Search results for 'Admissible sets' (try it on Scholar)

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  1. Diego Rojas-Rebolledo (2012). Bounds on the Strength of Ordinal Definable Determinacy in Small Admissible Sets. Notre Dame Journal of Formal Logic 53 (3):351-371.score: 224.0
    We give upper and lower bounds for the strength of ordinal definable determinacy in a small admissible set. The upper bound is roughly a premouse with a measurable cardinal $\kappa$ of Mitchell order $\kappa^{++}$ and $\omega$ successors. The lower bound are models of ZFC with sequences of measurable cardinals, extending the work of Lewis, below a regular limit of measurable cardinals.
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  2. Jon Barwise (1975). Admissible Sets and Structures: An Approach to Definability Theory. Springer-Verlag.score: 210.0
  3. Gerhard Jäger (1986). Theories for Admissible Sets: A Unifying Approach to Proof Theory. Bibliopolis.score: 210.0
     
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  4. Jacob Lurie (1999). Anti-Admissible Sets. Journal of Symbolic Logic 64 (2):407-435.score: 180.0
    Aczel's theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the "circular logic" of [3]. This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical "extension" to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of (...)
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  5. Barbara Majcher‐Iwanow (2008). Polish Group Actions, Nice Topologies, and Admissible Sets. Mathematical Logic Quarterly 54 (6):597-616.score: 178.0
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  6. Dieter Probst (2005). On the Relationship Between Fixed Points and Iteration in Admissible Set Theory Without Foundation. Archive for Mathematical Logic 44 (5):561-580.score: 160.0
    In this article we show how to use the result in Jäger and Probst [7] to adapt the technique of pseudo-hierarchies and its use in Avigad [1] to subsystems of set theory without foundation. We prove that the theory KPi0 of admissible sets without foundation, extended by the principle (Σ-FP), asserting the existence of fixed points of monotone Σ operators, has the same proof-theoretic ordinal as KPi0 extended by the principle (Σ-TR), that allows to iterate Σ operations along (...)
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  7. Jon Barwise (1969). Infinitary Logic and Admissible Sets. Journal of Symbolic Logic 34 (2):226-252.score: 150.0
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  8. H. Jerome Keisler & Julia F. Knight (2004). Barwise: Infinitary Logic and Admissible Sets. Bulletin of Symbolic Logic 10 (1):4-36.score: 150.0
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  9. Alan Adamson (1978). Admissible Sets and the Saturation of Structures. Annals of Mathematical Logic 14 (2):111-157.score: 150.0
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  10. Bernd I. Dahn (1978). Admissible Sets and Structures. Studia Logica 37 (3):297-299.score: 150.0
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  11. E. G. K. Lopez-Escobar (1971). Review: Jon Barwise, Infinitary Logic and Admissible Sets. [REVIEW] Journal of Symbolic Logic 36 (1):156-157.score: 150.0
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  12. Carl E. Bredlau (1979). Admissible Sets and Recursive Equivalence Types. Notre Dame Journal of Formal Logic 20 (2):355-365.score: 150.0
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  13. Nigel Cutland (1973). Model Theory on Admissible Sets. Annals of Mathematical Logic 5 (4):257-289.score: 150.0
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  14. Judy Green (1974). Σ1 Compactness for Next Admissible Sets. Journal of Symbolic Logic 39 (1):105 - 116.score: 150.0
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  15. Judy Green (1974). $Sigma_1$ Compactness for Next Admissible Sets. Journal of Symbolic Logic 39 (1):105-116.score: 150.0
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  16. Judy Green (1975). A Note on ${\Cal P}$-Admissible Sets with Urelements. Notre Dame Journal of Formal Logic 16 (3):415-417.score: 150.0
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  17. Judy Green (1977). Next $P$ Admissible Sets Are of Cofinality $\Omega$. Notre Dame Journal of Formal Logic 18 (1):175-176.score: 150.0
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  18. Menachem Magidor, Saharon Shelah & Jonathan Stavi (1984). Countably Decomposable Admissible Sets. Annals of Pure and Applied Logic 26 (3):287-361.score: 150.0
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  19. Mark Nadel (1978). Review: Jon Barwise, Admissible Sets and Structures. An Approach to Definability Theory. [REVIEW] Journal of Symbolic Logic 43 (1):139-144.score: 150.0
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  20. Mark Nadel (1974). Scott Sentences and Admissible Sets. Annals of Mathematical Logic 7 (2-3):267-294.score: 150.0
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  21. John S. Schlipf (1977). A Guide to the Identification of Admissible Sets Above Structures. Annals of Mathematical Logic 12 (2):151-192.score: 150.0
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  22. Steven Homer (1981). Review: Wolfgang Maass, Inadmissibility, Tame R.E. Sets and the Admissible Collapse; Wolfgang Maass, On $Alpha$- and $Beta$-Recursively Enumerable Degrees. [REVIEW] Journal of Symbolic Logic 46 (3):665-667.score: 120.0
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  23. Wolfgang Maass (1978). Inadmissibility, Tame R.E. Sets and the Admissible Collapse. Annals of Mathematical Logic 13 (2):149-170.score: 120.0
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  24. Michael Machtey (1971). Admissible Ordinals and Lattices of Α-R.E. Sets. Annals of Mathematical Logic 2 (4):379-417.score: 120.0
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  25. M. Makkai (1977). An “Admissible” Generalization of a Theorem on Countable ∑11 Sets of Reals with Applications. Annals of Mathematical Logic 11 (1):1-30.score: 120.0
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  26. Sy D. Friedman (1979). HC of an Admissible Set. Journal of Symbolic Logic 44 (1):95-102.score: 100.0
    If A is an admissible set, let HC(A) = {x∣ x ∈ A and x is hereditarily countable in A}. Then HC(A) is admissible. Corollaries are drawn characterizing the "real parts" of admissible sets and the analytical consequences of admissible set theory.
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  27. Jeremy Avigad (2002). An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations. Journal of Mathematical Logic 2 (01):91-112.score: 96.0
    The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides (...)
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  28. Michael Glanzberg (2004). A Contextual-Hierarchical Approach to Truth and the Liar Paradox. Journal of Philosophical Logic 33 (1):27-88.score: 90.0
    This paper presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context drawn from linguistics and philosophy of language, we can see the Liar sentence to be context dependent. Once this context dependence is properly understood, it is argued, a hierarchical structure emerges which is (...)
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  29. Barbara Majcher-Iwanow (2012). Polish Group Actions and Effectivity. Archive for Mathematical Logic 51 (5-6):563-573.score: 90.0
    We extend a theorem of Barwise and Nadel describing the relationship between approximations of canonical Scott sentences and admissible sets to the case of orbit equivalence relations induced on an arbitrary Polish space by a Polish group action.
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  30. Gerhard Jäger & Thomas Strahm (2001). Upper Bounds for Metapredicative Mahlo in Explicit Mathematics and Admissible Set Theory. Journal of Symbolic Logic 66 (2):935-958.score: 72.0
    In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established.
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  31. Robert Goldblatt & Michael Kane (2010). An Admissible Semantics for Propositionally Quantified Relevant Logics. Journal of Philosophical Logic 39 (1):73 - 100.score: 66.0
    The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p -instantiations of A . It is also shown that without the admissibility qualification many of the (...)
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  32. K. J. Barwise, R. O. Gandy & Y. N. Moschovakis (1971). The Next Admissible Set. Journal of Symbolic Logic 36 (1):108-120.score: 60.0
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  33. Heikki Heikkilä & Jouko Väänänen (1994). Reflection of Long Game Formulas. Mathematical Logic Quarterly 40 (3):381-392.score: 60.0
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  34. Matthias Schröder (2007). Admissible Representations for Probability Measures. Mathematical Logic Quarterly 53 (4):431-445.score: 58.0
    In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of type-2 theory of effectivity. This gives rise to a natural representation of the set of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel probability measure as a lower semicontinuous function from the open sets to the unit interval. We show that this canonical (...)
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  35. Angelika Kratzer (2005). Constraining Premise Sets for Counterfactuals. Journal of Semantics 22 (2):153-158.score: 54.0
    This note is a reply to ‘On the Lumping Semantics of Counterfactuals’ by Makoto Kanazawa, Stefan Kaufmann and Stanley Peters. It shows first that the first triviality result obtained by Kanazawa, Kaufmann, and Peters is already ruled out by the constraints on admissible premise sets listed in Kratzer (1989). Second, and more importantly, it points out that the results obtained by Kanazawa, Kaufmann, and Peters are obsolete in view of the revised analysis of counterfactuals in Kratzer (1990, 2002).
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  36. Theodore A. Slaman (1986). On the Kleene Degrees of Π11 Sets. Journal of Symbolic Logic 51 (2):352 - 359.score: 54.0
    Let A and B be subsets of the reals. Say that A κ ≥ B, if there is a real a such that the relation "x ∈ B" is uniformly Δ 1 (a, A) in L[ ω x,a,A 1 , x,a,A]. This reducibility induces an equivalence relation $\equiv_\kappa$ on the sets of reals; the $\equiv_\kappa$ -equivalence class of a set is called its Kleene degree. Let K be the structure that consists of the Kleene degrees and the induced partial (...)
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  37. Boniface Mbih (1995). On Admissible Strategies and Manipulation of Social Choice Procedures. Theory and Decision 39 (2):169-188.score: 48.0
    A collective choice mechanism can be viewed as a game in normal form; in this article it is shown, for very attractive rules and for sets with any number of alternatives, how individuals involved in a collective decision problem can construct the preferences they choose to express. An example is given with a version of plurality rule. Manipulability results are deduced from such a characterization.
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  38. Shaughan Lavine (1993). Generalized Reduction Theorems for Model-Theoretic Analogs of the Class of Coanalytic Sets. Journal of Symbolic Logic 58 (1):81-98.score: 44.0
    Let A be an admissible set. A sentence of the form ∀R̄φ is a ∀1(A) (∀s 1(A),∀1(Lω1ω)) sentence if φ ∈ A (φ is $\bigvee\Phi$ , where Φ is an A-r.e. set of sentences from A; φ ∈ Lω1ω). A sentence of the form ∃R̄φ is an ∃2(A) (∃s 2(A),∃2(Lω1ω)) sentence if φ is a ∀1(A) (∀s 1(A),∀1(Lω1ω)) sentence. A class of structures is, for example, a ∀1(A) class if it is the class of models of a ∀1(A) sentence. (...)
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  39. Noam Greenberg (2005). The Role of True Finiteness in the Admissible Recursively Enumerable Degrees. Bulletin of Symbolic Logic 11 (3):398-410.score: 42.0
    We show, however, that this is not always the case.
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  40. Fabio G. Cozman (2012). Sets of Probability Distributions, Independence, and Convexity. Synthese 186 (2):577-600.score: 38.0
    This paper analyzes concepts of independence and assumptions of convexity in the theory of sets of probability distributions. The starting point is Kyburg and Pittarelli’s discussion of “convex Bayesianism” (in particular their proposals concerning E-admissibility, independence, and convexity). The paper offers an organized review of the literature on independence for sets of probability distributions; new results on graphoid properties and on the justification of “strong independence” (using exchangeability) are presented. Finally, the connection between Kyburg and Pittarelli’s results and (...)
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  41. Alexander Citkin (2012). A Note on Admissible Rules and the Disjunction Property in Intermediate Logics. Archive for Mathematical Logic 51 (1-2):1-14.score: 38.0
    With any structural inference rule A/B, we associate the rule ${(A \lor p)/(B \lor p)}$ , providing that formulas A and B do not contain the variable p. We call the latter rule a join-extension ( ${\lor}$ -extension, for short) of the former. Obviously, for any intermediate logic with disjunction property, a ${\lor}$ -extension of any admissible rule is also admissible in this logic. We investigate intermediate logics, in which the ${\lor}$ -extension of each admissible rule is (...)
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  42. Solomon Feferman with with R. L. Vaught, Operational Set Theory and Small Large Cardinals.score: 36.0
    “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 (...)
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  43. Alexej P. Pynko (2009). Distributive-Lattice Semantics of Sequent Calculi with Structural Rules. Logica Universalis 3 (1):59-94.score: 36.0
    The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables (viz., metavariables for finite sets of formulas), upon the basis of the conception of model introduced in (Fuzzy Sets Syst 121(3):27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules (...)
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  44. P. D. Welch (2004). On Unfoldable Cardinals, Ω-Closed Cardinals, and the Beginning of the Inner Model Hierarchy. Archive for Mathematical Logic 43 (4):443-458.score: 30.0
    Let κ be a cardinal, and let H κ be the class of sets of hereditary cardinality less than κ ; let τ (κ) > κ be the height of the smallest transitive admissible set containing every element of {κ}∪H κ . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H κ is as long as τ. (It is known that (...)
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  45. Jens Erik Fenstad, R. O. Gandy & Gerald E. Sacks (eds.) (1978). Generalized Recursion Theory Ii: Proceedings of the 1977 Oslo Symposium. Sole Distributors for the U.S.A. And Canada, Elsevier North-Holland.score: 30.0
    GENERALIZED RECUBION THEORY II © North-Holland Publishing Company (1978) MONOTONE QUANTIFIERS AND ADMISSIBLE SETS Ion Barwise University of Wisconsin ...
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  46. Mark Nadel & Jonathan Stavi (1977). The Pure Part of HYP(M). Journal of Symbolic Logic 42 (1):33-46.score: 30.0
    Let M be a structure for a language L on a set M of urelements. HYP(M) is the least admissible set above M. In § 1 we show that pp(HYP(M)) [ = the collection of pure sets in HYP(M] is determined in a simple way by the ordinal α = ⚬(HYP(M)) and the $\mathscr{L}_{\propto\omega}$ theory of M up to quantifier rank α. In § 2 we consider the question of which pure countable admissible sets are of (...)
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  47. Mark Nadel & Jonathan Stavi (1977). The Pure Part of $Mathrm{HYP}(Mathscr{M}$). Journal of Symbolic Logic 42 (1):33-46.score: 30.0
    Let $\mathscr{M}$ be a structure for a language $\mathscr{L}$ on a set $M$ of urelements. $\mathrm{HYP}(\mathscr{M})$ is the least admissible set above $\mathscr{M}$. In $\S 1$ we show that $pp(\mathrm{HYP}(\mathscr{M})) \lbrack = \text{the collection of pure sets in} \mathrm{HYP}(\mathscr{M}\rbrack$ is determined in a simple way by the ordinal $\alpha = \circ(\mathrm{HYP}(\mathscr{M}))$ and the $\mathscr{L}_{\propto\omega}$ theory of $\mathscr{M}$ up to quantifier rank $\alpha$. In $\S 2$ we consider the question of which pure countable admissible sets are of (...)
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  48. Dieter Probst & Thomas Strahm (2011). Admissible Closures of Polynomial Time Computable Arithmetic. Archive for Mathematical Logic 50 (5-6):643-660.score: 28.0
    We propose two admissible closures ${\mathbb{A}({\sf PTCA})}$ and ${\mathbb{A}({\sf PHCA})}$ of Ferreira’s system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) ${\mathbb{A}({\sf PTCA})}$ is conservative over PTCA with respect to ${\forall\exists\Sigma^b_1}$ sentences, and (ii) ${\mathbb{A}({\sf PHCA})}$ is conservative over full bounded arithmetic PHCA for ${\forall\exists\Sigma^b_{\infty}}$ sentences. This yields that (i) the ${\Sigma^b_1}$ definable functions of ${\mathbb{A}({\sf PTCA})}$ are the polytime functions, and (ii) the ${\Sigma^b_{\infty}}$ (...)
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  49. T. A. Slaman (1986). ∑1 Definitions with Parameters. Journal of Symbolic Logic 51 (2):453 - 461.score: 26.0
    Let p be a set. A function φ is uniformly σ 1 (p) in every admissible set if there is a σ 1 formula φ in the parameter p so that φ defines φ in every σ 1 -admissible set which includes p. A theorem of Van de Wiele states that if φ is a total function from sets to sets then φ is uniformly σ 1R in every admissible set if anly only if it (...)
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  50. Shaughan Lavine (1992). A Spector-Gandy Theorem for cPCd(A) Classes. Journal of Symbolic Logic 57 (2):478 - 500.score: 26.0
    Let U be an admissible structure. A cPCd(U) class is the class of all models of a sentence of the form $\neg\exists\bar{K} \bigwedge \Phi$ , where K̄ is an U-r.e. set of relation symbols and φ is an U-r.e. set of formulas of L∞ω that are in U. The main theorem is a generalization of the following: Let U be a pure countable resolvable admissible structure such that U is not Σ-elementarily embedded in HYP(U). Then a class K (...)
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