This category needs an editor. We encourage you to help if you are qualified.
Volunteer, or read more about what this involves.
Related categories

794 found
Order:
1 — 50 / 794
Material to categorize
  1. A Fixed Point Theorem Equivalent to the Axiom of Choice.Alexander Abian - 1985 - Archive for Mathematical Logic 25 (1):173-174.
    Remove from this list   Direct download  
     
    Export citation  
     
    My bibliography  
  2. Flipping Properties: A Unifying Thread in the Theory of Large Cardinals.F. G. Abramson, L. A. Harrington, E. M. Kleinberg & W. S. Zwicker - 1977 - Annals of Mathematical Logic 12 (1):25-58.
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    My bibliography   6 citations  
  3. Tarski Alfred. Axiomatic and Algebraic Aspects of Two Theorems on Sums of Cardinals. Ebd., S. 79–104.W. Ackermann - 1950 - Journal of Symbolic Logic 14 (4):257-258.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  4. Hierarchies and the Axiom of Constructibility.J. W. Addison - 1966 - Journal of Symbolic Logic 31 (1):137-138.
    Remove from this list   Direct download  
     
    Export citation  
     
    My bibliography  
  5. Some Consequences of the Axiom of Constructibility.J. W. Addison - 1963 - Journal of Symbolic Logic 28 (4):293-293.
    Remove from this list   Direct download  
     
    Export citation  
     
    My bibliography   6 citations  
  6. Department of Computer Science. Eotvos University, Rakoczi Ut 5, H-1088 Budapest VIII, Hungary, Kope@ Cs. Elte. Hu. Ten Papers by Arthur Apter on Large Cardinals Arthur W. After. On the Least Strongly Compact Cardinal. Israeljournal of Mathematics, Vol. 35 (1980). Pp. 225-233. [REVIEW]S. Aharon Shelah - 2000 - Bulletin of Symbolic Logic 6:86.
  7. Sameness of Age Cohorts in the Mathematics of Population Growth.Abraham Akkerman - 1994 - British Journal for the Philosophy of Science 45 (2):679-691.
    The axiom of extensionality of set theory states that any two classes that have identical members are identical. Yet the class of persons age i at time t and the class of persons age i + 1 at t + l, both including same persons, possess different demographic attributes, and thus appear to be two different classes. The contradiction could be resolved by making a clear distinction between age groups and cohorts. Cohort is a multitude of individuals, which is constituted (...)
    Remove from this list   Direct download (9 more)  
     
    Export citation  
     
    My bibliography  
  8. The Axiom Scheme of Acyclic Comprehension.Zuhair Al-Johar, M. Randall Holmes & Nathan Bowler - 2014 - Notre Dame Journal of Formal Logic 55 (1):11-24.
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  9. The Simplest Axiom System for Hyperbolic Geometry Revisited, Again.Jesse Alama - 2014 - Studia Logica 102 (3):609-615.
    Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  10. Baire Category on Cardinals.C. Alkor & B. Intrigila - 1983 - Mathematical Logic Quarterly 29 (4):245-252.
  11. On Bourbaki's Axiomatic System for Set Theory.Maribel Anacona, Luis Carlos Arboleda & F. Javier Pérez-Fernández - 2014 - Synthese 191 (17):4069-4098.
    In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to Bourbaki’s system the axiom (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  12. On the Axiomatizability of the Notion of an Automorphism of a Finite Order.D. A. Anapolitanos & J. Väänänen - 1980 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 26 (28-30):433-437.
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  13. Connections Between Axioms of Set Theory and Basic Theorems of Universal Algebra.H. Andréka, Á Kurucz & I. Németi - 1994 - Journal of Symbolic Logic 59 (3):912-923.
    One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class K of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of K. G. Gratzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in ZF. Surprisingly, the Axiom of Foundation plays a crucial role (...)
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    My bibliography  
  14. Games, Scales, and Suslin Cardinals. The Cabal Seminar, Volume I. [REVIEW]Alessandro Andretta - 2012 - Bulletin of Symbolic Logic 18 (1):122-125.
  15. REVIEWS-AS Kechris, B. Löwe, and JR Steel (Editors), Games, Scales, and Suslin Cardinals. The Cabal Seminar, Volume I.Alessandro Andretta - 2012 - Bulletin of Symbolic Logic 18 (1):122.
  16. Large Cardinals and Iteration Trees of Height Ω.Alessandro Andretta - 1991 - Annals of Pure and Applied Logic 54 (1):1-15.
    In this paper we continue the line of work initiated in “Building iteration trees”. It is shown that the existence of a certain kind of iteration tree of height ω is equivalent to the existence of a cardinal δ that is Woodin with respect to functions in the next admissible.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  17. Removing Laver Functions From Supercompactness Arguments.A. W. Apter - 2005 - Mathematical Logic Quarterly 51 (2):154.
    We show how the use of a Laver function in the proof of the consistency, relative to the existence of a supercompact cardinal, of both the Proper Forcing Axiom and the Semiproper Forcing Axiom can be eliminated via the use of lottery sums of the appropriate partial orderings.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  18. Level by Level Equivalence and Strong Compactness.A. W. Apter - 2004 - Mathematical Logic Quarterly 50 (1):51.
    We force and construct models in which there are non-supercompact strongly compact cardinals which aren't measurable limits of strongly compact cardinals and in which level by level equivalence between strong compactness and supercompactness holds non-trivially except at strongly compact cardinals. In these models, every measurable cardinal κ which isn't either strongly compact or a witness to a certain phenomenon first discovered by Menas is such that for every regular cardinal λ > κ, κ is λ strongly compact iff κ is (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  19. Failure of GCH and the Level by Level Equivalence Between Strong Compactness and Supercompactness.A. W. Apter - 2003 - Mathematical Logic Quarterly 49 (6):587.
    We force and obtain three models in which level by level equivalence between strong compactness and supercompactness holds and in which, below the least supercompact cardinal, GCH fails unboundedly often. In two of these models, GCH fails on a set having measure 1 with respect to certain canonical measures. There are no restrictions in all of our models on the structure of the class of supercompact cardinals.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  20. Strong Cardinals Can Be Fully Laver Indestructible.A. W. Apter - 2002 - Mathematical Logic Quarterly 48 (4):499-507.
    We prove three theorems which show that it is relatively consistent for any strong cardinal κ to be fully Laver indestructible under κ-directed closed forcing.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  21. Strong Compactness and a Global Version of a Theorem of Ben-David and Magidor.A. W. Apter - 2000 - Mathematical Logic Quarterly 46 (4):453-460.
    Starting with a model in which κ is the least inaccessible limit of cardinals δ which are δ+ strongly compact, we force and construct a model in which κ remains inaccessible and in which, for every cardinal γ < κ, □γ+ω fails but □γ+ω, ω holds. This generalizes a result of Ben-David and Magidor and provides an analogue in the context of strong compactness to a result of the author and Cummings in the context of supercompactness.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  22. Indestructibility and Destructible Measurable Cardinals.Arthur W. Apter - 2016 - Archive for Mathematical Logic 55 (1-2):3-18.
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  23. Inaccessible Cardinals, Failures of GCH, and Level-by-Level Equivalence.Arthur W. Apter - 2014 - Notre Dame Journal of Formal Logic 55 (4):431-444.
    We construct models for the level-by-level equivalence between strong compactness and supercompactness containing failures of the Generalized Continuum Hypothesis at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal $\delta $, $2^{\delta }\gt \delta ^{++}$. In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inaccessible cardinals at which GCH holds are also measurable. These results extend and generalize earlier work of (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  24. Indestructible Strong Compactness and Level by Level Inequivalence.Arthur W. Apter - 2013 - Mathematical Logic Quarterly 59 (4-5):371-377.
  25. Indestructibility, Measurability, and Degrees of Supercompactness.Arthur W. Apter - 2012 - Mathematical Logic Quarterly 58 (1):75-82.
    Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ+ supercompact} is unbounded in κ. If in addition λ is 2λ supercompact, then A2 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is δ+ supercompact} is unbounded in κ as well. The large cardinal (...)
    Remove from this list   Direct download (6 more)  
     
    Export citation  
     
    My bibliography  
  26. On Some Questions Concerning Strong Compactness.Arthur W. Apter - 2012 - Archive for Mathematical Logic 51 (7-8):819-829.
    A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ < κ, then must GCH fail at some regular cardinal δ ≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We get a negative (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  27. A Remark on the Tree Property in a Choiceless Context.Arthur W. Apter - 2011 - Archive for Mathematical Logic 50 (5-6):585-590.
    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 (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  28. Indestructibility, HOD, and the Ground Axiom.Arthur W. Apter - 2011 - Mathematical Logic Quarterly 57 (3):261-265.
    Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, …, 4 are the theories “ZFC + φ1 + φ2,” “ZFC + ¬φ1 + φ2,” “ZFC + φ1 + ¬φ2,” and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, Ai = df{δ κ is inaccessible. We show it is also (...)
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  29. Level by Level Inequivalence Beyond Measurability.Arthur W. Apter - 2011 - Archive for Mathematical Logic 50 (7-8):707-712.
    We construct models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In each model, above the supercompact cardinal, there are finitely many strongly compact cardinals, and the strongly compact and measurable cardinals precisely coincide.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  30. How Many Normal Measures Can ℵ Ω 1+1 Carry?Arthur W. Apter - 2010 - Mathematical Logic Quarterly 56 (2):164-170.
    Relative to the existence of a supercompact cardinal with a measurable cardinal above it, we show that it is consistent for ℵ1 to be regular and for ℵmath image to be measurable and to carry precisely τ normal measures, where τ ≥ ℵmath image is any regular cardinal. This extends the work of [2], in which the analogous result was obtained for ℵω +1 using the same hypotheses.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  31. Indestructibility, Instances of Strong Compactness, and Level by Level Inequivalence.Arthur W. Apter - 2010 - Archive for Mathematical Logic 49 (7-8):725-741.
    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 ${A = \emptyset}$ . The first of these contains a supercompact cardinal κ and (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  32. Tallness and Level by Level Equivalence and Inequivalence.Arthur W. Apter - 2010 - Mathematical Logic Quarterly 56 (1):4-12.
    We construct two models containing exactly one supercompact cardinal in which all non-supercompact measurable cardinals are strictly taller than they are either strongly compact or supercompact. In the first of these models, level by level equivalence between strong compactness and supercompactness holds. In the other, level by level inequivalence between strong compactness and supercompactness holds. Each universe has only one strongly compact cardinal and contains relatively few large cardinals.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  33. Indestructibility and Stationary Reflection.Arthur W. Apter - 2009 - Mathematical Logic Quarterly 55 (3):228-236.
    If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ -strategically closed forcing and λ is weakly compact, then we show thatA = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets}must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  34. Indestructibility Under Adding Cohen Subsets and Level by Level Equivalence.Arthur W. Apter - 2009 - Mathematical Logic Quarterly 55 (3):271-279.
    We construct a model for the level by level equivalence between strong compactness and supercompactness in which the least supercompact cardinal κ has its strong compactness indestructible under adding arbitrarily many Cohen subsets. There are no restrictions on the large cardinal structure of our model.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  35. Indestructibility and Measurable Cardinals with Few and Many Measures.Arthur W. Apter - 2008 - Archive for Mathematical Logic 47 (2):101-110.
    If κ < λ are such that κ is indestructibly supercompact and λ is measurable, then we show that both A = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries the maximal number of normal measures} and B = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries fewer than the maximal number of normal measures} are unbounded (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  36. Reducing the Consistency Strength of an Indestructibility Theorem.Arthur W. Apter - 2008 - Mathematical Logic Quarterly 54 (3):288-293.
    Using an idea of Sargsyan, we show how to reduce the consistency strength of the assumptions employed to establish a theorem concerning a uniform level of indestructibility for both strong and supercompact cardinals.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  37. Indestructibility and Level by Level Equivalence and Inequivalence.Arthur W. Apter - 2007 - Mathematical Logic Quarterly 53 (1):78-85.
    If κ < λ are such that κ is indestructibly supercompact and λ is 2λ supercompact, it is known from [4] that {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ violates level by level equivalence between strong compactness and supercompactness}must be unbounded in κ. On the other hand, using a variant of the argument used to establish this fact, it is possible to prove that if κ < λ are (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   4 citations  
  38. Supercompactness and Level by Level Equivalence Are Compatible with Indestructibility for Strong Compactness.Arthur W. Apter - 2007 - Archive for Mathematical Logic 46 (3-4):155-163.
    It is known that if $\kappa < \lambda$ are such that κ is indestructibly supercompact and λ is 2λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal, there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact cardinal κ in which κ’s strong compactness is indestructible (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  39. Failures of SCH and Level by Level Equivalence.Arthur W. Apter - 2006 - Archive for Mathematical Logic 45 (7):831-838.
    We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is a stationary set of cardinals on which SCH fails. In this model, the structure of the class of supercompact cardinals can be arbitrary.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  40. Supercompactness and Measurable Limits of Strong Cardinals II: Applications to Level by Level Equivalence.Arthur W. Apter - 2006 - Mathematical Logic Quarterly 52 (5):457-463.
    We construct models for the level by level equivalence between strong compactness and supercompactness in which for κ the least supercompact cardinal and δ ≤ κ any cardinal which is either a strong cardinal or a measurable limit of strong cardinals, 2δ > δ+ and δ is < 2δ supercompact. In these models, the structure of the class of supercompact cardinals can be arbitrary, and the size of the power set of κ can essentially be made as large as desired. (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  41. The Least Strongly Compact Can Be the Least Strong and Indestructible.Arthur W. Apter - 2006 - Annals of Pure and Applied Logic 144 (1):33-42.
    We construct two models in which the least strongly compact cardinal κ is also the least strong cardinal. In each of these models, κ satisfies indestructibility properties for both its strong compactness and strongness.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  42. Universal Partial Indestructibility and Strong Compactness.Arthur W. Apter - 2005 - Mathematical Logic Quarterly 51 (5):524-531.
    For any ordinal δ, let λδ be the least inaccessible cardinal above δ. We force and construct a model in which the least supercompact cardinal κ is indestructible under κ-directed closed forcing and in which every measurable cardinal δ < κ is < λδ strongly compact and has its < λδ strong compactness indestructible under δ-directed closed forcing of rank less than λδ. In this model, κ is also the least strongly compact cardinal. We also establish versions of this result (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  43. Diamond, Square, and Level by Level Equivalence.Arthur W. Apter - 2004 - Archive for Mathematical Logic 44 (3):387-395.
    We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, ♦ δ holds for every regular uncountable cardinal δ, and below the least supercompact cardinal κ, □ δ holds on a stationary subset of κ. There are no restrictions in our model on the structure of the class of supercompact cardinals.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  44. Characterizing Strong Compactness Via Strongness.Arthur W. Apter - 2003 - Mathematical Logic Quarterly 49 (4):375.
    We construct a model in which the strongly compact cardinals can be non-trivially characterized via the statement “κ is strongly compact iff κ is a measurable limit of strong cardinals”. If our ground model contains large enough cardinals, there will be supercompact cardinals in the universe containing this characterization of the strongly compact cardinals.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  45. Some Remarks on Indestructibility and Hamkins' Lottery Preparation.Arthur W. Apter - 2003 - Archive for Mathematical Logic 42 (8):717-735.
  46. On a Problem of Woodin.Arthur W. Apter - 2000 - Archive for Mathematical Logic 39 (4):253-259.
    A question of Woodin asks if $\kappa$ is strongly compact and GCH holds for all cardinals $\delta < \kappa$ , then must GCH hold everywhere. We get a negative answer to Woodin's question in the context of the negation of the Axiom of Choice.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  47. Forcing the Least Measurable to Violate GCH.Arthur W. Apter - 1999 - Mathematical Logic Quarterly 45 (4):551-560.
    Starting with a model for “GCH + k is k+ supercompact”, we force and construct a model for “k is the least measurable cardinal + 2k = K+”. This model has the property that forcing over it with Add preserves the fact k is the least measurable cardinal.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  48. More on the Least Strongly Compact Cardinal.Arthur W. Apter - 1997 - Mathematical Logic Quarterly 43 (3):427-430.
    We show that it is consistent, relative to a supercompact limit of supercompact cardinals, for the least strongly compact cardinal k to be both the least measurable cardinal and to be > 2k supercompact.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  49. Patterns of Compact Cardinals.Arthur W. Apter - 1997 - Annals of Pure and Applied Logic 89 (2-3):101-115.
    We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + ƒ : Ω → 2 is a function”, then there is a partial ordering P V so that for , There is a proper class of compact cardinals + If (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   11 citations  
  50. A Cardinal Pattern Inspired by AD.Arthur W. Apter - 1996 - Mathematical Logic Quarterly 42 (1):211-218.
    Assuming Con, a model in which there are unboundedly many regular cardinals below Θ and in which the only regular cardinals below Θ are limit cardinals was previously constructed. Using a large cardinal hypothesis far beyond Con, we construct in this paper a model in which there is a proper class of regular cardinals and in which the only regular cardinals in the universe are limit cardinals.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography   1 citation  
1 — 50 / 794