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  1. 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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  2. 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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  3. 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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  4. 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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  5. 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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  6. 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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  7. 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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  8. 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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  9. 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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  10. 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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  11. 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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  12. Ronald Jensen (1995). Inner Models and Large Cardinals. Bulletin of Symbolic Logic 1 (4):393-407.
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  13. 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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  14. Peter Koellner, Independence and Large Cardinals. Stanford Encyclopedia of Philosophy.
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  15. 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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  16. 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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  17. E. Montenegro (1992). Combinatorics on Large Cardinals. Journal of Symbolic Logic 57 (2):617-643.
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  18. Carl F. Morgenstern (1979). On the Ordering of Certain Large Cardinals. Journal of Symbolic Logic 44 (4):563-565.
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  19. 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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  20. Michael Sheard (1983). Indecomposable Ultrafilters Over Small Large Cardinals. Journal of Symbolic Logic 48 (4):1000-1007.
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  21. 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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  22. 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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  23. R. J. Watro (1984). On Partitioning the Infinite Subsets of Large Cardinals. Journal of Symbolic Logic 49 (2):539-541.
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