The Core Model Iterability Problem

Author(s):  
John R. Steel
Keyword(s):  
2008 ◽  
Vol 73 (2) ◽  
pp. 369-390 ◽  
Author(s):  
J. R. Steel

In this note we shall proveTheorem 0.1. Letbe a countably ω-iterable-mouse which satisfies AD, and [α, β] a weak gap of. Supposeis captured by mice with iteration strategies in ∣α. Let n be least such that ; then we have that believes that has the Scale Property.This complements the work of [5] on the construction of scales of minimal complexity on sets of reals in K(ℝ). Theorem 0.1 was proved there under the stronger hypothesis that all sets definable over are determined, although without the capturing hypothesis. (See [5, Theorem 4.14].) Unfortunately, this is more determinacy than would be available as an induction hypothesis in a core model induction. The capturing hypothesis, on the other hand, is available in such a situation. Since core model inductions are one of the principal applications of the construction of optimal scales, it is important to prove 0.1 as stated.Our proof will incorporate a number of ideas due to Woodin which figure prominently in the weak gap case of the core model induction. It relies also on the connection between scales and iteration strategies with the Dodd-Jensen property first discovered in [3]. Let be the pointclass at the beginning of the weak gap referred to in 0.1. In section 1, we use Woodin's ideas to construct a Γ-full a mouse having ω Woodin cardinals cofinal in its ordinals, together with an iteration strategy Σ which condenses well in the sense of [4, Def. 1.13]. In section 2, we construct the desired scale from and Σ.


2011 ◽  
Vol 76 (2) ◽  
pp. 541-560 ◽  
Author(s):  
Victoria Gitman ◽  
P. D. Welch

AbstractThis paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.


2018 ◽  
Vol 83 (3) ◽  
pp. 920-938
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER

AbstractIt is shown that $K|{\omega _1}$ need not be solid in the sense previously introduced by the authors: it is consistent that there is no inner model with a Woodin cardinal yet there is an inner model W and a Cohen real x over W such that $K|{\omega _1}\,\, \in \,\,W[x] \setminus W$. However, if ${0^{\rm{\P}}}$ does not exist and $\kappa \ge {\omega _2}$ is a cardinal, then $K|\kappa$ is solid. We draw the conclusion that solidity is not forcing absolute in general, and that under the assumption of $\neg {0^{\rm{\P}}}$, the core model is contained in the solid core, previously introduced by the authors.It is also shown, assuming ${0^{\rm{\P}}}$ does not exist, that if there is a forcing that preserves ${\omega _1}$, forces that every real has a sharp, and increases $\delta _2^1$, then ${\omega _1}$ is measurable in K.


2015 ◽  
Vol 31 (6) ◽  
pp. 414-425 ◽  
Author(s):  
Anne Mette Ølholm ◽  
Kristian Kidholm ◽  
Mette Birk-Olsen ◽  
Janne Buck Christensen

Objectives: There is growing interest in implementing hospital-based health technology assessment (HB-HTA) as a tool to facilitate decision making based on a systematic and multidisciplinary assessment of evidence. However, the decision-making process, including the informational needs of hospital decision makers, is not well described. The objective was to review empirical studies analysing the information that hospital decision makers need when deciding about health technology (HT) investments.Methods: A systematic review of empirical studies published in English or Danish from 2000 to 2012 was carried out. The literature was assessed by two reviewers working independently. The identified informational needs were assessed with regard to their agreement with the nine domains of EUnetHTA's Core Model.Results: A total of 2,689 articles were identified and assessed. The review process resulted in 14 relevant studies containing 74 types of information that hospital decision makers found relevant. In addition to information covered by the Core Model, other types of information dealing with political and strategic aspects were identified. The most frequently mentioned types of information in the literature related to clinical, economic and political/strategic aspects. Legal, social, and ethical aspects were seldom considered most important.Conclusions: Hospital decision makers are able to describe their information needs when deciding on HT investments. The different types of information were not of equal importance to hospital decision makers, however, and full agreement between EUnetHTA's Core Model and the hospital decision-makers’ informational needs was not observed. They also need information on political and strategic aspects not covered by the Core Model.


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