AN EQUICONSISTENCY RESULT ON PARTIAL SQUARES

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

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NOWHERE PRECIPITOUSNESS OF THE NON-STATIONARY IDEAL OVER ${\mathcal P}_\kappa \lambda$

Journal of Mathematical Logic ◽

10.1142/s021906130200014x ◽

2002 ◽

Vol 02 (01) ◽

pp. 81-89 ◽

Cited By ~ 6

Author(s):

YO MATSUBARA ◽

SAHARON SHELAH

Keyword(s):

Stationary Subset ◽

Singular Cardinal ◽

Strong Limit ◽

Uncountable Cardinal

We prove that if λ is a strong limit singular cardinal and κ a regular uncountable cardinal < λ, then NSκλ, the non-stationary ideal over [Formula: see text], is nowhere precipitous. We also show that under the same hypothesis every stationary subset of [Formula: see text] can be partitioned into λκ disjoint stationary sets.

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Power-like models of set theory

Journal of Symbolic Logic ◽

10.2307/2694973 ◽

2001 ◽

Vol 66 (4) ◽

pp. 1766-1782 ◽

Cited By ~ 3

Author(s):

Ali Enayat

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Mahlo Cardinal ◽

Uncountable Cardinal ◽

Consistent Extension ◽

Elementary Submodel ◽

Finite Language ◽

The Universe ◽

Models Of Set Theory ◽

The Continuum

Abstract.A model = (M. E, …) of Zermelo-Fraenkel set theory ZF is said to be 0-like. where E interprets ∈ and θ is an uncountable cardinal, if ∣M∣ = θ but ∣{b ∈ M: bEa}∣ < 0 for each a ∈ M, An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ1-like model. Coupled with Chang's two cardinal theorem this implies that if θ is a regular cardinal 0 such that 2<0 = 0 then every consistent extension of ZF also has a 0+-like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ2-like model. Here we prove:Theorem A. If 0 has the tree property then the following are equivalent for any completion T of ZFC:(i) T has a 0-like model.(ii) Ф ⊆ T. where Ф is the recursive set of axioms {∃κ (κ is n-Mahlo and “Vκis a Σn-elementary submodel of the universe”): n ∈ ω}.(iii) T has a λ-like model for every uncountable cardinal λ.Theorem B. The following are equiconsistent over ZFC:(i) “There exists an ω-Mahlo cardinal”.(ii) “For every finite language , all ℵ2-like models of ZFC() satisfy the schemeФ().

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THE HARRINGTON–SHELAH MODEL WITH LARGE CONTINUUM

Journal of Symbolic Logic ◽

10.1017/jsl.2018.74 ◽

2019 ◽

Vol 84 (02) ◽

pp. 684-703

Author(s):

THOMAS GILTON ◽

JOHN KRUEGER

Keyword(s):

Stationary Subset ◽

Mahlo Cardinal ◽

Large Continuum ◽

Image Position

AbstractWe prove from the existence of a Mahlo cardinal the consistency of the statement that 2ω = ω3 holds and every stationary subset of ${\omega _2}\mathop \cap \nolimits {\rm{cof}}\left( \omega \right)$ reflects to an ordinal less than ω2 with cofinality ω1.

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Ideals and combinatorial principles

Journal of Symbolic Logic ◽

10.2307/2275734 ◽

1997 ◽

Vol 62 (1) ◽

pp. 117-122 ◽

Cited By ~ 2

Author(s):

Douglas Burke ◽

Yo Matsubara

Keyword(s):

Regular Cardinal ◽

Stationary Subset ◽

Strongly Compact Cardinal ◽

Compact Cardinal ◽

Uncountable Cardinal ◽

Combinatorial Principles ◽

Stationary Subsets

It is well known that if σ is a strongly compact cardinal and λ a regular cardinal ≥ σ, then for every stationary subset X of {α < λ: cof (α) = ω} there is some β < λ such that X ⋂ β is stationary in β. In fact the existence of a uniform, countably complete ultrafilter over λ is sufficient to prove the same conclusion about stationary subsets of {α < λ: cof (α) = ω}. See [13] or [10]. By analyzing the proof of this theorem as presented in [10], we realized the same conclusion will follow from the existence of a certain ideal, not necessarily prime, on . Throughout we will assume that σ is a regular uncountable cardinal and use the word “ideal” to mean fine ideal.

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A Forcing Axiom Deciding the Generalized Souslin Hypothesis

Canadian Journal of Mathematics ◽

10.4153/cjm-2017-058-2 ◽

2019 ◽

Vol 71 (2) ◽

pp. 437-470

Author(s):

Chris Lambie-Hanson ◽

Assaf Rinot

Keyword(s):

Mahlo Cardinal ◽

Uncountable Cardinal ◽

Souslin Trees ◽

Forcing Axiom

AbstractWe derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal$\unicode[STIX]{x1D706}$, if$\unicode[STIX]{x1D706}^{++}$is not a Mahlo cardinal in Gödel’s constructible universe, then$2^{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}^{+}$entails the existence of a$\unicode[STIX]{x1D706}^{+}$-complete$\unicode[STIX]{x1D706}^{++}$-Souslin tree.

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Full reflection at a measurable cardinal

Journal of Symbolic Logic ◽

10.2307/2275413 ◽

1994 ◽

Vol 59 (2) ◽

pp. 615-630 ◽

Cited By ~ 2

Author(s):

Thomas Jech ◽

Jiří Witzany

Keyword(s):

Measurable Cardinal ◽

Higher Order ◽

Stationary Subset ◽

Regular Cardinals ◽

Uncountable Cardinal ◽

Stationary Set ◽

Full Reflection

AbstractA stationary subset S of a regular uncountable cardinal κreflects fully at regular cardinals if for every stationary set T ⊆ κ of higher order consisting of regular cardinals there exists an α Є T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection holds at every λ ≤ κ and κ is measurable.

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Co-stationarity of the ground model

Journal of Symbolic Logic ◽

10.2178/jsl/1154698589 ◽

2006 ◽

Vol 71 (3) ◽

pp. 1029-1043 ◽

Cited By ~ 3

Author(s):

Natasha Dobrinen ◽

Sy-David Friedman

Keyword(s):

Large Cardinals ◽

Partial Ordering ◽

Proper Class ◽

Ground Model ◽

Covering Theorem ◽

Uncountable Cardinal ◽

Cohen Forcing ◽

Measurable Cardinals

AbstractThis paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

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The spectrum of resplendency

Journal of Symbolic Logic ◽

10.2307/2274652 ◽

1990 ◽

Vol 55 (2) ◽

pp. 626-636

Author(s):

John T. Baldwin

Keyword(s):

Homogeneous Model ◽

Order Theory ◽

First Order ◽

First Order Theory ◽

Uncountable Cardinal ◽

Recursive Theory ◽

Finite Language

AbstractLet T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

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Homogeneous changes in cofinalities with applications to HOD

Journal of Mathematical Logic ◽

10.1142/s0219061317500076 ◽

2017 ◽

Vol 17 (02) ◽

pp. 1750007 ◽

Cited By ~ 1

Author(s):

Omer Ben-Neria ◽

Spencer Unger

Keyword(s):

Large Cardinals ◽