Results for 'forcing'

999 found
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  1.  9
    Forcing with Sequences of Models of Two Types.Itay Neeman - 2014 - Notre Dame Journal of Formal Logic 55 (2):265-298.
    We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than $\aleph_{1}$, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. (...)
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  2.  17
    The Subjective Roots of Forcing Theory and Their Influence in Independence Results.Stathis Livadas - 2015 - Axiomathes 25 (4):433-455.
    This article attempts a subjectively based approach, in fact one phenomenologically motivated, toward some key concepts of forcing theory, primarily the concepts of a generic set and its global properties and the absoluteness of certain fundamental relations in the extension to a forcing model M[G]. By virtue of this motivation and referring both to the original and current formulation of forcing I revisit certain set-theoretical notions serving as underpinnings of the theory and try to establish their deeper (...)
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  3. Supercompact Extender Based Prikry Forcing.Carmi Merimovich - 2011 - Archive for Mathematical Logic 50 (5-6):591-602.
    The extender based Prikry forcing notion is being generalized to super compact extenders.
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  4. The Short Extenders Gap Three Forcing Using a Morass.Carmi Merimovich - 2011 - Archive for Mathematical Logic 50 (1-2):115-135.
    We show how to construct Gitik’s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type.
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  5.  45
    Inner Models with Large Cardinal Features Usually Obtained by Forcing.Arthur W. Apter, Victoria Gitman & Joel David Hamkins - 2012 - Archive for Mathematical Logic 51 (3-4):257-283.
    We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 κ = κ +, another for which 2 κ = κ ++ and another in which the least strongly compact cardinal is (...)
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  6.  67
    The Necessary Maximality Principle for C. C. C. Forcing is Equiconsistent with a Weakly Compact Cardinal.Joel D. Hamkins & W. Hugh Woodin - 2005 - Mathematical Logic Quarterly 51 (5):493-498.
    The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal.
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  7.  18
    Semi-Proper Forcing, Remarkable Cardinals, and Bounded Martin's Maximum.Ralf Schindler - 2004 - Mathematical Logic Quarterly 50 (6):527-532.
    We show that L absoluteness for semi-proper forcings is equiconsistent with the existence of a remarkable cardinal, and hence by [6] with L absoluteness for proper forcings. By [7], L absoluteness for stationary set preserving forcings gives an inner model with a strong cardinal. By [3], the Bounded Semi-Proper Forcing Axiom is equiconsistent with the Bounded Proper Forcing Axiom , which in turn is equiconsistent with a reflecting cardinal. We show that Bounded Martin's Maximum is much stronger than (...)
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  8. The Short Extenders Gap Two Forcing is of Prikry Type.Carmi Merimovich - 2009 - Archive for Mathematical Logic 48 (8):737-747.
    We show that Gitik’s short extender gap-2 forcing is of Prikry type.
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  9.  17
    Forcing Under Anti‐Foundation Axiom: An Expression of the Stalks.Sato Kentaro - 2006 - Mathematical Logic Quarterly 52 (3):295-314.
    We introduce a new simple way of defining the forcing method that works well in the usual setting under FA, the Foundation Axiom, and moreover works even under Aczel's AFA, the Anti-Foundation Axiom. This new way allows us to have an intuition about what happens in defining the forcing relation. The main tool is H. Friedman's method of defining the extensional membership relation ∈ by means of the intensional membership relation ε .Analogously to the usual forcing and (...)
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  10.  12
    A Lifting Argument for the Generalized Grigorieff Forcing.Radek Honzík & Jonathan Verner - 2016 - Notre Dame Journal of Formal Logic 57 (2):221-231.
    In this short paper, we describe another class of forcing notions which preserve measurability of a large cardinal $\kappa$ from the optimal hypothesis, while adding new unbounded subsets to $\kappa$. In some ways these forcings are closer to the Cohen-type forcings—we show that they are not minimal—but, they share some properties with treelike forcings. We show that they admit fusion-type arguments which allow for a uniform lifting argument.
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  11.  25
    Creature Forcing and Large Continuum: The Joy of Halving.Jakob Kellner & Saharon Shelah - 2012 - Archive for Mathematical Logic 51 (1-2):49-70.
    For ${f,g\in\omega^\omega}$ let ${c^\forall_{f,g}}$ be the minimal number of uniform g-splitting trees needed to cover the uniform f-splitting tree, i.e., for every branch ν of the f-tree, one of the g-trees contains ν. Let ${c^\exists_{f,g}}$ be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that it is consistent that ${c^\exists_{f_\epsilon,g_\epsilon}{=}c^\forall_{f_\epsilon,g_\epsilon}{=}\kappa_\epsilon}$ for continuum many pairwise different cardinals ${\kappa_\epsilon}$ and suitable pairs ${(f_\epsilon,g_\epsilon)}$ . For the proof we introduce a new mixed-limit creature (...) construction. (shrink)
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  12.  33
    Fatal Heyting Algebras and Forcing Persistent Sentences.Leo Esakia & Benedikt Löwe - 2012 - Studia Logica 100 (1-2):163-173.
    Hamkins and Löwe proved that the modal logic of forcing is S4.2 . In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.
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  13.  8
    A Cofinality-Preserving Small Forcing May Introduce a Special Aronszajn Tree.Assaf Rinot - 2009 - Archive for Mathematical Logic 48 (8):817-823.
    It is relatively consistent with the existence of two supercompact cardinals that a special Aronszajn tree of height ${\aleph_{\omega_1+1}}$ is introduced by a cofinality-preserving forcing of size ${\aleph_3}$.
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  14.  15
    Sweet & Sour and Other Flavours of Ccc Forcing Notions.Andrzej Rosłanowski & Saharon Shelah - 2004 - Archive for Mathematical Logic 43 (5):583-663.
    We continue developing the general theory of forcing notions built with the use of norms on possibilities, this time concentrating on ccc forcing notions and classifying them.
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  15.  17
    Proper Forcing Extensions and Solovay Models.Joan Bagaria & Roger Bosch - 2003 - Archive for Mathematical Logic 43 (6):739-750.
    We study the preservation of the property of being a Solovay model under proper projective forcing extensions. We show that every strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a -Mahlo cardinal. We also show that the consistency strength of (...)
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  16. Combinatorics on Ideals and Forcing with Trees.Marcia J. Groszek - 1987 - Journal of Symbolic Logic 52 (3):582-593.
    Classes of forcings which add a real by forcing with branching conditions are examined, and conditions are found which guarantee that the generic real is of minimal degree over the ground model. An application is made to almost-disjoint coding via a real of minimal degree.
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  17.  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 existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable (...)
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  18.  11
    More About Λ-Support Iterations of (<Λ)-Complete Forcing Notions.Andrzej Rosłanowski & Saharon Shelah - 2013 - Archive for Mathematical Logic 52 (5-6):603-629.
    This article continues Rosłanowski and Shelah (Int J Math Math Sci 28:63–82, 2001; Quaderni di Matematica 17:195–239, 2006; Israel J Math 159:109–174, 2007; 2011; Notre Dame J Formal Logic 52:113–147, 2011) and we introduce here a new property of (<λ)-strategically complete forcing notions which implies that their λ-support iterations do not collapse λ + (for a strongly inaccessible cardinal λ).
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  19.  37
    Projective Absoluteness for Sacks Forcing.Daisuke Ikegami - 2009 - Archive for Mathematical Logic 48 (7):679-690.
    We show that ${{\bf \Sigma}^1_3}$ -absoluteness for Sacks forcing is equivalent to the non-existence of a ${{\bf \Delta}^1_2}$ Bernstein set. We also show that Sacks forcing is the weakest forcing notion among all of the preorders that add a new real with respect to ${{\bf \Sigma}^1_3}$ forcing absoluteness.
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  20.  29
    The Bounded Axiom A Forcing Axiom.Thilo Weinert - 2010 - Mathematical Logic Quarterly 56 (6):659-665.
    We introduce the Bounded Axiom A Forcing Axiom . It turns out that it is equiconsistent with the existence of a regular ∑2-correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom.
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  21. On the Equivalence of Certain Consequences of the Proper Forcing Axiom.Peter Nyikos & Leszek Piątkiewicz - 1995 - Journal of Symbolic Logic 60 (2):431-443.
    We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω 1 with ω 1 generators, then there exists an uncountable $X \subseteq \omega_1$ , such that either [ X] ω ∩ I = ⊘ or $\lbrack X\rbrack^\omega \subseteq I$.
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  22.  10
    Forcing Properties of Ideals of Closed Sets.Marcin Sabok & Jindřich Zapletal - 2011 - Journal of Symbolic Logic 76 (3):1075 - 1095.
    With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also (...)
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  23.  5
    Unsupported Boolean Algebras and Forcing.Miloš S. Kurilić - 2004 - Mathematical Logic Quarterly 50 (6):594-602.
    If κ is an infinite cardinal, a complete Boolean algebra B is called κ-supported if for each sequence 〈bβ : β αbβ = equation imagemath imageequation imageβ∈Abβ holds. Combinatorial and forcing equivalents of this property are given and compared with the other forcing related properties of Boolean algebras . The set of regular cardinals κ for which B is not κ-supported is investigated.
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  24.  49
    A Reduction Theorem for the Kripke–Joyal Semantics: Forcing Over an Arbitrary Category Can Always Be Replaced by Forcing Over a Complete Heyting Algebra. [REVIEW]Imants Barušs & Robert Woodrow - 2013 - Logica Universalis 7 (3):323-334.
    It is assumed that a Kripke–Joyal semantics \({\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}\) has been defined for a first-order language \({\mathcal{L}}\) . To transform \({\mathbb{C}}\) into a Heyting algebra \({\overline{\mathbb{C}}}\) on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of \({\mathbb{C}}\) . A pretopology \({\overline{{\rm Cov}}}\) is defined on \({\overline{\mathbb{C}}}\) using the pretopology on \({\mathbb{C}}\) . A sheaf \({\overline{{\it F}}}\) is made up of sections (...)
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  25.  31
    Forcing with the Anti‐Foundation Axiom.Olivier Esser - 2012 - Mathematical Logic Quarterly 58 (1-2):55-62.
    In this paper we define the forcing relation and prove its basic properties in the context of the theory ZFCA, i.e., ZFC minus the Foundation axiom and plus the Anti-Foundation axiom.
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  26.  9
    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.
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  27.  28
    Arithmetical Sacks Forcing.Rod Downey & Liang Yu - 2006 - Archive for Mathematical Logic 45 (6):715-720.
    We answer a question of Jockusch by constructing a hyperimmune-free minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented.
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  28.  52
    Essential Forcing Generics.Stephanie Cawthorne & David Kueker - 2000 - Notre Dame Journal of Formal Logic 41 (1):41-52.
    We use model theoretic forcing to study and generalize the construction of ()-generic models introduced by Kueker and Laskowski. We characterize the ()-generic models in terms of forcing and introduce a more general class of models, called essential forcing generics, which have many of the same properties.
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  29.  23
    A Variant of Mathias Forcing That Preserves {\ Mathsf {ACA} _0}.François G. Dorais - 2012 - Archive for Mathematical Logic 51 (7-8):751-780.
    We present and analyze ${F_\sigma}$ -Mathias forcing, which is similar but tamer than Mathias forcing. In particular, we show that this forcing preserves certain weak subsystems of second-order arithmetic such as ${\mathsf{ACA}_0}$ and ${\mathsf{WKL}_0 + \mathsf{I}\Sigma^0_2}$ , whereas Mathias forcing does not. We also show that the needed reals for ${F_\sigma}$ -Mathias forcing (in the sense of Blass in Ann Pure Appl Logic 109(1–2):77–88, 2001) are just the computable reals, as opposed to the hyperarithmetic reals (...)
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  30.  21
    Distributive Proper Forcing Axiom and Cardinal Invariants.Huiling Zhu - 2013 - Archive for Mathematical Logic 52 (5-6):497-506.
    In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.
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  31.  23
    Forcing, Downward Löwenheim-Skolem and Omitting Types Theorems, Institutionally.Daniel Găină - 2014 - Logica Universalis 8 (3-4):469-498.
    In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove two abstract results: Downward Löwenheim-Skolem Theorem and Omitting Types Theorem . We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These include first-order logic , (...)
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  32.  15
    Universal Forcing Notions and Ideals.Andrzej Rosłanowski & Saharon Shelah - 2007 - Archive for Mathematical Logic 46 (3-4):179-196.
    Our main result states that a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ-ideals determined by those universality parameters.
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  33.  11
    Generalized Prikry Forcing and Iteration of Generic Ultrapowers.Hiroshi Sakai - 2005 - Mathematical Logic Quarterly 51 (5):507-523.
    It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈Mn, jm,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j0,n | n ∈ ω〉 is a Prikry generic sequence over Mω. In this paper we generalize this for normal precipitous filters.
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  34.  11
    Forcing Operators on MTL-Algebras.George Georgescu & Denisa Diaconescu - 2011 - Mathematical Logic Quarterly 57 (1):47-64.
    We study the forcing operators on MTL-algebras, an algebraic notion inspired by the Kripke semantics of the monoidal t -norm based logic . At logical level, they provide the notion of the forcing value of an MTL-formula. We characterize the forcing operators in terms of some MTL-algebras morphisms. From this result we derive the equality of the forcing value and the truth value of an MTL-formula.
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  35.  11
    Gap Structure After Forcing with a Coherent Souslin Tree.Carlos Martinez-Ranero - 2013 - Archive for Mathematical Logic 52 (3-4):435-447.
    We investigate the effect after forcing with a coherent Souslin tree on the gap structure of the class of coherent Aronszajn trees ordered by embeddability. We shall show, assuming the relativized version PFA(S) of the proper forcing axiom, that the Souslin tree S forces that the class of Aronszajn trees ordered by the embeddability relation is universal for linear orders of cardinality at most ${\aleph_1}$.
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  36.  6
    A Formalism for Some Class of Forcing Notions.Piotr Koszmider & P. Koszmider - 1992 - Mathematical Logic Quarterly 38 (1):413-421.
    We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. We prove that (...)
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  37.  3
    Grigorieff Forcing on Uncountable Cardinals Does Not Add a Generic of Minimal Degree.Brooke M. Andersen & Marcia J. Groszek - 2009 - Notre Dame Journal of Formal Logic 50 (2):195-200.
    Grigorieff showed that forcing to add a subset of ω using partial functions with suitably chosen domains can add a generic real of minimal degree. We show that forcing with partial functions to add a subset of an uncountable κ without adding a real never adds a generic of minimal degree. This is in contrast to forcing using branching conditions, as shown by Brown and Groszek.
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  38.  14
    Intuitionistic Logic, Model Theory and Forcing.Melvin Fitting - 1969 - Amsterdam: North-Holland Pub. Co..
  39.  9
    Proper Forcing and Remarkable Cardinals II.Ralf-Dieter Schindler - 2001 - Journal of Symbolic Logic 66 (3):1481-1492.
    The current paper proves the results announced in [5]. We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and ω-Erdos cardinals. They are characterized by the existence of "O # -like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness for proper forcings. In particular, said absoluteness does not imply Π 1 1 determinacy.
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  40.  22
    Some Results About (+) Proved by Iterated Forcing.Tetsuya Ishiu & Paul B. Larson - 2012 - Journal of Symbolic Logic 77 (2):515-531.
    We shall show the consistency of CH+ᄀ(+) and CH+(+)+ there are no club guessing sequences on ω₁. We shall also prove that ◊⁺ does not imply the existence of a strong club guessing sequence ω₁.
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  41.  7
    Mad Families, Forcing and the Suslin Hypothesis.Miloš S. Kurilić - 2004 - Archive for Mathematical Logic 44 (4):499-512.
    Let κ be a regular cardinal and P a partial ordering preserving the regularity of κ. If P is (κ-Baire and) of density κ, then there is a mad family on κ killed in all generic extensions (if and) only if below each p∈P there exists a κ-sized antichain. In this case a mad family on κ is killed (if and) only if there exists an injection from κ onto a dense subset of Ult(P) mapping the elements of onto nowhere (...)
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  42.  8
    Understanding Preservation Theorems: Chapter VI of Proper and Improper Forcing, I.Chaz Schlindwein - 2014 - Archive for Mathematical Logic 53 (1-2):171-202.
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  43. Iterated Forcing and Coherent Sequences.M. C. Mcdermott - 1983
     
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  44.  9
    The Subcompleteness of Magidor Forcing.Gunter Fuchs - 2018 - Archive for Mathematical Logic 57 (3-4):273-284.
    It is shown that the Magidor forcing to collapse the cofinality of a measurable cardinal that carries a length \ sequence of normal ultrafilters, increasing in the Mitchell order, to \, is subcomplete.
    No categories
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  45. Gap Forcing: Generalizing the Lévy-Solovay Theorem.Joel David Hamkins - 1999 - Bulletin of Symbolic Logic 5 (2):264-272.
    The Lévy-Solovay Theorem [8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.
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  46.  11
    Bounded Forcing Axioms as Principles of Generic Absoluteness.Joan Bagaria - 2000 - Archive for Mathematical Logic 39 (6):393-401.
    We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a $\Sigma_1$ sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters (...)
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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:Vλ→Vλ, the existence of such a j which is moreover , and the existence of such a j which extends to an elementary j:Vλ+1→Vλ+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 (...)
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  48. A Characterization of Permutation Models in Terms of Forcing.Eric J. Hall - 2002 - Notre Dame Journal of Formal Logic 43 (3):157-168.
    We show that if N and M are transitive models of ZFA such that N M, N and M have the same kernel and same set of atoms, and M AC, then N is a Fraenkel-Mostowski-Specker (FMS) submodel of M if and only if M is a generic extension of N by some almost homogeneous notion of forcing. We also develop a slightly modified notion of FMS submodels to characterize the case where M is a generic extension of N (...)
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  49. No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics.Ehtibar N. Dzhafarov & Janne V. Kujala - 2014 - Foundations of Physics 44 (3):248-265.
    Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov’s probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice–Bob EPR paradigm, non-contextuality means that the identity of Alice’s spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis $\alpha _{i}$ chosen by Alice, irrespective of Bob’s axis $\beta _{j}$ (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice’s (...)
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  50.  6
    Forcing Absoluteness and Regularity Properties.Daisuke Ikegami - 2010 - Annals of Pure and Applied Logic 161 (7):879-894.
    For a large natural class of forcing notions, we prove general equivalence theorems between forcing absoluteness statements, regularity properties, and transcendence properties over and the core model . We use our results to answer open questions from set theory of the reals.
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