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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 is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper we reconcile the two. We prove from large cardinals that the tree property at κ + is consistent with failure of the singular cardinal hypothesis at κ. §1. Introduction. In the early 1980s Woodin asked whether failure of the singular cardinal hypothesis (SCH) at ℵω implies the existence of an Aronszajn tree on ℵω+1. More generally, in 1989 Woodin and others asked whether failure of the SCH at a cardinal κ of cofinality ω, implies the existence of an Aronszajn tree on κ +, see Foreman [7, §2]. To understand the motivation for the question let us
Singular Cardinals And The PCF Theory
 Bull. Symbolic Logic
, 1995
"... this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main re ..."
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this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory.
Possible pcf algebras
 J. Symb. Logic
, 1996
"... Abstract. There exists a family {Bα}α<ω1 of sets of countable ordinals such that (1) max Bα = α, (2) if α ∈ Bβ then Bα ⊆ Bβ, (3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ, β < α, and by the bounded subsets of λ, (4) there is a partition {An} ∞ n=0 o ..."
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Abstract. There exists a family {Bα}α<ω1 of sets of countable ordinals such that (1) max Bα = α, (2) if α ∈ Bβ then Bα ⊆ Bβ, (3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ, β < α, and by the bounded subsets of λ, (4) there is a partition {An} ∞ n=0 of ω1 such that for every α and every n, Bα∩An is finite. 1. Introduction. 476 revision:19950323 modified:19950324 In [3], [4], [5] and [6] the second author developed the theory of possible cofinalities (pcf), and proved, among others, that if ℵω is a strong limit cardinal then
On Some Configurations Related to the Shelah Weak Hypothesis
"... We show that some cardinal arithmetic configurations related to the negation of the Shelah Weak Hypothesis and natural from the forcing point of view are impossible. 708 revision:20001004 modified:20001004 1 ..."
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We show that some cardinal arithmetic configurations related to the negation of the Shelah Weak Hypothesis and natural from the forcing point of view are impossible. 708 revision:20001004 modified:20001004 1
The pcf theorem revisited
, 1995
"... The pcf theorem (of the possible cofinability theory) was proved for reduced products ∏ i<κ λi/I, where κ < mini<κ λi. Here we prove this theorem under weaker assumptions such as wsat(I) < mini<κ λi, where wsat(I) is the minimal θ such that κ cannot be delivered to θ sets / ∈ I (or e ..."
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The pcf theorem (of the possible cofinability theory) was proved for reduced products ∏ i<κ λi/I, where κ < mini<κ λi. Here we prove this theorem under weaker assumptions such as wsat(I) < mini<κ λi, where wsat(I) is the minimal θ such that κ cannot be delivered to θ sets / ∈ I (or even slightly weaker condition). We also look at the existence of exact upper bounds relative to <I (<I −eub) as well as cardinalities of reduced products and the cardinals TD(λ). Finally we apply this to the problem of the depth of ultraproducts (and reduced products) of Boolean algebras.
FALLEN CARDINALS
"... Abstract. We prove that for every singular cardinal µ of cofinality ω, the complete Boolean algebra comp Pµ(µ) contains a complete subalgebra which is isomorphic to the collapse algebra Comp Col(ω1, µ ℵ0). Consequently, adding a generic filter to the quotient algebra Pµ(µ) = P(µ)/[µ] <µ collapse ..."
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Abstract. We prove that for every singular cardinal µ of cofinality ω, the complete Boolean algebra comp Pµ(µ) contains a complete subalgebra which is isomorphic to the collapse algebra Comp Col(ω1, µ ℵ0). Consequently, adding a generic filter to the quotient algebra Pµ(µ) = P(µ)/[µ] <µ collapses µ ℵ0 to ℵ1. Another corollary is that the Baire number of the space U(µ) of all uniform ultrafilters over µ is equal to ω2. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory. 720 revision:20010627 modified:20020227 1.
Halfway Up To the Mathematical Infinity I: On the Ontological & Epistemic Sustainability of Georg Cantor’s Transfinite Design
, 2009
"... Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously – albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of ..."
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Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously – albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability and wellordering – and implying an ordinal recreation of the continuum. During the last hundred years, the mainstream settheoretical research – all insights and adjustments due to Kurt Gödel’s revolutionary insights and discoveries notwithstanding – has compliantly centered its efforts on ad hoc axiomatizations of Cantor’s intuitive transfinite design. We demonstrate here that the ontological and epistemic sus
Retrieving the Mathematical Mission of the Continuum Concept from the Transfinitely Reductionist Debris of Cantor’s Paradise (Extended Abstract)
, 2011
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