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  1. Strongly Compact Cardinals, Elementary Embeddings and Fixed Points.Yoshihiro Abe - 1984 - Journal of Symbolic Logic 49 (3):808-812.
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  2. Supercompactness and Measurable Limits of Strong Cardinals.Arthur W. Apter - 2001 - Journal of Symbolic Logic 66 (2):629-639.
    In this paper, two theorems concerning measurable limits of strong cardinals and supercompactness are proven. This generalizes earlier work, both individual and joint with Shelah.
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  3. Some Structural Results Concerning Supercompact Cardinals.Arthur W. Apter - 2001 - Journal of Symbolic Logic 66 (4):1919-1927.
    We show how the forcing of [5] can be iterated so as to get a model containing supercompact cardinals in which every measurable cardinal δ is δ + supercompact. We then apply this iteration to prove three additional theorems concerning the structure of the class of supercompact cardinals.
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  4. On the Consistency Strength of Two Choiceless Cardinal Patterns.Arthur W. Apter - 1999 - Notre Dame Journal of Formal Logic 40 (3):341-345.
    Using work of Devlin and Schindler in conjunction with work on Prikry forcing in a choiceless context done by the author, we show that two choiceless cardinal patterns have consistency strength of at least one Woodin cardinal.
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  5. On Measurable Limits of Compact Cardinals.Arthur W. Apter - 1999 - Journal of Symbolic Logic 64 (4):1675-1688.
    We extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and (...)
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  6. Laver Indestructibility and the Class of Compact Cardinals.Arthur W. Apter - 1998 - Journal of Symbolic Logic 63 (1):149-157.
    Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in (...)
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  7. 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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  8. Exactly Controlling the Non-Supercompact Strongly Compact Cardinals.Arthur W. Apter & Joel David Hamkins - 2003 - Journal of Symbolic Logic 68 (2):669-688.
    We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and (...)
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  9. Indestructibility and the Level-by-Level Agreement Between Strong Compactness and Supercompactness.Arthur W. Apter & Joel David Hamkins - 2002 - Journal of Symbolic Logic 67 (2):820-840.
    Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.
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  10. Large Cardinal Structures Below ℵω.Arthur W. Apter & James M. Henle - 1986 - Journal of Symbolic Logic 51 (3):591 - 603.
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  11. Review: Saharon Shelah, Hugh Woodin, Large Cardinals Imply That Every Reasonably Definable Set of Reals Is Lebesgue Measurable. [REVIEW]Joan Bagaria - 2002 - Bulletin of Symbolic Logic 8 (4):543-545.
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  12. Richness and Reflection.Neil Barton - 2016 - Philosophia Mathematica 24 (3):330-359.
    A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, (...)
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  13. On Splitting Stationary Subsets of Large Cardinals.James E. Baumgartner, Alan D. Taylor & Stanley Wagon - 1977 - Journal of Symbolic Logic 42 (2):203-214.
    Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ + -saturated, i.e., are there κ + stationary subsets of κ with pairwise intersections nonstationary? Our first observation is: Theorem. NS is κ + -saturated iff for every normal ideal J on κ there is a stationary set $A \subseteq \kappa$ such that $J = NS \mid A = \{X \subseteq (...)
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  14. On the Consistency of ZF Set Theory and Its Large Cardinal Extensions.Luca Bellotti - 2006 - Epistemologia 29 (1):41-60.
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  15. Large Cardinals and Definable Well-Orders on the Universe.Andrew D. Brooke-Taylor - 2009 - Journal of Symbolic Logic 74 (2):641-654.
    We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ◊ $_{k^ - }^* $ at a proper class of cardinals k. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.
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  16. Inaccessible Set Axioms May Have Little Consistency Strength.L. Crosilla & M. Rathjen - 2002 - Annals of Pure and Applied Logic 115 (1-3):33-70.
    The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form Vκ where κ is a strongly inaccessible cardinal and Vκ denotes the κth 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 (...)
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  17. Review: Ten Papers by Arthur Apter on Large Cardinals. [REVIEW]James W. Cummings - 2000 - Bulletin of Symbolic Logic 6 (1):86 - 89.
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  18. Laver’s Results and Low-Dimensional Topology.Patrick Dehornoy - 2016 - Archive for Mathematical Logic 55 (1-2):49-83.
    In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimen\-sional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.
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  19. Does Mathematics Need New Axioms?Solomon Feferman, Harvey M. Friedman, Penelope Maddy & John R. Steel - 2000 - Bulletin of Symbolic Logic 6 (4):401-446.
    Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...)
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  20. Applications of Large Cardinals to Borel Functions.Harvey Friedman - manuscript
    The space CS(R) has a unique “Borel structure” in the following sense. Note that there is a natural mapping from R¥ onto CS(R}; namely, taking ranges. We can combine this with any Borel bijection from R onto R¥ in order to get a “preferred” surjection F:R ® CS(R).
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  21. Countable Model Theory and Large Cardinals.Harvey Friedman - manuscript
    We can look at this model theoretically as follows. By the linearly ordered predicate calculus, we simply mean ordinary predicate calculus with equality and a special binary relation symbol <. It is required that in all interpretations, < be a linear ordering on the domain. Thus we have the usual completeness theorem provided we add the axioms that assert that < is a linear ordering.
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  22. Extremely Large Cardinals in the Rationals.Harvey Friedman - manuscript
    In 1995 we gave a new simple principle of combinatorial set theory and showed that it implies the existence of a nontrivial elementary embedding from a rank into itself, and follows from the existence of a nontrivial elementary embedding from V into M, where M contains the rank at the first fixed point above the critical point. We then gave a “diamondization” of this principle, and proved its relative consistency by means of a standard forcing argument.
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  23. Finite Trees and the Necessary Use of Large Cardinals.Harvey Friedman - manuscript
    We introduce insertion domains that support the placement of new, higher, vertices into finite trees. We prove that every nonincreasing insertion domain has an element with simple structural properties in the style of classical Ramsey theory. This result is proved using standard large cardinal axioms that go well beyond the usual axioms for mathematics. We also establish that this result cannot be proved without these large cardinal axioms. We also introduce insertion rules that specify the placement of new, higher, vertices (...)
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  24. Applications of Large Cardinals to Graph Theory.Harvey M. Friedman - unknown
    Since then we have been engaged in the development of such results of greater relevance to mathematical practice. In January, 1997 we presented some new results of this kind involving what we call “jump free” classes of finite functions. This Jump Free Theorem is treated in section 2.
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  25. Concrete Incompleteness From Efa Through Large Cardinals.Harvey M. Friedman - unknown
    Normal mathematical culture is overwhelmingly concerned with finite structures, finitely generated structures, discrete structures (countably infinite), continuous and piecewise continuous functions between complete separable metric spaces, with lesser consideration of pointwise limits of sequences of such functions, and Borel measurable functions between complete separable metric spaces.
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  26. Combined Maximality Principles Up to Large Cardinals.Gunter Fuchs - 2009 - Journal of Symbolic Logic 74 (3):1015-1046.
    The motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for < κ -closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of (...)
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  27. Is the Dream Solution of the Continuum Hypothesis Attainable?Joel David Hamkins - 2015 - Notre Dame Journal of Formal Logic 56 (1):135-145.
    The dream solution of the continuum hypothesis would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to $\mathrm {CH}$ is unattainable.
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  28. Maximality Principles in Set Theory.Luca Incurvati - forthcoming - Philosophia Mathematica:nkw011.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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  29. Restrictiveness Relative to Notions of Interpretation.Luca Incurvati & Benedikt Löwe - 2016 - Review of Symbolic Logic 9 (2): 238-250.
    Maddy gave a semi-formal account of restrictiveness by defining a formal notion based on a class of interpretations and explaining how to handle false positives and false negatives. Recently, Hamkins pointed out some structural issues with Maddy's definition. We look at Maddy's formal definitions from the point of view of an abstract interpretation relation. We consider various candidates for this interpretation relation, including one that is close to Maddy's original notion, but fixes the issues raised by Hamkins. Our work brings (...)
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  30. Inner Models and Large Cardinals.Ronald Jensen - 1995 - Bulletin of Symbolic Logic 1 (4):393-407.
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  31. Stacking Mice.Ronald Jensen, Ernest Schimmerling, Ralf Schindler & John Steel - 2009 - Journal of Symbolic Logic 74 (1):315-335.
    We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal k ≥ N₃ such that □k and □(k) fail. 2) There is a cardinal k such that k is weakly compact in the generic extension by Col(k, k⁺). Of special interest is 1) with k = N₃ since it follows from PFA by theorems of (...)
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  32. On Large Cardinals and Partition Relations.E. M. Kleinberg & R. A. Shore - 1971 - Journal of Symbolic Logic 36 (2):305-308.
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  33. Independence and Large Cardinals.Peter Koellner - 2010 - Stanford Encyclopedia of Philosophy.
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  34. On the Question of Absolute Undecidability.Peter Koellner - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Philosophia Mathematica. Association for Symbolic Logic. pp. 153-188.
    The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH (...)
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  35. Large Cardinals and Large Dilators.Andy Lewis - 1998 - Journal of Symbolic Logic 63 (4):1496-1510.
    Applying Woodin's non-stationary tower notion of forcing, I prove that the existence of a supercompact cardinal κ in V and a Ramsey dilator in some small forcing extension V[G] implies the existence in V of a measurable dilator of size κ, measurable by κ-complete measures.
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  36. A Universal Extender Model Without Large Cardinals in V.William Mitchell & Ralf Schindler - 2004 - Journal of Symbolic Logic 69 (2):371 - 386.
    We construct, assuming that there is no inner model with a Woodin cardinal but without any large cardinal assumption, a model $K^{c}$ which is iterable for set length iterations, which is universal with respect to all weasels with which it can be compared, and (assuming GCH) is universal with respect to set sized premice.
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  37. Combinatorics on Large Cardinals.E. Montenegro - 1992 - Journal of Symbolic Logic 57 (2):617-643.
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  38. On the Ordering of Certain Large Cardinals.Carl F. Morgenstern - 1979 - Journal of Symbolic Logic 44 (4):563-565.
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  39. Counterexamples to the Unique and Cofinal Branches Hypotheses.Itay Neeman & John Steel - 2006 - Journal of Symbolic Logic 71 (3):977 - 988.
    We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal.
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  40. On the Reality of the Continuum Discussion Note: A Reply to Ormell, 'Russell's Moment of Candour', "Philosophy".Anne Newstead & James Franklin - 2008 - Philosophy 83 (323):117-127.
    This paper discusses an argument for the reality of the classical mathematical continuum. An inference to the best explanation type of argument is used to defend the idea that real numbers exist even when they cannot be constructively specified as with the "indefinable numbers".
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  41. Philosophical Perspectives on Infinity.Graham Oppy - 2006 - Cambridge University Press.
    This book is an exploration of philosophical questions about infinity. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world. He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Even simple notions, such as counting, adding and maximising present serious difficulties. Other topics examined include the nature of space and time, infinities in physical science, infinities in theories of probability and decision, the nature of (...)
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  42. The PCF Conjecture and Large Cardinals.Luís Pereira - 2008 - Journal of Symbolic Logic 73 (2):674 - 688.
    We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.
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  43. The Abc's of Mice.Ernest Schimmerling - 2001 - Bulletin of Symbolic Logic 7 (4):485-503.
  44. Indecomposable Ultrafilters Over Small Large Cardinals.Michael Sheard - 1983 - Journal of Symbolic Logic 48 (4):1000-1007.
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  45. All Things Must Pass Away.Joshua Spencer - 2012 - Oxford Studies in Metaphysics 7:67.
    Are there any things that are such that any things whatsoever are among them. I argue that there are not. My thesis follows from these three premises: (1) There are two or more things; (2) for any things, there is a unique thing that corresponds to those things; (3) for any two or more things, there are fewer of them than there are pluralities of them.
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  46. Distinct Iterable Branches.John R. Steel - 2005 - Journal of Symbolic Logic 70 (4):1127 - 1136.
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  47. Abstract Logic and Set Theory. II. Large Cardinals.Jouko Väänänen - 1982 - Journal of Symbolic Logic 47 (2):335-346.
    The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals.
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  48. Operational Set Theory and Small Large Cardinals.Solomon Feferman with with R. L. Vaught - manuscript
    “Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such disaparate and (...)
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  49. On Partitioning the Infinite Subsets of Large Cardinals.R. J. Watro - 1984 - Journal of Symbolic Logic 49 (2):539-541.
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  50. Large Cardinals, Inner Models, and Determinacy: An Introductory Overview.P. D. Welch - 2015 - Notre Dame Journal of Formal Logic 56 (1):213-242.
    The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines of development.
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