53 found
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  1.  21
    On Certain Indestructibility of Strong Cardinals and a Question of Hajnal.Moti Gitik & Saharon Shelah - 1989 - Archive for Mathematical Logic 28 (1):35-42.
    A model in which strongness ofκ is indestructible under κ+ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of {λ δ |2 δ <λ}.
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  2.  12
    The Negation of the Singular Cardinal Hypothesis From o=K++.Moti Gitik - 1989 - Annals of Pure and Applied Logic 43 (3):209-234.
  3. On Changing Cofinality of Partially Ordered Sets.Moti Gitik - 2010 - Journal of Symbolic Logic 75 (2):641-660.
    It is shown that under GCH every poset preserves its confinality in any cofinality preserving extension. On the other hand, starting with ω measurable cardinals, a model with a partial ordered set which can change its cofinality in a cofinality preserving extension is constructed.
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  4.  33
    The Least Measurable Can Be Strongly Compact and Indestructible.Arthur W. Apter & Moti Gitik - 1998 - Journal of Symbolic Logic 63 (4):1404-1412.
    We show the consistency, relative to a supercompact cardinal, of the least measurable cardinal being both strongly compact and fully Laver indestructible. We also show the consistency, relative to a supercompact cardinal, of the least strongly compact cardinal being somewhat supercompact yet not completely supercompact and having both its strong compactness and degree of supercompactness fully Laver indestructible.
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  5.  14
    Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis.Moti Gitik & William J. Mitchell - 1996 - Annals of Pure and Applied Logic 82 (3):273-316.
    We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem. Suppose κ is a singular strong limit cardinal and 2κ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf > ω then it follows that o λ, and if cf = ωthen either o λ or {α: K o α+n} is confinal in κ for (...)
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  6. On the Strength of No Normal Precipitous Filter.Moti Gitik & Liad Tal - 2011 - Archive for Mathematical Logic 50 (1-2):223-243.
    We consider a question of T. Jech and K. Prikry that asks if the existence of a precipitous filter implies the existence of a normal precipitous filter. The aim of this paper is to improve a result of Gitik (Israel J Math, 175:191–219, 2010) and to show that measurable cardinals of a higher order rather than just measurable cardinals are necessary in order to have a model with a precipitous filter but without a normal one.
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  7.  7
    Blowing Up Power of a Singular Cardinal—Wider Gaps.Moti Gitik - 2002 - Annals of Pure and Applied Logic 116 (1-3):1-38.
    The paper is concerned with methods for blowing power of singular cardinals using short extenders. Thus, for example, starting with κ of cofinality ω with {α<κ oα+n} cofinal in κ for every n<ω we construct a cardinal preserving extension having the same bounded subsets of κ and satisfying 2κ=κ+δ+1 for any δ<1.
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  8.  12
    Some Applications of Supercompact Extender Based Forcings to Hod.Moti Gitik & Carmi Merimovich - 2018 - Journal of Symbolic Logic 83 (2):461-476.
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  9.  4
    Silver Type Theorems for Collapses.Moti Gitik - 2020 - Annals of Pure and Applied Logic 171 (9):102825.
    Let κ be a cardinal of cofinality \omega_1 witnessed by a club of cardinals (κ_\alpha | \alpha < \omega_1) . We study Silver's type effects of collapsing of κ^+_\alphas 's on κ^+ . A model in which κ^+_\alphas 's (and also κ^+) are collapsed on a stationary co-stationary set is constructed.
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  10.  5
    The Negation of the Singular Cardinal Hypothesis From< I> o(< I> K_)=< I> K< Sup>++.Moti Gitik - 1989 - Annals of Pure and Applied Logic 43 (3):209-234.
  11.  12
    The Strenght of the Failure of the Singular Cardinal Hypothesis.Moti Gitik - 1991 - Annals of Pure and Applied Logic 51 (3):215-240.
    We show that o = k++ is necessary for ¬SCH. Together with previous results it provides the exact strenght of ¬SCH.
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  12.  41
    Nonsplitting Subset of Κ.Moti Gitik - 1985 - Journal of Symbolic Logic 50 (4):881-894.
    Assuming the existence of a supercompact cardinal, we construct a model of ZFC + ). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes P ℵ 1 - V stationary.
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  13.  11
    On Measurable Cardinals Violating the Continuum Hypothesis.Moti Gitik - 1993 - Annals of Pure and Applied Logic 63 (3):227-240.
    Gitik, M., On measurable cardinals violating the continuum hypothesis, Annals of Pure and Applied Logic 63 227-240. It is shown that an extender used uncountably many times in an iteration is reconstructible. This together with the Weak Covering Lemma is used to show that the assumption o=κ+α is necessary for a measurable κ with 2κ=κ+α.
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  14. Short Extenders Forcings I.Moti Gitik - 2012 - Journal of Mathematical Logic 12 (2):1250009.
    The purpose of the present paper is to present new methods of blowing up the power of a singular cardinal κ of cofinality ω. New PCF configurations are obtained. The techniques developed here will be used in a subsequent paper to construct a model with a countable set which pcf has cardinality ℵ1.
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  15.  3
    Short Extenders Forcings – Doing Without Preparations.Moti Gitik - 2020 - Annals of Pure and Applied Logic 171 (5):102787.
    We introduce certain morass type structures and apply them to blowing up powers of singular cardinals. As a bonus, a forcing for adding clubs with finite conditions to higher cardinals is obtained.
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  16. A Model with a Measurable Which Does Not Carry a Normal Measure.Eilon Bilinsky & Moti Gitik - 2012 - Archive for Mathematical Logic 51 (7-8):863-876.
    We construct a model of ZF in which there is a measurable cardinal but there is no normal ultrafilter over it.
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  17.  34
    Extender Based Forcings.Moti Gitik & Menachem Magidor - 1994 - Journal of Symbolic Logic 59 (2):445-460.
    The paper is a continuation of [The SCH revisited]. In § 1 we define a forcing with countably many nice systems. It is used, for example, to construct a model "GCH below κ, c f κ = ℵ0, and $2^\kappa > \kappa^{+\omega}$" from 0(κ) = κ+ω. In § 2 we define a triangle iteration and use it to construct a model satisfying "{μ ≤ λ∣ c f μ = ℵ0 and $pp(\mu) > \lambda\}$ is countable for some λ". The question (...)
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  18.  52
    No Bound for the First Fixed Point.Moti Gitik - 2005 - Journal of Mathematical Logic 5 (02):193-246.
    Our aim is to show that it is impossible to find a bound for the power of the first fixed point of the aleph function.
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  19.  15
    On Almost Precipitous Ideals.Asaf Ferber & Moti Gitik - 2010 - Archive for Mathematical Logic 49 (3):301-328.
    With less than 0# two generic extensions ofL are identified: one in which ${\aleph_1}$ , and the other ${\aleph_2}$ , is almost precipitous. This improves the consistency strength upper bound of almost precipitousness obtained in Gitik M, Magidor M (On partialy wellfounded generic ultrapowers, in Pillars of Computer Science, 2010), and answers some questions raised there. Also, main results of Gitik (On normal precipitous ideals, 2010), are generalized—assumptions on precipitousness are replaced by those on ∞-semi precipitousness. As an application it (...)
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  20.  37
    Indestructible Strong Compactness but Not Supercompactness.Arthur W. Apter, Moti Gitik & Grigor Sargsyan - 2012 - Annals of Pure and Applied Logic 163 (9):1237-1242.
  21.  9
    Blowing Up the Power of a Singular Cardinal.Moti Gitik - 1996 - Annals of Pure and Applied Logic 80 (1):17-33.
  22.  2
    Some Constructions of Ultrafilters Over a Measurable Cardinal.Moti Gitik - forthcoming - Annals of Pure and Applied Logic:102821.
    Some non-normal κ-complete ultrafilters over a measurable κ with special properties are constructed. Questions by A. Kanamori [4] about infinite Rudin-Frolik sequences, discreteness and products are answered.
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  23.  24
    On Hidden Extenders.Moti Gitik - 1996 - Archive for Mathematical Logic 35 (5-6):349-369.
    We prove the following theorem: Suppose that there is a singular $\kappa$ with the set of $\alpha$ 's with $o(\alpha)=\alpha^{+n}$ unbounded in it for every $n < \omega$ . Then in a generic extesion there are two precovering sets which disagree about common indiscernibles unboundedly often.
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  24.  21
    The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly $${\Theta}$$ Θ -Supercompact.Brent Cody, Moti Gitik, Joel David Hamkins & Jason A. Schanker - 2015 - Archive for Mathematical Logic 54 (5-6):491-510.
    We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta}$$\end{document}-supercompact, for any desired θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta}$$\end{document}. In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or (...)
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  25.  24
    Possible Values for 2ℵn and 2ℵω.Moti Gitik & Carmi Merimovich - 1997 - Annals of Pure and Applied Logic 90 (1-3):193-241.
  26.  8
    On the Splitting Number at Regular Cardinals.Omer Ben-Neria & Moti Gitik - 2015 - Journal of Symbolic Logic 80 (4):1348-1360.
    Letκ, λ be regular uncountable cardinals such that λ >κ+is not a successor of a singular cardinal of low cofinality. We construct a generic extension withs = λ starting from a ground model in whicho = λ and prove that assuming ¬0¶,s = λ implies thato ≥ λ in the core model.
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  27.  14
    On Closed Unbounded Sets Consisting of Former Regulars.Moti Gitik - 1999 - Journal of Symbolic Logic 64 (1):1-12.
    A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with o(κ) = κ + 1. On the other hand, it is shown that the strength of above is at least o(κ) = κ.
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  28.  13
    Possible Values for 2 (Aleph N) and 2 (Aleph Omega).Moti Gitik & Carmi Merimovich - 1997 - Annals of Pure and Applied Logic 90 (1-3):193-241.
  29.  21
    Some Pathological Examples of Precipitous Ideals.Moti Gitik - 2008 - Journal of Symbolic Logic 73 (2):492 - 511.
    We construct a model with an indecisive precipitous ideal and a model with a precipitous ideal with a non precipitous normal ideal below it. Such kind of examples were previously given by M. Foreman [2] and R. Laver [4] respectively. The present examples differ in two ways: first- they use only a measurable cardinal and second- the ideals are over a cardinal. Also a precipitous ideal without a normal ideal below it is constructed. It is shown in addition that if (...)
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  30.  5
    On Gaps Under GCH Type Assumptions.Moti Gitik - 2003 - Annals of Pure and Applied Logic 119 (1-3):1-18.
    We prove equiconsistency results concerning gaps between a singular strong limit cardinal κ of cofinality 0 and its power under assumptions that 2κ=κ+δ+1 for δ<κ and some weak form of the Singular Cardinal Hypothesis below κ. Together with the previous results this basically completes the study of consistency strength of the various gaps between such κ and its power under GCH type assumptions below.
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  31.  13
    Non-Homogeneity of Quotients of Prikry Forcings.Moti Gitik & Eyal Kaplan - 2019 - Archive for Mathematical Logic 58 (5-6):649-710.
    We study non-homogeneity of quotients of Prikry and tree Prikry forcings with non-normal ultrafilters over some natural distributive forcing notions.
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  32.  73
    Approachability at the Second Successor of a Singular Cardinal.Moti Gitik & John Krueger - 2009 - Journal of Symbolic Logic 74 (4):1211 - 1224.
    We prove that if μ is a regular cardinal and ℙ is a μ-centered forcing poset, then ℙ forces that $(I[\mu ^{ + + } ])^V $ generates I[µ⁺⁺] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.
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  33.  19
    On Generic Elementary Embeddings.Moti Gitik - 1989 - Journal of Symbolic Logic 54 (3):700-707.
  34.  10
    On the Mitchell and Rudin-Kiesler Orderings of Ultrafilters.Moti Gitik - 1988 - Annals of Pure and Applied Logic 39 (2):175-197.
  35.  7
    On Non-Minimal P-Points Over a Measurable Cardinal.Moti Gitik - 1981 - Annals of Mathematical Logic 20 (3):269-288.
  36.  53
    A Cardinal Preserving Extension Making the Set of Points of Countable V Cofinality Nonstationary.Moti Gitik, Itay Neeman & Dima Sinapova - 2007 - Archive for Mathematical Logic 46 (5-6):451-456.
    Assuming large cardinals we produce a forcing extension of V which preserves cardinals, does not add reals, and makes the set of points of countable V cofinality in κ+ nonstationary. Continuing to force further, we obtain an extension in which the set of points of countable V cofinality in ν is nonstationary for every regular ν ≥ κ+. Finally we show that our large cardinal assumption is optimal.
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  37.  31
    Cardinal Preserving Ideals.Moti Gitik & Saharon Shelah - 1999 - Journal of Symbolic Logic 64 (4):1527-1551.
    We give some general criteria, when κ-complete forcing preserves largeness properties-like κ-presaturation of normal ideals on λ (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength "NS λ is ℵ 1 -preserving". for λ > ℵ 2.
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  38.  25
    On Some Configurations Related to the Shelah Weak Hypothesis.Moti Gitik & Saharon Shelah - 2001 - Archive for Mathematical Logic 40 (8):639-650.
    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.
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  39.  1
    Extender-Based Forcings with Overlapping Extenders and Negations of the Shelah Weak Hypothesis.Moti Gitik - forthcoming - Journal of Mathematical Logic.
    Extender-based Prikry–Magidor forcing for overlapping extenders is introduced. As an application, models with strong forms of negations of the Shelah Weak Hypothesis for various cofinalities are constructed.
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  40.  34
    Applications of Pcf for Mild Large Cardinals to Elementary Embeddings.Moti Gitik & Saharon Shelah - 2013 - Annals of Pure and Applied Logic 164 (9):855-865.
    The following pcf results are proved:1. Assume thatκ>ℵ0κ>ℵ0is a weakly compact cardinal. Letμ>2κμ>2κbe a singular cardinal of cofinality κ. Then for every regularView the MathML sourceλ sup{suppcfσ⁎-complete|a⊆Reg∩and|a|<μ}.Turn MathJax onAs an application we show that:if κ is a measurable cardinal andj:V→Mj:V→Mis the elementary embedding by a κ-complete ultrafilter over κ, then for every τ the following holds:1. ifjjis a cardinal thenj=τj=τ;2. |j|=|j)||j|=|j)|;3. for any κ-complete ultrafilter W on κ, |j|=|jW||j|=|jW|.The first two items provide affirmative answers to questions from Gitik and Shelah (...)
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  41.  2
    Possible Values for 2K-and 2K.Moti Gitik & Carmi Merimovich - 1997 - Annals of Pure and Applied Logic 90 (1):193-242.
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  42.  28
    On a Question of Pereira.Moti Gitik - 2008 - Archive for Mathematical Logic 47 (1):53-64.
    Answering a question of Pereira we show that it is possible to have a model violating the Singular Cardinal Hypothesis without a tree-like continuous scale.
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  43.  8
    Possible Values for 2< Sup> and 2.Moti Gitik & Carmi Merimovich - 1997 - Annals of Pure and Applied Logic 90 (1):193-241.
  44.  8
    A Note on Sequences Witnessing Singularity, Following Magidor and Sinapova.Moti Gitik - 2018 - Mathematical Logic Quarterly 64 (3):249-253.
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  45.  9
    Strange Ultrafilters.Moti Gitik - 2019 - Archive for Mathematical Logic 58 (1-2):35-52.
    We deal with some natural properties of ultrafilters which trivially fail for normal ultrafilters.
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  46.  24
    A Model with a Precipitous Ideal, but No Normal Precipitous Ideal.Moti Gitik - 2013 - Journal of Mathematical Logic 13 (1):1250008.
    Starting with a measurable cardinal κ of the Mitchell order κ++ we construct a model with a precipitous ideal on ℵ1 but without normal precipitous ideals. This answers a question by T. Jech and K. Prikry. In the constructed model there are no Q-point precipitous filters on ℵ1, i. e. those isomorphic to extensions of Cubℵ1.
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  47.  22
    On Precipitousness of the Nonstationary Ideal Over a Supercompact.Moti Gitik - 1986 - Journal of Symbolic Logic 51 (3):648-662.
  48.  1
    Blowing Up the Power of a Singular Cardinal of Uncountable Cofinality.Moti Gitik - 2019 - Journal of Symbolic Logic 84 (4):1722-1743.
    A new method for blowing up the power of a singular cardinal is presented. It allows to blow up the power of a singular in the core model cardinal of uncountable cofinality. The method makes use of overlapping extenders.
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  49.  7
    Power Function on Stationary Classes.Moti Gitik & Carmi Merimovich - 2006 - Annals of Pure and Applied Logic 140 (1):75-103.
    We show that under certain large cardinal requirements there is a generic extension in which the power function behaves differently on different stationary classes. We achieve this by doing an Easton support iteration of the Radin on extenders forcing.
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  50.  7
    Nonsplitting Subset of $Mathscr{P}_kappa(Kappa^+)$.Moti Gitik - 1985 - Journal of Symbolic Logic 50 (4):881-894.
    Assuming the existence of a supercompact cardinal, we construct a model of ZFC + (There exists a nonsplitting stationary subset of $\mathscr{P}_|kappa(\kappa^+)$). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes $\mathscr{P}_{\aleph_1}(\aleph_2) - V$ stationary.
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