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Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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A small ultrafilter number at smaller Cardinals
Archive for Mathematical Logic, 2020Co-Authors: Dilip Raghavan, Saharon ShelahAbstract:It is proved to be consistent relative to a Measurable Cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact Cardinal that there is a uniform ultrafilter on $${\aleph }_{\omega +1}$$ ℵ ω + 1 which is generated by fewer than $${2}^{{\aleph }_{\omega +1}}$$ 2 ℵ ω + 1 sets.
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constructing regular ultrafilters from a model theoretic point of view
Transactions of the American Mathematical Society, 2015Co-Authors: M Malliaris, Saharon ShelahAbstract:This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultralters which aect saturation of unstable theories: the lower conality lcf(@0,D) of @0 modulo D, saturation of the minimum unstable theory (the random graph), exi- bility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultralters thus assume a Measurable car- dinal. The main results are as follows. First, we investigate the strength of exibility, known to be detected by non-low theories. Assuming� > @0 is Measurable, we construct a regular ultralter on � � 2 � which is exible but not good, and which moreover has large lcf(@0) but does not even saturate models of the random graph. This implies (a) that exibility alone cannot characterize saturation of any theory, however (b) by separating exibility from goodness, we remove a main obstacle to proving non-low does not imply maximal. Since exible is precisely OK, this also shows that (c) from a set-theoretic point of view, consistently, ok need not imply good, addressing a prob- lem from Dow 1985. Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for D regular onand M a model of an unstable theory, M � /D is not (2 � ) + -saturated; and for M a model of a non-simple theory and � = � <� , M � /D is not � ++ -saturated. In the third part of the paper, we investigate realization and omission of symmetric cuts, signicant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on SOP2. We prove that if D is a �-complete ultralter on �, any ultrapower of a suciently saturated model of linear or der will have no (�,�)-cuts, and that ifD is also normal, it will have a (� + ,� + )-cut. We apply this to prove that for any n < !, assuming the existence of n Measurable Cardinals below �, there is a regular ultralter D onsuch that any D-ultrapower of a model of linear order will haven alternations of cuts, as dened below. Moreover, D will � + -saturate all stable theories but will not (2 � ) + -saturate any unstable theory, whereis the smallest Measurable Cardinal used in the construction.
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model theoretic properties of ultrafilters built by independent families of functions
Journal of Symbolic Logic, 2014Co-Authors: Mary Malliaris, Saharon ShelahAbstract:Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite Cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regular ultrafilter on λ (i.e., a point after which all antichains of have Cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filter on λ so that any ultrafilter extending fails to -saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a Measurable Cardinal κ, we construct a regular ultrafilter on which is λ-flexible but not -good, improving our previous answer to a question raised in Dow (1985). Third, assuming a weakly compact Cardinal κ, we construct an ultrafilter to show that may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory.
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COMPLETE QUOTIENT BOOLEAN ALGEBRAS
2013Co-Authors: Akihiro Kanamori, Saharon ShelahAbstract:Abstract. For / a proper, countably complete ideal on the power set â"(X) for some set X, can the quotient Boolean algebra ^(X)/! be complete? We first show that, if the Cardinality of X is at least u¡-¡, then having completeness implies the existence of an inner model with a Measurable Cardinal. A well-known situation that entails completeness is when the ideal / is a (nontrivial) ideal over a Cardinal k which is /c+-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over k = iO \ relative to the existence of a Woodin Cardinal. Augmenting his proof by interlacing forcings that adjoin Boolean suprema, we establish, relative to the same large Cardinal hypothesis, the consistency of: 2<ui = «3 and there is an ideal ideal / over u> \ suchthat â°((Oi)II is complete. (The Cardinality assertion implies that there is no ideal over w \ which is a>2-saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.) For / a proper, countably complete ideal on ¿P(X) for some set X, ca
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More on real-valued Measurable Cardinals and forcing with ideals
2012Co-Authors: Moti Gitik, Saharon ShelahAbstract:(1) It is shown that if c is real-valued Measurable then the Maharam type of (c, P(c), σ) is 2c. This answers a question of D. Fremlin [Fr,(P2f)]. (2) A different construction of a model with a real-valued Measurable Cardinal is given from that of R. Solovay [So]. This answers a question of D. Fremlin [Fr,(P1)]. (3) The forcing with a κ-complete ideal over a set X, |X | ≥ κ cannot be isomorphic to Random×Cohen or Cohen×Random. The result for X = κ was proved in [Gi-Sh1] but as was pointed out to us by M. Burke the application of it in [Gi-Sh2] requires dealing with any X.In Section 1 we deal with the Maharam types of real-valued Measurable Cardinals. The result (1) stated in the abstract and its stronger version are proved. The proofs are based on Shelah’s strong covering lemmas and his revised power set operation. In Section 2 a model with a real-valued Measurable which is not obtained as the Solovay one by forcing random reals over a model with a Measurable. In Section 3, the result (3) stated in the abstract is proved. Theorem 1.1 and the construction of Section 2 is due to the first author. Theorem 1.2 is joint and the result of Section 3 is due to the second author. We are grateful to David Fremlin for bringing the questions on real-valued measurability to our attention. His excellent survey article [Fr] gave the inspiration for the present paper. We wish to thank the Max Burke for pointing out a missing stage in the argument of [Gi-Sh2]
Peter Koepke - One of the best experts on this subject based on the ideXlab platform.
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A MINIMAL PRIKRY-TYPE FORCING FOR SINGULARIZING A Measurable Cardinal
2015Co-Authors: Peter Koepke, Philipp SchlichtAbstract:Abstract. Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by Pp휔q{finite. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a Measurable Cardinal but which is minimal, i.e. there are no intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality of the associated measures. §1. Introduction. The classical Prikry forcing first appeared in Prikry’s dis-sertation [9] in 1970. It gave a positive answer the following question of Silver and Solovay: Is there a forcing preserving all Cardinals while some cofinality changes? In fact, the singularization of regular Cardinals by some forcing is necessaril
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The consistency strength of choiceless failures of SCH
2015Co-Authors: Arthur W Apter, Peter KoepkeAbstract:We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one Measurable Cardinal, including a surjective failure of Shelah’s pcf theorem about the size of the power set of ℵω. Using symmetric collapses t
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Embedding Normal Forms and ¦11{Determinacy
2014Co-Authors: Peter KoepkeAbstract:We give a new proof of D. Martin's theorem that ¦11{sets of re-als are determined if there is a Measurable Cardinal. The argument is based on representing ¦11{sets using systems of elementary embed-dings of models of set theory. 1 Games on Trees We consider games whose positions are ¯nite sequences and where two players called I and II alternately try to lengthen a position by one move. Thereby, they determine a maximal path through the tree of positions. Player I's aim is to get this path into a ¯xed winning set while player II tries to prevent this. Accordingly we de¯ne: A tree is a nonempty set of ¯nite sequences, T µ <!V, closed under the formation of initial segments. T is partially ordered by µ. A path through T is a sequence p of length · ! such that 8n <! : p " n 2 T; p is maximal if there is no path through T properly extending p. A maxima
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Forcing a mutual stationarity property in cofinality ω1
2014Co-Authors: Peter Koepke, Communicated Julia KnightAbstract:Abstract. We show that the consistency strength, relative to the system ZFC, of the mutual stationarity property MS(ℵ3,ℵ5,ℵ7,...;ω1) is equal to the existence of one Measurable Cardinal. We also discuss mutual stationarity for some other configurations of small Cardinal parameters. 1
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the consistency strength of choiceless failures of sch
Journal of Symbolic Logic, 2010Co-Authors: Arthur W Apter, Peter KoepkeAbstract:We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis(SCH)inthesettingoftheZermelo-FraenkelaxiomsystemZFwithouttheAxiomofChoice (AC). BythenewnotionofparallelPrikryforcingthatweintroduce,weobtainsurjectivefailuresof SCH using only one Measurable Cardinal, including a surjectivefailure of Shelah's pcf theorem about the size ofthepower setof @u. Usingsymmetric collapsesto @u, @u1,or @u2,weshowthat injective failuresat @u, @u1,or @u2 canhaverelatively mildconsistency strengths intermsofMitchellorders ofMeasurable Cardinals. InjectivefailuresofboththeaforementionedtheoremofShelahandSilver'stheoremthatGCH cannotfirst fail at a singular strong limit Cardinal of uncountable cofinality are also obtained. Lower boundsareshownbycoremodeltechniquesandmethodsduetoGitikandMitchell.
Arthur W Apter - One of the best experts on this subject based on the ideXlab platform.
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Indestructibility, Instances of Strong Compactness, and Level by Level Inequivalence
2015Co-Authors: Arthur W ApterAbstract:Suppose λ> κ is Measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is Measurable, yet δ is neither δ+ strongly compact nor a limit of Measurable Cardinals} must be unbounded in κ. The large Cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which A = ∅. The first of these contains a supercompact Cardinal κ and is such that no Cardinal δ> κ is Measurable, κ’s supercompactness is indestructible under κ-directed closed, (κ+,∞)-distributive forcing, and every Measurable Cardinal δ < κ is δ+ strongly compact. The second of these contains a strong Cardinal κ and is such that no Cardinal δ> κ is Measurable, κ’s strongness is indestructible under <κ-strategically closed, (κ+,∞)-distributive forcing, and level by level inequivalence between strong compactness and supercompactness holds. The model from the first of our forcing constructions is used to show that it is consistent, relative to a supercompact Cardinal, for the least Cardinal κ which is both strong and has its strongness indestructible under κ-directed closed, (κ+,∞)-distributive forcing to be the same as the least supercompact Cardinal, which has its supercompactness indestructible unde
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The consistency strength of choiceless failures of SCH
2015Co-Authors: Arthur W Apter, Peter KoepkeAbstract:We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one Measurable Cardinal, including a surjective failure of Shelah’s pcf theorem about the size of the power set of ℵω. Using symmetric collapses t
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the consistency strength of choiceless failures of sch
Journal of Symbolic Logic, 2010Co-Authors: Arthur W Apter, Peter KoepkeAbstract:We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis(SCH)inthesettingoftheZermelo-FraenkelaxiomsystemZFwithouttheAxiomofChoice (AC). BythenewnotionofparallelPrikryforcingthatweintroduce,weobtainsurjectivefailuresof SCH using only one Measurable Cardinal, including a surjectivefailure of Shelah's pcf theorem about the size ofthepower setof @u. Usingsymmetric collapsesto @u, @u1,or @u2,weshowthat injective failuresat @u, @u1,or @u2 canhaverelatively mildconsistency strengths intermsofMitchellorders ofMeasurable Cardinals. InjectivefailuresofboththeaforementionedtheoremofShelahandSilver'stheoremthatGCH cannotfirst fail at a singular strong limit Cardinal of uncountable cofinality are also obtained. Lower boundsareshownbycoremodeltechniquesandmethodsduetoGitikandMitchell.
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the consistency strength of aleph_ omega and aleph_ omega _1 being rowbottom Cardinals without the axiom of choice
Archive for Mathematical Logic, 2006Co-Authors: Arthur W Apter, Peter KoepkeAbstract:We show that for all natural numbers n, the theory “ZF + DC $$_{\aleph_n}$$ + $$\aleph_{\omega}$$ is a Rowbottom Cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a Measurable Cardinal”. In addition, we show that the theory “ZF + $$\aleph_{\omega_1}$$ is an ω 2-Rowbottom Cardinal carrying an ω 2-Rowbottom filter and ω 1 is regular” has the same consistency strength as the theory “ZFC + There exist ω 1 Measurable Cardinals”. We also discuss some generalizations of these results.
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blowing up the power set of the least Measurable
Journal of Symbolic Logic, 2002Co-Authors: Arthur W Apter, James CummingsAbstract:We prove some results related to the problem of blowing up the power set of the least Measurable Cardinal. Our forcing results improve those of [1] by using the optimal hypothesis. ?
Golshani Mohammad - One of the best experts on this subject based on the ideXlab platform.
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Changing Measurable into small accessible Cardinals
2019Co-Authors: Golshani MohammadAbstract:We give a detailed proof of the properties of the usual Prikry type forcing notion for turning a Measurable Cardinal into $\aleph_\omega$
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The tree property at all regular even Cardinals
2018Co-Authors: Golshani MohammadAbstract:Assuming the existence of a strong Cardinal and a Measurable Cardinal above it, we construct a model of $ZFC$ in which for every singular Cardinal $\delta$, $\delta$ is strong limit, $2^\delta=\delta^{+3}$ and the tree property at $\delta^{++}$ holds. This answers a question of Friedman, Honzik and Stejskalova [8]. We also produce, relative to the existence of a strong Cardinal and two Measurable Cardinals above it, a model of $ZFC$ in which the tree property holds at all regular even Cardinals. The result answers questions of Friedman-Halilovic [5] and Friedman-Honzik [6].Comment: Comments are welcom
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The tree property at double successors of singular Cardinals of uncountable cofinality
2017Co-Authors: Golshani Mohammad, Mohammadpour RahmanAbstract:Assuming the existence of a strong Cardinal $\kappa$ and a Measurable Cardinal above it, we force a generic extension in which $\kappa$ is a singular strong limit Cardinal of any prescribed cofinality, and such that the tree property holds at $\kappa^{++}$.Comment: 16 page
Sydavid Friedman - One of the best experts on this subject based on the ideXlab platform.
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LARGE CardinalS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH
2015Co-Authors: Sydavid Friedman, Peter Holy, Philipp LückeAbstract:Abstract. This paper deals with the question whether the assumption that for every inaccessible Cardinal κ there is a well-order of H(κ+) definable over the structure 〈H(κ+),∈ 〉 by a formula without parameters is consistent with the existence of (large) large Cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the i-function and construct a class forcing that adds such a well-order at every inaccessible Cardinal and preserves ZFC, all cofinalities, the continuum function and all supercompact Cardinals. Even in the absence of a proper class of inaccessible Cardinals, this forcing produces a model of “V = HOD ” and can therefore be used to force this axiom while preserving large Cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong Cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a Measurable Cardinal. 1
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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