is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the Jónsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text].
We develop the theory of abstract fine structural operators and operator-premice. We identify properties, which we require of operatorpremice and operators, which ensure that certain basic facts about standard premice generalize. We define fine condensation for operators F and show that fine condensation and iterability together ensure that F-mice have the fundamental fine structural properties including universality and solidity of the standard parameter.
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