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Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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Double weakness
Acta Mathematica Hungarica, 2021Co-Authors: S. Garti, Saharon ShelahAbstract:We prove that, consistently, there exists a weakly but not Strongly Inaccessible Cardinal $$\lambda$$ λ for which the sequence $$\langle 2^\theta:\theta
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Not collapsing Cardinals ≤ κ in (< κ)-support iterations: Part II
2012Co-Authors: Saharon ShelahAbstract:We deal with the problem of preserving various versions of completeness in (< κ)–support iterations of forcing notions, generalizing the case “S–complete proper is preserved by CS iterations for a stationary costationary S ⊆ ω1”. We give applications to Uniformization and the Whitehead problem. In particular, for a Strongly Inaccessible Cardinal κ and a stationary set S ⊆ κ with fat complement we can have uniformization for 〈Aδ: δ ∈ S ′ 〉, Aδ ⊆ δ = supAδ, cf(δ) = otp(Aδ) and a stationary non-reflecting set S ′ ⊆ S
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ORDER POLYNOMIALLY COMPLETE LATTICES MUST BE LARGE
2011Co-Authors: Martin Goldstern, Saharon ShelahAbstract:Abstract. If L is an o.p.c. (order polynomially complete) lattice, then the Cardinality of L is a Strongly Inaccessible Cardinal. In particular, the existence of o.p.c. lattices is not provable in ZFC, not even from ZFC+GCH
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ON CON(dλ> COVλ(MEAGRE))
2011Co-Authors: Saharon ShelahAbstract:Abstract. We prove the consistency of: for suitable Strongly Inaccessible Cardinal λ the dominating number, i.e., the cofinality of λ λ, is strictly bigger than covλ(meagre), i.e. the minimal number of nowhere dense subsets of λ 2 needed to cover it. This answers a question of Matet. 945 revision:2010-01-15 modified:2010-01-17 Cardinal invariants on the continuum have a long tradition of research. For a topologist, it can be viewed as investigating the space β(ω), the Stone Čzec
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Possible Size of an Ultrapower of ω 1
2011Co-Authors: Renling Jin, Saharon ShelahAbstract:Let ω be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In §1 we show that, assuming the consistency of a supercompact Cardinal, there may exist an ultrapower of ω, whose Cardinality is (1) a singular strong limit Cardinal, (2) a Strongly Inaccessible Cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In §2 we construct several λ-Archimedean ultrapowers of ω under some large Cardinal assumptions. For example, we show that, assuming the consistency of a measurable Cardinal, there may exist a λ-Archimedean ultrapower of ω for some uncountable Cardinal λ. This answers a question in [8], modulo the assumption of measurability. 0. On Notation and Boolean Algebras An important way of constructing a desired ultrafilter on κ is to use the construction of an ultrafilter E on the Boolean algebra B = P(κ)/D for some filter D on κ as an intermediate step. The construction of E has a great deal of flexibility when B 626 revision:1998-01-12 modified:1998-01-13 contains a large free (or κ-free) subalgebra. In this paper we always construct an ultrafilter E on B first such that ω B /E, the Boolean ultrapower of ω modulo E, has some desired properties, and then use E to define an ultrafilter F on κ so that ω κ /F, the ultrapower of ω modulo F, is isomorphic to ω B /E. In each case a large Cardinal is used to construct D so that B = P(κ)/D always contains a large free (or κ-free) subalgebra
Shelah Saharon - One of the best experts on this subject based on the ideXlab platform.
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On CON(${\mathfrak d}_\lambda >$ cov$_\lambda$(meagre))
2020Co-Authors: Shelah SaharonAbstract:We prove the consistency of: for suitable Strongly Inaccessible Cardinal lambda the dominating number, i.e., the cofinality of ^{lambda}lambda, is strictly bigger than cov_lambda(meagre), i.e. the minimal number of nowhere dense subsets of ^{lambda}2 needed to cover it. This answers a question of Matet.Comment: The paper was multiply submitted by mistake. Correct number arXiv:0904.081
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Can a small forcing create Kurepa trees
Published by Elsevier B.V., 1997Co-Authors: Jin Renling, Shelah SaharonAbstract:AbstractIn this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In Section 1 we show that in the Lévy model obtained by collapsing all Cardinals between ω1 and a Strongly Inaccessible Cardinal by forcing with a countable support Lévy collapsing order, many ω1-preserving forcing notions of size at most ω1 including all ω-proper forcing notions and some proper but not ω-proper forcing notions of size at most ω1 do not create Kurepa trees. In Section 2 we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions
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Not collapsing Cardinals
1997Co-Authors: Shelah SaharonAbstract:We deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S subseteq omega_1''. We give applications to Uniformization and the Whitehead problem. In particular, for a Strongly Inaccessible Cardinal kappa and a stationary set S subseteq kappa with fat complement we can have uniformization for , A_delta subseteq delta = sup A_delta, cf(delta)=otp(A_delta) and a stationary non-reflecting set S' subseteq S
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Order polynomially complete lattices must be LARGE
1997Co-Authors: Goldstern Martin, Shelah SaharonAbstract:If L is an order polynomially complete lattice, (that is: every monotone function from L^n to L is induced by a lattice-theoretic polynomial) then the Cardinality of L is a Strongly Inaccessible Cardinal. In particular, the existence of such lattices is not provable in ZFC, nor from ZFC+GCH. Although the problem originates in algebra, the proof is purely set-theoretical. The main tools are partition and canonisation theorems. It is still open if the existence of infinite o.p.c. lattices can be refuted in ZFC.Comment: This is paper number GoSh:633 in Shelah's list. The paper is to appear in Algebra Universalis
Renling Jin - One of the best experts on this subject based on the ideXlab platform.
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Possible Size of an Ultrapower of ω 1
2011Co-Authors: Renling Jin, Saharon ShelahAbstract:Let ω be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In §1 we show that, assuming the consistency of a supercompact Cardinal, there may exist an ultrapower of ω, whose Cardinality is (1) a singular strong limit Cardinal, (2) a Strongly Inaccessible Cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In §2 we construct several λ-Archimedean ultrapowers of ω under some large Cardinal assumptions. For example, we show that, assuming the consistency of a measurable Cardinal, there may exist a λ-Archimedean ultrapower of ω for some uncountable Cardinal λ. This answers a question in [8], modulo the assumption of measurability. 0. On Notation and Boolean Algebras An important way of constructing a desired ultrafilter on κ is to use the construction of an ultrafilter E on the Boolean algebra B = P(κ)/D for some filter D on κ as an intermediate step. The construction of E has a great deal of flexibility when B 626 revision:1998-01-12 modified:1998-01-13 contains a large free (or κ-free) subalgebra. In this paper we always construct an ultrafilter E on B first such that ω B /E, the Boolean ultrapower of ω modulo E, has some desired properties, and then use E to define an ultrafilter F on κ so that ω κ /F, the ultrapower of ω modulo F, is isomorphic to ω B /E. In each case a large Cardinal is used to construct D so that B = P(κ)/D always contains a large free (or κ-free) subalgebra
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Possible Size of an Ultrapower of ω
2007Co-Authors: Renling Jin, Saharon ShelahAbstract:Let ω be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In §1 we show that, assuming the consistency of a supercompact Cardinal, there may exist an ultrapower of !, whose Cardinality is (1) a singular strong limit Cardinal, (2) a Strongly Inaccessible Cardinal. This answers two questions in #1#, modulo the assumption of supercompactness. In §2 we construct several #-Archimedean ultrapowers of ω under some large Cardinal assumptions. For example, we show that, assuming the consistency of a measurable Cardinal, there may exist a #-Archimedean ultrapower of ω for some uncountable Cardinal #. This answers a question in #8#, modulo the assumption of measurability
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Can a Small Forcing Create Kurepa Trees
2007Co-Authors: Renling Jin, Saharon ShelahAbstract:In the paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees byan! 1 -preserving forcing notion of size at most ! 1 . In the #rst section we show that in the L#evy model obtained by collapsing all Cardinals between ! 1 and a Strongly Inaccessible Cardinal by forcing with a countable support L#evy collapsing order, many ! 1 -preserving forcing notions of size at most ! 1 including all !-proper forcing notions and some proper but not !-proper forcing notions of size at most ! 1 do not create Kurepa trees. In the second section we construct a model of CH plus no Kurepa trees, in which there is an !-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions. 0 Introduction By a model we mean a model of ZFC. By a forcing notion we mean a separative partially ordered set P with a largest element 1 P ..
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Can a Small Forcing Create Kurepa Trees
2002Co-Authors: Renling Jin, Saharon ShelahAbstract:In the paper we probe the possibilities of creating a Kurepa tree in a generic extension of a model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In the first section we show that in the Lévy model obtained by collapsing all Cardinals between ω1 and a Strongly Inaccessible Cardinal by forcing with a countable support Lévy collapsing order many ω1preserving forcing notions of size at most ω1 including all ω-proper forcing notions and some proper but not ω-proper forcing notions of size at most ω1 do not create Kurepa trees. In the second section we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions
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Can a small forcing create Kurepa trees
arXiv: Logic, 1995Co-Authors: Renling Jin, Saharon ShelahAbstract:In the paper we probe the possibilities of creating a Kurepa tree in a generic extension of a model of CH plus no Kurepa trees by an omega_1-preserving forcing notion of size at most omega_1. In the first section we show that in the Levy model obtained by collapsing all Cardinals between omega_1 and a Strongly Inaccessible Cardinal by forcing with a countable support Levy collapsing order many omega_1-preserving forcing notions of size at most omega_1 including all omega-proper forcing notions and some proper but not omega-proper forcing notions of size at most omega_1 do not create Kurepa trees. In the second section we construct a model of CH plus no Kurepa trees, in which there is an omega-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions.
Sydavid Friedman - One of the best experts on this subject based on the ideXlab platform.
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THE CONSISTENCY STRENGTH OF THE TREE PROPERTY AT THE DOUBLE SUCCESSOR OF A MEASURABLE
2014Co-Authors: Natasha Dobrinen, Sydavid FriedmanAbstract:Abstract. The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable Cardinal and the tree property holds at κ++; (2) κ is a weakly compact hypermeasurable Cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable Cardinal and a measurable Cardinal far enough above it, then there is an inner model in which there is a proper class of measurable Cardinals, and in which the tree property holds at the double successor of each Strongly Inaccessible Cardinal. If 0 # exists, then we can construct an inner model in which the tree property holds at the double successor of each Strongly inacces-sible Cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double succes-sor of a measurable Cardinal. The upper and lower bounds differ only by 1 in the Mitchell order. 1
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the consistency strength of the tree property at the double successor of a measurable cardina
Fundamenta Mathematicae, 2010Co-Authors: Natasha Dobrinen, Sydavid FriedmanAbstract:The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable Cardinal and the tree property holds at κ; (2) κ is a weakly compact hypermeasurable Cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable Cardinal and a measurable Cardinal far enough above it, then there is an inner model in which there is a proper class of measurable Cardinals, and in which the tree property holds at the double successor of each Strongly Inaccessible Cardinal. If 0 exists, then we can construct an inner model in which the tree property holds at the double successor of each Strongly Inaccessible Cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double successor of a measurable Cardinal. The upper and lower bounds differ only by 1 in the Mitchell order.
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The consistency strength of the tree property at the double successor of a measurable, preprint
2008Co-Authors: Natasha Dobrinen, Sydavid FriedmanAbstract:Abstract. The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable Cardinal and the tree property holds at κ++; (2) κ is a weakly compact hypermeasurable Cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable Cardinal and a measurable Cardinal far enough above it, then there is an inner model in which there is a proper class of measurable Cardinals, and in which the tree property holds at the double successor of each Strongly Inaccessible Cardinal. If 0 # exists, then we can construct an inner model in which the tree property holds at the double successor of each Strongly inacces-sible Cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double succes-sor of a measurable Cardinal. The upper and lower bounds differ only by 1 in the Mitchell order. 1
Michael Rathjen - One of the best experts on this subject based on the ideXlab platform.
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Inaccessible Set Axioms May Have Little Consistency Strength
2007Co-Authors: Laura Crosilla, Michael RathjenAbstract:. The paper investigates Inaccessible set axioms and their consistency strength in constructive set theory. In ZFC Inaccessible sets are of the form V where is a Strongly Inaccessible Cardinal and V denotes the - th level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of Inaccessible set axioms heavily depends on the context in which they are embedded. The context here will be the theory CZF \Gamma of constructive Zermelo Fraenkel set theory but without 2 - Induction (foundation). Let INAC be the statement that for every set there is an Inaccessible set containing it. CZF \Gamma +INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF \Gamma + INAC also has a realizability interpretation in type theory whic..
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Inaccessible set axioms may have little consistency strength
Annals of Pure and Applied Logic, 2002Co-Authors: Laura Crosilla, Michael RathjenAbstract:Abstract The paper investigates Inaccessible set axioms and their consistency strength in constructive set theory. In ZFC Inaccessible sets are of the form V κ where κ is a Strongly Inaccessible Cardinal and V κ denotes the κ th level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of Inaccessible set axioms heavily depend on the context in which they are embedded. The context here will be the theory CZF − of constructive Zermelo–Fraenkel set theory but without ∈ -Induction (foundation). Let INAC be the statement that for every set there is an Inaccessible set containing it. CZF − + INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF − + INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF − + INAC is a small ordinal known as the Feferman–Schutte ordinal Γ 0 .
