Results on the Generic Kurepa Hypothesis

Jensen, Ronald Björn; Schlechta, Karl

doi:10.1007/bf01793783

Cited by 17 publications

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“…Hence Max(CCC) + implies the existence of a Kurepa tree. On the other hand, it is consistent relative to the consistency of a Mahlo cardinal, that there are no Kurepa trees in CCC-extensions [4]. Thus Max(CCC) + "there are no Kurepa trees" is consistent relative to the consistency of a Mahlo cardinal.…”

Section: Nowmentioning

confidence: 95%

“…Remark: Max(C) is consistent with , the existence of a Souslin tree and the existence of a Kurepa tree, for we may assume V = L in the ground model in the proof of Theorem 18. On the other hand, Jensen and Schlechta [4] prove that it is consistent, relative to the consistency of a Mahlo cardinal, that there are no Kurepa trees in CCC-extensions of the universe. Thus we get the consistency of Max(C) with the non-existence of Kurepa trees, relative to the consistency of a Mahlo cardinal.…”

Section: Cohen-extensionsmentioning

confidence: 99%

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Reflection principles for the continuum

Stavi¹,

Väänánen²

2002

Logic and Algebra

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Let HC denote the set of sets of hereditary cardinality less than 2 ω . We consider reflection principles for HC in analogy with the Levy reflection principle for HC. Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x 1 , . . . , x n ) is a property which is provably persistent in extensions by elements of B, then R(a 1 , . . . , a n ) holds whenever a 1 , . . . , a n ∈ HC and R(a 1 , . . . , a n ) has a positive IB-value for some IB ∈ B. Suppose C is the class of Cohen algebras. We prove that Con(ZF ) implies Con(ZF C + Max(C)). For a different principle, let CCC be the class of all CCC algebras. We prove that ZF + Levy schema, and ZF C + Max(CCC) are equiconsistent. Max(CCC) implies M A, while Max(C) implies ¬M A. We give applications of these reflection principles to Löwenheim-Skolem theorems of extensions of first order logic. For example, Max(C) implies that the Löwenheim number of the extension of first order logic by the Härtig quantifier is less than 2 ω . * MSC2000 classification numbers 03E50, 03E35 and 03E40. Keywords: Reflections principle, Martin's axiom, Boolean valued models.† The second author has for many years been unable to reach the first author to get his approval of the final manuscript. Therefore the second author takes all resposibility for any errors or shortcomings of the paper.One of the fundamental properties of the universe of sets is the fact that HC = {x | x is hereditarily countable} reflects all Σ 1 -properties, that is, if a ∈ HC and P (a) is a true Σ 1 -property of a then HC |= P (a). If 2 ω > ω 1 , there is an interesting variant of HC:The basic observation underlying this paper is that while HC trivially reflects all Σ 1properties, it may, in a suitable model of set theory, reflect much more. Typically, it may reflect all properties which are Σ 1 with respect to the class of all cardinals.The strongest and perhaps the most interesting reflection principle to be considered is the following: Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x 1 , . . . , x n ) is a property which is provably persistent in extensions by elements of B, then R(a 1 , . . . , a n ) holds whenever a 1 , . . . , a n ∈ HC and R(a 1 , . . . , a n ) has a positive IB-value for some IB ∈ B.The relevant classes B to be considered here are the class C of all Cohen algebras (for exploding 2 ω ) and the class CCC of all CCC algebras. The main results are: 1. Con(ZF ) ↔ Con(ZF C + Max(C)). 2. Con(ZF + Levy schema) ↔ Con(ZF C + Max(CCC)). Max(CCC) → M A.The principle Max(C) is inconsistent with M A and hence inconsistent with Max(CCC). Thus we have two mutually inconsistent reflection principles which both make HC reflect all properties Σ 1 with respect to the class of all cardinals.Abstract logics relevant in the applications of these reflection principles are logics the satisfaction relation of which is preserved by C or CCC-extensions. An example of such a logic is the logic L(I) with the Härtig-quantifier IxyA(x)B(y) ↔ |{ a | A(a) }| = |{ b | B(b) }|...

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Section: Nowmentioning

confidence: 95%

Section: Cohen-extensionsmentioning

confidence: 99%

Reflection principles for the continuum

Stavi¹,

Väänánen²

2002

Logic and Algebra

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“…P r o o f. In [10], Jensen considered the generic Kurepa hypothesis GKH:= "there is a c.c.c. forcing that forces the Kurepa hypothesis KH in the generic extension".…”

Section: Corollary 94 Assume That There Is a Mahlo Cardinal Then Thmentioning

confidence: 99%

“…Jensen's constructions use ♦ and , cf. [3] or [10]. In [6, Section 5] Jech gives a forcing that adjoins a so-called Souslin mess.…”

Section: Corollary 94 Assume That There Is a Mahlo Cardinal Then Thmentioning

confidence: 99%

Chain homogeneous Souslin algebras

Scharfenberger-Fabian

2011

Mathematical Logic Quarterly

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“…Also this property has several different names, e.g. full (Jensen, Todorçević, [Jen,Tod84]) or 'Souslin and all derived trees Souslin' (Abraham and Shelah, [AS85, AS93]). In the context of [FH09] (cf.…”

Section: Strongly Homogeneous and Free Souslin Treesmentioning

confidence: 99%

Optimal matrices of partitions and an application to Souslin trees

Scharfenberger-Fabian

2010

Fund. Math.

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The basic result of this note is a statement about the existence of families of partitions of the set of natural numbers with some favourable properties, the n-optimal matrices of partitions. We use this to improve a decomposition result for strongly homogeneous Souslin trees. The latter is in turn applied to separate strong notions of rigidity of Souslin trees thereby answering a considerable portion of a question of Fuchs and Hamkins.2000 Mathematics Subject Classification. Primary 03E05, Secondary 05A18.

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Results on the Generic Kurepa Hypothesis

Cited by 17 publications

References 5 publications

Reflection principles for the continuum

Reflection principles for the continuum

Chain homogeneous Souslin algebras

Optimal matrices of partitions and an application to Souslin trees

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