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Counterexamples to the unique and cofinal branches hypotheses
 The Journal of Symbolic Logic
, 2006
"... Abstract. We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal. Martin–Steel [3] introduced the notion of an iteration tree, and with it the question of iterability: the existence of ..."
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Abstract. We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal. Martin–Steel [3] introduced the notion of an iteration tree, and with it the question of iterability: the existence
Comment.Math.Univ.Carolin. 46,4 (2005)721–734 721
"... A tree pibase for R ∗ without cofinal branches ..."
Iterated Forcing with ${}^{\omega}\omega$bounding and Semiproper Preorders
"... Assume the Continuum Hypothesis (CH) in the ground model. If we iteratively force with preorders which are $\omega\omega$bounding and semiproper taking suitable limits, then so is the final preorder constructed. Therefore we may show that the Cofinal Branch Principle (CBP) of [F] is strictly weaker ..."
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Assume the Continuum Hypothesis (CH) in the ground model. If we iteratively force with preorders which are $\omega\omega$bounding and semiproper taking suitable limits, then so is the final preorder constructed. Therefore we may show that the Cofinal Branch Principle (CBP) of [F] is strictly
On The Consistency Of The Definable Tree Property On ℵ1
, 1998
"... In this paper we prove the equiconsistency of "Every ! 1 \Gammatree which is first order definable over (H!1 ; ") has a cofinal branch" with the existence of a \Pi 1 1 reflecting cardinal. We also prove that the addition of MA to the definable tree property increases the consistency ..."
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In this paper we prove the equiconsistency of "Every ! 1 \Gammatree which is first order definable over (H!1 ; ") has a cofinal branch" with the existence of a \Pi 1 1 reflecting cardinal. We also prove that the addition of MA to the definable tree property increases the consistency
Extender based forcings, fresh sets and Aronszajn trees
, 2011
"... Extender based forcings are studied with respect of adding branches to Aronszajn trees. We construct a model with no Aronszajn tree over ℵω+2 from the optimal assumptions. This answers a question of Friedman and Halilovic ́ [1]. The reader interested only in Friedman and Halilovic ́ question may ski ..."
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skip the first section and go directly to the second. 1 No branches to κ+ Aronszajn trees. We deal here with Extender Based Prikry forcing, Long and short extenders Prikry forcing. Let us refer to [2] for definitions. Theorem 1.1 Extender based Prikry forcing over κ cannot add a cofinal branch to a κ
Souslin Trees Which Are Hard To Specialise
"... . We construct some + Souslin trees which cannot be specialised by any forcing which preserves cardinals and cofinalities. For a regular cardinal we use the principle, for singular we use squares and diamonds. 1. Introduction We start by recalling a few basic definitions concerning trees. ..."
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= f x 2 T : ht T (x) = ff g. (c) ht(T ) is the least ff such that T ff = ;. (d) T ff = S fi!ff T fi . (e) A cofinal branch of T is a set B ` T such that B is linearly ordered by ! T , and 8ff ! ht(T ) 9b 2 B ht T (b) ff. 3. T is a tree iff ht(T ) = and jT ff j ! for all ff ! . 4. T is a
Aronszajn trees and failure of the singular cardinal hypothesis
 J. Math. Log
"... Abstract. The tree property at κ + states that there are no Aronszajn trees on κ +, or, equivalently, that every κ + tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ + and failure of the singular cardinal hypothesis at κ; the former is ..."
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Abstract. The tree property at κ + states that there are no Aronszajn trees on κ +, or, equivalently, that every κ + tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ + and failure of the singular cardinal hypothesis at κ; the former
Determinacy and the tree property
, 2003
"... The following theorems are shown. (A) If ω2 and ω3 have the tree property and either 2 ℵ0 = ℵ2 or Ω is measurable then Π 1 2 Determinacy holds. (B) If ωn has the tree property for all n < ω, n ≥ 2, and 2 <ℵω = ℵω then Projective Determinacy holds. 1 Introduction. A cardinal δ is said to have t ..."
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the tree property if there is no Aronszajn δtree, i.e., if every tree of height δ all of whose levels have size < δ has a cofinal branch. Hence König’s Lemma states that ω has the tree property, whereas Aronszajn himself discovered that there is an Aronszajn ω1tree
On patterns of cardinals with the tree property
, 2008
"... shown in [2], starting from ω many supercompact cardinals, that consistently the following holds. (⋆) For every n < ω, 2 ℵn = ℵn+2 and ℵn+1 has the tree property. Recall that a cardinal κ is said to have the tree property if there is no Aronszajn κtree, i.e. if every tree of height κ all of whos ..."
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of whose levels have size < κ admits a cofinal branch. We here show (in a certain sense of ”show”): Theorem 0.1 Suppose that there are δ1 < δ2 < δ3 <... with supremum σ such that σ is a strong limit cardinal, and for all n < ω, δ2n+2 = (δ2n+1) + and δn+1 has the tree property. Let G be Col
Results 1  10
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