Results for 'ZFC'

314 found
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  1.  8
    Large Transitive Models in Local ZFC.Athanassios Tzouvaras - 2014 - Archive for Mathematical Logic 53 (3-4):233-260.
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  2.  9
    A Strictly Finitary Non-Triviality Proof for a Paraconsistent System of Set Theory Deductively Equivalent to Classical ZFC Minus Foundation.Arief Daynes - 2000 - Archive for Mathematical Logic 39 (8):581-598.
    The paraconsistent system CPQ-ZFC/F is defined. It is shown using strong non-finitary methods that the theorems of CPQ-ZFC/F are exactly the theorems of classical (...) ZFC minus foundation. The proof presented in the paper uses the assumption that a strongly inaccessible cardinal exists. It is then shown using strictly finitary methods that CPQ-ZFC/F is non-trivial. CPQ-ZFC/F thus provides a formulation of set theory that has the same deductive power as the corresponding classical system but is more reliable in that non-triviality is provable by strictly finitary methods. This result does not contradict Gödel's incompleteness theorem because the proof of the deductive equivalence of the paraconsistent and classical systemss use non-finitary methods. (shrink)
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  3.  17
    More on Regular and Decomposable Ultrafilters in ZFC.Paolo Lipparini - 2010 - Mathematical Logic Quarterly 56 (4):340-374.
    We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: If m1 and the ultrafilter D is , equation imagem)-regular (...), then D is κ -decomposable for some κ with λκ ). If λ is a strong limit cardinal and D is , equation imagem)-regular, then either D is -regular or there are arbitrarily large κ < λ for which D is κ -decomposable ). Suppose that λ is singular, λ < κ, cf κcf λ and D is -regular. Then: D is either -regular, or -regular for some λ' < λ . If κ is regular, then D is either -regular, or -regular for every κ' < κ . If either λ is a strong limit cardinal and λ<λ < , or λ<λ < κ, then D is either λ -decomposable, or -regular for some λ' < λ . If λ is singular, D is -regular and there are arbitrarily large ν < λ for which D is ν -decomposable, then D is κ -decomposable for some κ with λκλ<μ . D × D' is -regular if and only if there is a ν such that D is -regular and D' is -regular for all ν∼ < ν .We also list some problems, and furnish applications to topological spaces and to extended logics. (shrink)
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  4.  30
    W. Hugh Woodin. AD and the Uniqueness of the Supercompact Measures on Pω1 . Cabal Seminar 7981, Proceedings, Caltech-UCLA Logic Seminar 197981, Edited by A. S. Kechris, D. A. Martin, and Y. N. Moschavokis, Lecture Notes in Mathematics, Vol. 1019, Springer-Verlag, Berlin Etc. 1983, Pp. 6771. - W. Hugh Woodin. Some Consistency Results in ZFC Using AD. Cabal Seminar 7981, Proceedings, Caltech-UCLA Logic Seminar 197981, Edited by A. S. Kechris, D. A. Martin, and Y. N. Moschavokis, Lecture Notes in Mathematics, Vol. 1019, Springer-Verlag, Berlin Etc. 1983, Pp. 172198. - Alexander S. Kechris. Subsets of ℵ1 Constructihle From Areal. Cabal Seminar 8185, Proceedings, Caltech-UCLA Logic Seminar 198185, Edited by A. S. Kechris, D. A. Martin, and J. R. Steel, Lecture Notes in Mathematics, Vol. 1333, Springer-Verlag, Berlin Etc. 1988, Pp. 110116[REVIEW]Andreas Blass - 1992 - Journal of Symbolic Logic 57 (1):259-261.
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  5.  5
    What is the Theory ZFC Without Power Set?Victoria Gitman, Joel David Hamkins & Thomas A. Johnstone - 2016 - Mathematical Logic Quarterly 62 (4-5):391-406.
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  6.  65
    On Arbitrary Sets and ZFC.José Ferreirós - 2011 - Bulletin of Symbolic Logic 17 (3):361-393.
    Set theory deals with the most fundamental existence questions in mathematicsquestions which affect other areas of mathematics, from the real numbers to structures of all kinds, (...)but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-calledarbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what is meant by definability and byarbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the ZermeloFraenkel system goes in laying out principles that capture the idea ofarbitrary sets”. We argue that the theory is rather poor in this respect. (shrink)
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  7. On the Structure of $\Operatorname{Ext}(a, \Mathbf{Z})$ in ZFC+.G. Sageev & S. Shelah - 1985 - Journal of Symbolic Logic 50 (2):302 - 315.
  8.  5
    The Consistency of ZFC + 2ℵ0 > ℵω + = .Martin Gilchrist & Saharon Shelah - 1997 - Journal of Symbolic Logic 62 (4):1151-1160.
  9.  10
    Monadic Theory of Order and Topology in ZFC.Yuri Gurevich & Saharon Shelah - 1982 - Annals of Mathematical Logic 23 (2-3):179-198.
  10. Natural Internal Forcing Schemata Extending ZFC: Truth in the Universe?Garvin Melles - 1994 - Journal of Symbolic Logic 59 (2):461-472.
  11.  3
    Minimal Collapsing Extensions of Models of Zfc.Lev Bukovský & Eva Copláková-Hartová - 1990 - Annals of Pure and Applied Logic 46 (3):265-298.
  12.  47
    Is There a Simple, Pedestrian Arithmetic Sentence Which is Independent of Zfc?Francisco Antonio Doria - 2000 - Synthese 125 (1-2):69-76.
    We show that the P 2 0 sentence, and explore some of theconsequences of that fact. This paper summarizes recent workby the author with N. C. A. (...)
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  13.  24
    Extending Standard Models of ZFC to Models of Nonstandard Set Theories.Vladimir Kanovei & Michael Reeken - 2000 - Studia Logica 64 (1):37-59.
    We study those models of ZFCwhich are embeddable, as the class of all standard sets, in a model of internal set theory >ISTor models of some other (...)
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  14.  6
    A Reflection Principle As a Reverse-Mathematical Fixed Point Over the Base Theory ZFC.Sakaé Fuchino - 2017 - Annals of the Japan Association for Philosophy of Science 25:67-77.
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  15.  11
    Review: W. Hugh Woodin, A. S. Kechris, D. A. Martin, Y. N. Moschavokis, Ad and the Uniqueness of the Supercompact Measures on $Pomega1 (Lambda)$; W. Hugh Woodin, Some Consistency Results in ZFC Using AD; Alexander S. Kechris, D. A. Martin, J. R. Steel, Subsets of $Aleph1$ Constructible From a Real[REVIEW]Andreas Blass - 1992 - Journal of Symbolic Logic 57 (1):259-261.
  16.  1
    Zfc Proves That the Class of Ordinals is Not Weakly Compact for Definable Classes.Ali Enayat & Joel David Hamkins - 2018 - Journal of Symbolic Logic 83 (1):146-164.
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  17.  11
    A New Inner Model for ZFC.Wlodzimierz Zadrozny - 1981 - Journal of Symbolic Logic 46 (2):393-396.
    Assume $(\exists\kappa) \lbrack\kappa \rightarrow (\kappa)^{ . Then a new inner model H exists and has the following properties: (1) HHOD; (2) Th(H) = Th(HOD); ( (...)3) there is j: HH; (4) there is a c.u.b. class of indiscernibles for H. (shrink)
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  18.  4
    11 M + φ L interpretiert parametrisiert ZFC.Philipp Werner - 2015 - In David Lewis Und Seine Mereologische Interpretation der Zermelo-Fraenkelschen Mengenlehre: Eine Rekonstruktion. De Gruyter. pp. 103-126.
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  19.  3
    On the Structure of Ext in ZFC+.G. Sageev & S. Shelah - 1985 - Journal of Symbolic Logic 50 (2):302-315.
  20.  1
    ZfcModels as KripkeModels.Franco Montagna - 1983 - Mathematical Logic Quarterly 29 (3):163-168.
  21. Heights of Models of ZFC and the Existence of End Elementary Extensions II.Andrés Villaveces - 1999 - Journal of Symbolic Logic 64 (3):1111-1124.
    The existence of End Elementary Extensions of models M of ZFC is related to the ordinal height of M, according to classical results due to Keisler, Morley (...)
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  22.  45
    After All, There Are Some Inequalities Which Are Provable in ZFC.Tomek Bartoszyński, Andrzej Rosłanowski & Saharon Shelah - 2000 - Journal of Symbolic Logic 65 (2):803-816.
    We address ZFC inequalities between some cardinal invariants of the continuum, which turned out to be true in spite of strong expectations given by [11].
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  23.  13
    The Consistency of $\Mathrm{ZFC} + 2^{\Aleph0} > \Aleph\Omega + \Mathscr{J} = \Mathscr{J}$.Martin Gilchrist & Saharon Shelah - 1997 - Journal of Symbolic Logic 62 (4):1151-1160.
  24. Well- and Non-Well-Founded Fregean Extensions.Ignacio Jané & Gabriel Uzquiano - 2004 - Journal of Philosophical Logic 33 (5):437-465.
    George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom (...) V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)
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  25.  70
    A Natural Model of the Multiverse Axioms.Victoria Gitman & Joel David Hamkins - 2010 - Notre Dame Journal of Formal Logic 51 (4):475-484.
    If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins.
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  26.  16
    Localizing the Axioms.Athanassios Tzouvaras - 2010 - Archive for Mathematical Logic 49 (5):571-601.
    We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to (...) a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}$ consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ0-Collection and minus ${\in}$ -induction scheme. ZFC+ “there is an inaccessible cardinalproves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and ${\Pi_1^1}$ -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form ${Loc({\rm ZFC}+\phi)}$ are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved. (shrink)
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  27.  23
    Models as Universes.Brice Halimi - 2017 - Notre Dame Journal of Formal Logic 58 (1):47-78.
    Kreisels set-theoretic problem is the problem as to whether any logical consequence of ZFC is ensured to be true. Kreisel and Boolos both proposed an answer (...), taking truth to mean truth in the background set-theoretic universe. This article advocates another answer, which lies at the level of models of set theory, so that truth remains the usual semantic notion. The article is divided into three parts. It first analyzes Kreisels set-theoretic problem and proposes one way in which any model of set theory can be compared to a background universe and shown to contain internal models. It then defines logical consequence with respect to a model of ZFC, solves the model-scaled version of Kreisels set-theoretic problem, and presents various further results bearing on internal models. Finally, internal models are presented as accessible worlds, leading to an internal modal logic in which internal reflection corresponds to modal reflexivity, and resplendency corresponds to modal axiom 4. (shrink)
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  28. The Concept of Strong and Weak Virtual Reality.Andreas Martin Lisewski - 2006 - Minds and Machines 16 (2):201-219.
    We approach the virtual reality phenomenon by studying its relationship to set theory. This approach offers a characterization of virtual reality in set theoretic terms, and we (...)
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  29.  40
    Background Independence in Quantum Gravity and Forcing Constructions.Jerzy Król - 2004 - Foundations of Physics 34 (3):361-403.
    A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the (...)
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  30. Justifying Induction Mathematically: Strategies and Functions.Alexander Paseau - 2008 - Logique Et Analyse 51 (203):263.
    If the total state of the universe is encodable by a real number, Hardin and Taylor have proved that there is a solution to one version of (...)
     
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  31.  61
    True or False? A Case in the Study of Harmonic Functions.Fausto di Biase - 2009 - Topoi 28 (2):143-160.
    Recent mathematical results, obtained by the author, in collaboration with Alexander Stokolos, Olof Svensson, and Tomasz Weiss, in the study of harmonic functions, have prompted the following (...)
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  32.  19
    Quotients of Boolean Algebras and Regular Subalgebras.B. Balcar & T. Pazák - 2010 - Archive for Mathematical Logic 49 (3):329-342.
    Let ${\mathbb{B}}$ and ${\mathbb{C}}$ be Boolean algebras and ${e: \mathbb{B}\rightarrow \mathbb{C}}$ an embedding. We examine the hierarchy of ideals on ${\mathbb{C}}$ for which (...)
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  33.  65
    The Monadic Theory of Ω2.Yuri Gurevich, Menachem Magidor & Saharon Shelah - 1983 - Journal of Symbolic Logic 48 (2):387-398.
    Assume ZFC + "There is a weakly compact cardinal" is consistent. Then: (i) For every $S \subseteq \omega, \mathrm{ZFC} +$ "S and the monadic theory of ω 2 (...) are recursive each in the other" is consistent; and (ii) ZFC + "The full second-order theory of ω 2 is interpretable in the monadic theory of ω 2 " is consistent. (shrink)
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  34.  49
    PCF Structures of Height Less Than Ω3.Karim Er-Rhaimini & Boban Veličković - 2010 - Journal of Symbolic Logic 75 (4):1231-1248.
    We show that it is relatively consistent with ZFC to have PCF structures of height δ, for all ordinals δ < ω3.
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  35.  12
    The Downward Directed Grounds Hypothesis and Very Large Cardinals.Toshimichi Usuba - 2017 - Journal of Mathematical Logic 17 (2):1750009.
    A transitive model M of ZFC is called a ground if the universe V is a set forcing extension of M. We show that the grounds ofV (...)
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  36. Wide Sets, ZFCU, and the Iterative Conception.Christopher Menzel - 2014 - Journal of Philosophy 111 (2):57-83.
    The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets (...)have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence ofwidesets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the existence of wide sets. Drawing upon Cantors notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)
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  37. Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics.Vladimir Kanovei, Mikhail G. Katz & Thomas Mormann - 2013 - Foundations of Science 18 (2):259-296.
    We examine some of Connescriticisms of Robinsons infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but (...) the model tends to boomerang, undercutting Connesown earlier work in functional analysis. Connes described the hyperreals as both avirtual theoryand achimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwingthought experiment, but reach an opposite conclusion. In S , all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as beingvirtualif it is not definable in a suitable model of ZFC. If so, Connesclaim that a theory of the hyperreals isvirtualis refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters arent definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connescriticism of virtuality. We analyze the philosophical underpinnings of Connesargument based on Gödels incompleteness theorem, and detect an apparent circularity in Conneslogic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connesmagnum opus) and the HahnBanach theorem, in Connesown framework. We also note an inaccuracy in Machovers critique of infinitesimal-based pedagogy. (shrink)
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  38. Toward a Theory of Second-Order Consequence.Augustín Rayo & Gabriel Uzquiano - 1999 - Notre Dame Journal of Formal Logic 40 (3):315-325.
    There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization (...) is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form , ∈ ∩ ( × ) , for κ a strongly inaccessible ordinal. (shrink)
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  39. Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism.Hasen Khudairi - 2019 - In Matteo Vincenzo D'Alfonso & Don Berkich (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence. Springer.
    This essay examines the philosophical significance of Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic (...)
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  40. E Pluribus Unum: Plural Logic and Set Theory.John P. Burgess - 2004 - Philosophia Mathematica 12 (3):193-221.
    A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only (...)positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory. (shrink)
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  41.  5
    Chain Conditions of Products, and Weakly Compact Cardinals.Assaf Rinot - 2014 - Bulletin of Symbolic Logic 20 (3):293-314,.
    The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of (...) a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > א1, the principleis equivalent to the existence of a certain strong coloring c : [κ]2κ for which the family of fibers T is a nonspecial κ-Aronszajn tree. The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the κ-chain condition is productive for a given regular cardinal κ > א1, then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ-chain condition is productive. (shrink)
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  42. Category Theory as an Autonomous Foundation.Øystein Linnebo & Richard Pettigrew - 2011 - Philosophia Mathematica 19 (3):227-254.
    Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three (...)
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  43.  7
    The Ground Axiom.Jonas Reitz - 2007 - Journal of Symbolic Logic 72 (4):1299 - 1317.
    A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is (...)
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  44.  11
    Combinatorial Principles in the Core Model for One Woodin Cardinal.Ernest Schimmerling - 1995 - Annals of Pure and Applied Logic 74 (2):153-201.
    We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form (...)
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  45. Does Homotopy Type Theory Provide a Foundation for Mathematics?James Ladyman & Stuart Presnell - 2016 - British Journal for the Philosophy of Science:axw006.
    Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set (...)
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  46.  23
    Iteration One More Time.R. Cook - 2003 - Notre Dame Journal of Formal Logic 44 (2):63--92.
    A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. (...)The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that there are infinitely many nonsets, captures all (or enough) of standard second-order ZFC. Issues pertaining to the axiom of foundation are also investigated, and I conclude by arguing that this treatment provides the neologicist with the most viable reconstruction of set theory he is likely to obtain. (shrink)
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  47.  14
    Certain Very Large Cardinals Are Not Created in Small Forcing Extensions.Richard Laver - 2007 - Annals of Pure and Applied Logic 149 (1):1-6.
    The large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j:, the existence of such a j which is moreover (...) , and the existence of such a j which extends to an elementary j:+1+1. It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of these preservations are proved. Also the following is shown : if V is a model of ZFC and V[G] is a -generic forcing extension of V, then in V[G], V is definable using the parameter +1, where. (shrink)
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  48.  5
    Strongly Uplifting Cardinals and the Boldface Resurrection Axioms.Joel David Hamkins & Thomas A. Johnstone - 2017 - Archive for Mathematical Logic 56 (7-8):1115-1133.
    We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we (...)show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing. (shrink)
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  49.  22
    The Tree Property at Successors of Singular Cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
    Assuming some large cardinals, a model of ZFC is obtained in which $\aleph_{\omega+1}$ carries no Aronszajn trees. It is also shown that if $\lambda$ is a (...) singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees. (shrink)
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  50.  21
    Canonical Structure in the Universe of Set Theory: Part Two.James Cummings, Matthew Foreman & Menachem Magidor - 2006 - Annals of Pure and Applied Logic 142 (1):55-75.
    We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of (...)
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