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  1. Yoshihiro Abe (1984). Strongly Compact Cardinals, Elementary Embeddings and Fixed Points. Journal of Symbolic Logic 49 (3):808-812.
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  2. Arthur W. Apter (2001). Supercompactness and Measurable Limits of Strong Cardinals. 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. Arthur W. Apter (2001). Some Structural Results Concerning Supercompact Cardinals. 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. Arthur W. Apter (1999). On the Consistency Strength of Two Choiceless Cardinal Patterns. 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. Arthur W. Apter (1999). On Measurable Limits of Compact Cardinals. 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. Arthur W. Apter (1998). Laver Indestructibility and the Class of Compact Cardinals. 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. Arthur W. Apter & Moti Gitik (1998). The Least Measurable Can Be Strongly Compact and Indestructible. 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. Arthur W. Apter & Joel David Hamkins (2003). Exactly Controlling the Non-Supercompact Strongly Compact Cardinals. 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. Arthur W. Apter & Joel David Hamkins (2002). Indestructibility and the Level-by-Level Agreement Between Strong Compactness and Supercompactness. 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. Arthur W. Apter & James M. Henle (1986). Large Cardinal Structures Below ℵω. Journal of Symbolic Logic 51 (3):591 - 603.
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  11. Joan Bagaria (2002). Review: Saharon Shelah, Hugh Woodin, Large Cardinals Imply That Every Reasonably Definable Set of Reals Is Lebesgue Measurable. [REVIEW] Bulletin of Symbolic Logic 8 (4):543-545.
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  12. James E. Baumgartner, Alan D. Taylor & Stanley Wagon (1977). On Splitting Stationary Subsets of Large Cardinals. 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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  13. Luca Bellotti (2006). On the Consistency of ZF Set Theory and Its Large Cardinal Extensions. Epistemologia 29 (1):41-60.
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  14. Andrew D. Brooke-Taylor (2009). Large Cardinals and Definable Well-Orders on the Universe. 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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  15. James W. Cummings (2000). Review: Ten Papers by Arthur Apter on Large Cardinals. [REVIEW] Bulletin of Symbolic Logic 6 (1):86 - 89.
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  16. Solomon Feferman, Harvey M. Friedman, Penelope Maddy & John R. Steel (2000). Does Mathematics Need New Axioms? 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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  17. Harvey Friedman, Countable Model Theory and Large Cardinals.
    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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  18. Harvey Friedman, Extremely Large Cardinals in the Rationals.
    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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  19. Harvey Friedman, Finite Trees and the Necessary Use of Large Cardinals.
    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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  20. Harvey Friedman, Applications of Large Cardinals to Borel Functions.
    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. Harvey M. Friedman, Applications of Large Cardinals to Graph Theory.
    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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  22. Harvey M. Friedman, Concrete Incompleteness From Efa Through Large Cardinals.
    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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  23. Gunter Fuchs (2009). Combined Maximality Principles Up to Large Cardinals. 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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  24. Joel David Hamkins (2015). Is the Dream Solution of the Continuum Hypothesis Attainable? 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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  25. Ronald Jensen (1995). Inner Models and Large Cardinals. Bulletin of Symbolic Logic 1 (4):393-407.
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  26. Ronald Jensen, Ernest Schimmerling, Ralf Schindler & John Steel (2009). Stacking Mice. 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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  27. E. M. Kleinberg & R. A. Shore (1971). On Large Cardinals and Partition Relations. Journal of Symbolic Logic 36 (2):305-308.
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  28. Peter Koellner, Independence and Large Cardinals. Stanford Encyclopedia of Philosophy.
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  29. Peter Koellner (2010). On the Question of Absolute Undecidability. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Philosophia Mathematica. Association for Symbolic Logic. 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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  30. Andy Lewis (1998). Large Cardinals and Large Dilators. 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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  31. William Mitchell & Ralf Schindler (2004). A Universal Extender Model Without Large Cardinals in V. 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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  32. E. Montenegro (1992). Combinatorics on Large Cardinals. Journal of Symbolic Logic 57 (2):617-643.
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  33. Carl F. Morgenstern (1979). On the Ordering of Certain Large Cardinals. Journal of Symbolic Logic 44 (4):563-565.
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  34. Itay Neeman & John Steel (2006). Counterexamples to the Unique and Cofinal Branches Hypotheses. 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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  35. Anne Newstead & James Franklin (2008). On the Reality of the Continuum Discussion Note: A Reply to Ormell, 'Russell's Moment of Candour', "Philosophy". 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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  36. Luís Pereira (2008). The PCF Conjecture and Large Cardinals. 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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  37. Ernest Schimmerling (2001). The Abc's of Mice. Bulletin of Symbolic Logic 7 (4):485-503.
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  38. Michael Sheard (1983). Indecomposable Ultrafilters Over Small Large Cardinals. Journal of Symbolic Logic 48 (4):1000-1007.
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  39. John R. Steel (2005). Distinct Iterable Branches. Journal of Symbolic Logic 70 (4):1127 - 1136.
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  40. Jouko Väänänen (1982). Abstract Logic and Set Theory. II. Large Cardinals. 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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  41. Solomon Feferman with with R. L. Vaught, Operational Set Theory and Small Large Cardinals.
    “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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  42. R. J. Watro (1984). On Partitioning the Infinite Subsets of Large Cardinals. Journal of Symbolic Logic 49 (2):539-541.
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  43. P. D. Welch (2015). Large Cardinals, Inner Models, and Determinacy: An Introductory Overview. 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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