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GENERIC ABSOLUTENESS AND THE CONTINUUM
 MATHEMATICAL RESEARCH LETTERS 9, 465–471
, 2002
"... Let Hω2 denote the collection of all sets whose transitive closure has size at most ℵ1. Thus, (Hω2, ∈) is a natural model of ZFC minus the powerset axiom which correctly estimates manyof the problems left open by the smaller and better understood structure (Hω1, ∈) of hereditarily countable sets. O ..."
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Let Hω2 denote the collection of all sets whose transitive closure has size at most ℵ1. Thus, (Hω2, ∈) is a natural model of ZFC minus the powerset axiom which correctly estimates manyof the problems left open by the smaller and better understood structure (Hω1, ∈) of hereditarily countable sets. One of such problems is, for example, the Continuum Hypothesis. It is largely for this reason that the structure (Hω2, ∈) has recently received a considerable amount of study (see e.g. [15] and [16]). Recall the wellknown LevySchoenfield absoluteness theorem ([10, §2]) which states that for everyΣ0−sentence ϕ(x, a) with one free variable x and parameter a from Hω2, if there is an x such that ϕ(x, a) holds then there is such an x in Hω2, or in other words, (1) (Hω2, ∈) ≺1 (V,∈). Strictly speaking, what is usuallycalled the LevySchoenfield absoluteness theorem is a bit stronger result than this, but this is the form of their absoluteness theorem that allows a variation of interest to us here. The generic absoluteness
Consistency Strengths of Modified Maximality Principles
"... The Maximality Principle mp is a scheme which states that if a sentence of the language of zfc is true in some forcing extension V P, and remains true in any further forcing extension of V P, then it is true in all forcing extensions of V. A modified maximality principle mpΓ arises when considering ..."
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The Maximality Principle mp is a scheme which states that if a sentence of the language of zfc is true in some forcing extension V P, and remains true in any further forcing extension of V P, then it is true in all forcing extensions of V. A modified maximality principle mpΓ arises when considering forcing with a particular class Γ of forcing notions. A parametrized form of such a principle, mpΓ(X), considers formulas taking parameters; to avoid inconsistency such parameters must be restricted to a specific set X which depends on the forcing class Γ being considered. A stronger necessary form of such a principle, ✷mpΓ(X), occurs when it continues to be true in all Γ forcing extensions. This study uses iterated forcing, modal logic, and other techniques to establish consistency strengths for various modified maximality principles restricted to various forcing classes, including ccc, cohen, coll (the forcing notions that collapse ordinals to ω), < κ directed closed forcing notions, etc., both with and without parameter sets. Necessary forms of these principles are also considered. Contents
ABSOLUTENESS FOR UNIVERSALLY BAIRE SETS AND THE UNCOUNTABLE I
, 2006
"... Cantor’s Continuum Hypothesis was proved to be independent from the usual ZFC axioms of Set Theory by Gödel and Cohen. The method of forcing, developed by Cohen to this end, has lead to a profusion of independence results in the following decades. Many other statements about infinite sets, such as t ..."
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Cited by 2 (1 self)
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Cantor’s Continuum Hypothesis was proved to be independent from the usual ZFC axioms of Set Theory by Gödel and Cohen. The method of forcing, developed by Cohen to this end, has lead to a profusion of independence results in the following decades. Many other statements about infinite sets, such as the Borel Conjecture, Whitehead’s problem, and automatic continuity for Banach Algebras, were proved independent, perhaps leaving an impression that most nontrivial statements about infinite sets can be neither proved nor refuted in ZFC. 1 Moreover, some classical statements imply the consistency of ZFC and stronger theories, and by Gödel’s incompleteness theorems the consistency of these statements with ZFC can be proved only by using strong axioms of infinity, socalled large cardinal axioms. A classical example is Banach’s ‘Lebesgue measure has a σadditive extension to all sets of reals. ’ While it is fairly easy to find a model in which this is false and there is no known ZFCproof of its negation, proving the consistency of this statement requires assuming the existence of a measurable cardinal ([32]). A remarkable result was proved by Shoenfield ([31]): every statement of the form
BAUMGARTNER’S CONJECTURE AND BOUNDED FORCING AXIOMS
"... Abstract. We study the spectrum of forcing notions between the iterations of closed followed by ccc forcings and the proper forcings. This includes the hierarchy of ↵proper forcings for indecomposable countable ordinals ↵, the Axiom A forcings and forcings completely embeddable into an iteration ..."
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Abstract. We study the spectrum of forcing notions between the iterations of closed followed by ccc forcings and the proper forcings. This includes the hierarchy of ↵proper forcings for indecomposable countable ordinals ↵, the Axiom A forcings and forcings completely embeddable into an iteration of a closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner’s Axiom A. This resolves a conjecture of Baumgartner from the 1980s. We also study the bounded forcing axioms for the hierarchy of ↵proper forcings. Following ideas of Shelah we separate them for distinct countable indecomposable ordinals. 1.