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Arthur W Apter - One of the best experts on this subject based on the ideXlab platform.
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An equiConsistency for universal indestructibility
Journal of Symbolic Logic, 2020Co-Authors: Arthur W Apter, Grigor SargsyanAbstract:We obtain an equiConsistency for a weak form of universal indestructibility for strongness. The equiConsistency is relative to a cardinal weaker in Consistency Strength than a Woodin cardinal, Stewart Baldwin’s notion of hyperstrong cardinal. We also brie∞y indicate how our methods are applicable to universal indestructibility for supercompactness and strong compactness.
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On the Consistency Strength of level by level inequivalence
Archive for Mathematical Logic, 2017Co-Authors: Arthur W ApterAbstract:We show that the theories “ZFC $$+$$ + There is a supercompact cardinal” and “ZFC $$+$$ + There is a supercompact cardinal $$+$$ + Level by level inequivalence between strong compactness and supercompactness holds” are equiconsistent.
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A remark on the tree property in a choiceless context
Archive for Mathematical Logic, 2011Co-Authors: Arthur W ApterAbstract:We show that the Consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the Consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the Consistency of the theory “ $${{\rm ZF} + \neg{\rm AC}_\omega}$$ + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from hypotheses stronger in Consistency Strength than a supercompact limit of supercompact cardinals. A lower bound in Consistency Strength is provided by a result of Busche and Schindler, who showed that the Consistency of the theory “ZF + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” implies the Consistency of AD^ L ( R ).
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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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sandwiching the Consistency Strength of two global choiceless cardinal patterns
Bulletin of The Polish Academy of Sciences Mathematics, 2009Co-Authors: Arthur W ApterAbstract:We provide upper and lower bounds in Consistency Strength for the theories “ZF + ¬ACω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω” and “ZF + ¬ACω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω1”. In particular, our models for both of these theories satisfy “ZF + ¬ACω + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal”. There are many instances in the literature where choiceless large cardinal patterns are initially forced from strong hypotheses which one later sees can be weakened somewhat. For example, it is shown in [4] that the models constructed in [11] from an almost huge cardinal can actually be ∗2000 Mathematics Subject Classifications: 03E25, 03E35, 03E45, 03E55. †
Peter Koepke - One of the best experts on this subject based on the ideXlab platform.
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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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Making all cardinals almost Ramsey
Archive for Mathematical Logic, 2008Co-Authors: Arthur W Apter, Peter KoepkeAbstract:We examine combinatorial aspects and Consistency Strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + $${\neg {\rm AC}_\omega}$$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiConsistency of the following theories: “ZFC + There is a proper class of regular almost Ramsey cardinals”, and “ZF + DC + All infinite cardinals except possibly successors of singular cardinals are almost Ramsey”.
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forcing a mutual stationarity property in cofinality
Proceedings of the American Mathematical Society, 2007Co-Authors: Peter KoepkeAbstract:We show that the Consistency Strength, relative to the system ZFC, of the mutual stationarity property MS(N 3 , N 5 , N 7 ,...;ω 1 ) is equal to the existence of one measurable cardinal. We also discuss mutual stationarity for some other configurations of small cardinal parameters.
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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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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.
D L Miller - One of the best experts on this subject based on the ideXlab platform.
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Research methods for promotion of lung health
International Journal of Tuberculosis and Lung Disease, 2004Co-Authors: Donald A. Enarson, S. M. Kennedy, D L MillerAbstract:The goal of health research is to develop new knowledge for action to improve health. Relevant health research includes study of diseases, their causes and their treatment/prevention as well as structure and utilisation of health services and policies to improve lung health. As resources are not infinite, priorities must be established. These are determined by the relative frequency of a condition, how much dysfunction or disability it produces and whether there are cost-effective means to deal with it. Epidemiology is the discipline used to address these issues. Using it, one can describe the distribution and relative importance of a condition (the descriptive study), identify determinants and define its natural history (the analytical study), assess methods of prevention, cure and amelioration (the experimental study) and evaluate the process and outcome of services (health services or operational research). Epidemiology addresses itself to determining causation among associated variables. Characteristics associated with causation include Consistency, Strength of association, specificity, dose response, temporal relationship, coherence, and experimental evidence. Epidemiology can truly be described as the 'basic science of public health.' When used strategically, it can create the new knowledge that is the cornerstone for improving the health of the whole population.
Roger Bosch - One of the best experts on this subject based on the ideXlab platform.
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Proper forcing extensions and Solovay models
Archive for Mathematical Logic, 2004Co-Authors: Joan Bagaria, Roger BoschAbstract:We study the preservation of the property of being a Solovay model under proper projective forcing extensions. We show that every strongly-proper forcing notion preserves this property. This yields that the Consistency Strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a -Mahlo cardinal. We also show that the Consistency Strength of the absoluteness of under forcing extensions with σ-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.
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Proper forcing extensions and Solovay models
Archive for Mathematical Logic, 2004Co-Authors: Joan Bagaria, Roger BoschAbstract:We study the preservation of the property of Open image in new window being a Solovay model under proper projective forcing extensions. We show that every Open image in new window strongly-proper forcing notion preserves this property. This yields that the Consistency Strength of the absoluteness of Open image in new window under Open image in new window strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of Open image in new window under projective strongly-proper forcing notions is consistent relative to the existence of a Open image in new window-Mahlo cardinal. We also show that the Consistency Strength of the absoluteness of Open image in new window under forcing extensions with σ-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.
Donald A. Enarson - One of the best experts on this subject based on the ideXlab platform.
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Research methods for promotion of lung health
International Journal of Tuberculosis and Lung Disease, 2004Co-Authors: Donald A. Enarson, S. M. Kennedy, D L MillerAbstract:The goal of health research is to develop new knowledge for action to improve health. Relevant health research includes study of diseases, their causes and their treatment/prevention as well as structure and utilisation of health services and policies to improve lung health. As resources are not infinite, priorities must be established. These are determined by the relative frequency of a condition, how much dysfunction or disability it produces and whether there are cost-effective means to deal with it. Epidemiology is the discipline used to address these issues. Using it, one can describe the distribution and relative importance of a condition (the descriptive study), identify determinants and define its natural history (the analytical study), assess methods of prevention, cure and amelioration (the experimental study) and evaluate the process and outcome of services (health services or operational research). Epidemiology addresses itself to determining causation among associated variables. Characteristics associated with causation include Consistency, Strength of association, specificity, dose response, temporal relationship, coherence, and experimental evidence. Epidemiology can truly be described as the 'basic science of public health.' When used strategically, it can create the new knowledge that is the cornerstone for improving the health of the whole population.