Search results for 'higher set theory' (try it on Scholar)

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  1. Harvey Friedman, Higher Set Theory.
    Russell’s way out of his paradox via the impre-dicative theory of types has roughly the same logical power as Zermelo set theory - which supplanted it as a far more flexible and workable axiomatic foundation for mathematics. We discuss some new formalisms that are conceptually close to Russell, yet simpler, and have the same logical power as higher set theory - as represented by the far more powerful Zermelo-Frankel set theory and beyond. END.
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  2.  8
    W. Degen & J. Johannsen (2000). Cumulative Higher-Order Logic as a Foundation for Set Theory. Mathematical Logic Quarterly 46 (2):147-170.
    The systems Kα of transfinite cumulative types up to α are extended to systems K∞α that include a natural infinitary inference rule, the so-called limit rule. For countable α a semantic completeness theorem for K∞α is proved by the method of reduction trees, and it is shown that every model of K∞α is equivalent to a cumulative hierarchy of sets. This is used to show that several axiomatic first-order set theories can be interpreted in K∞α, for suitable α.
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  3.  87
    S. Shapiro (2012). Higher-Order Logic or Set Theory: A False Dilemma. Philosophia Mathematica 20 (3):305-323.
    The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?
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  4.  5
    Harvey M. Friedman (1971). Higher Set Theory and Mathematical Practice. Annals of Mathematical Logic 2 (3):325-357.
  5. Akihiro Kanamori (1994). The Higher Infinite Large Cardinals in Set Theory From Their Beginnings.
  6.  1
    Richard Montague, J. N. Crossley & M. A. E. Dummett (1975). Set Theory and Higher-Order Logic. Journal of Symbolic Logic 40 (3):459-459.
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  7.  3
    Azriel Levy (1996). Review: Akihiro Kanamori, The Higher Infinite. Large Cardinals in Set Theory From Their Beginnings. [REVIEW] Journal of Symbolic Logic 61 (1):334-336.
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  8.  2
    Richard Mansfield (1975). Review: Richard Montague, J. N. Crossley, M. A. E. Dummett, Set Theory and Higher-Order Logic. [REVIEW] Journal of Symbolic Logic 40 (3):459-459.
  9. A. Hajnal & P. Komjáth (1987). Some Higher-Gap Examples in Combinatorial Set Theory. Annals of Pure and Applied Logic 33 (3):283-296.
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  10.  13
    Siegfried Gottwald (2006). Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches. [REVIEW] Studia Logica 82 (2):211 - 244.
    For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.
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  11. Harvey Friedman, Axiomatization of Set Theory by Extensionality, Separation, and Reducibility.
    We discuss several axiomatizations of set theory in first order predicate calculus with epsilon and a constant symbol W, starting with the simple system K(W) which has a strong equivalence with ZF without Foundation. The other systems correspond to various extensions of ZF by certain large cardinal hypotheses. These axiomatizations are unusually simple and uncluttered, and are highly suggestive of underlying philosophical principles that generate higher set theory.
     
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  12.  17
    Akihiro Kanamori (2009). Bernays and Set Theory. Bulletin of Symbolic Logic 15 (1):43-69.
    We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.
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  13.  9
    Sy D. Friedman (1983). Some Recent Developments in Higher Recursion Theory. Journal of Symbolic Logic 48 (3):629-642.
    In recent years higher recursion theory has experienced a deep interaction with other areas of logic, particularly set theory (fine structure, forcing, and combinatorics) and infinitary model theory. In this paper we wish to illustrate this interaction by surveying the progress that has been made in two areas: the global theory of the κ-degrees and the study of closure ordinals.
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  14.  14
    Scott Martin & Carl Pollard (2012). A Higher-Order Theory of Presupposition. Studia Logica 100 (4):727-751.
    So-called 'dynamic' semantic theories such as Kamp's discourse representation theory and Heim's file change semantics account for such phenomena as cross-sentential anaphora, donkey anaphora, and the novelty condition on indefinites, but compare unfavorably with Montague semantics in some important respects (clarity and simplicity of mathematical foundations, compositionality, handling of quantification and coordination). Preliminary efforts have been made by Muskens and by de Groote to revise and extend Montague semantics to cover dynamic phenomena. We present a new higher-order (...) of discourse semantics which improves on their accounts by incorporating a more articulated notion of context inspired by ideas due to David Lewis and to Craige Roberts. On our account, a context consists of a common ground of mutually accepted propositions together with a set of discourse referents preordered by relative salience. Employing a richer notion of contexts enables us to extend our coverage beyond pronominal anaphora to a wider range of presuppositional phenomena, such as the factivity of certain sentential-complement verbs, resolution of anaphora associated with arbitrarily complex definite descriptions, presupposition 'holes' such as negation, and the independence condition on the antecedents of conditionals. Formally, our theory is expressed within a higher-order logic with natural number type, separation-style subtyping, and dependent coproducts parameterized by the natural numbers. The system of semantic types builds on proposals due to Thomason and to Pollard in which the type of propositions (static meanings of sentential utterances) is taken as basic and worlds are constructed from propositions (rather than the other way around as in standard Montague semantics). (shrink)
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  15.  33
    Jerzy Król (2006). A Model for Spacetime II. The Emergence of Higher Dimensions and Field Theory/Strings Dualities. Foundations of Physics 36 (12):1778-1800.
    We show that in 4-spacetime modified at very short distances due to the weakening of classical logic, the higher dimensions emerge. We analyse the case of some smooth topoi, and the case of some class of pointless topoi. The pointless topoi raise the dimensionality due to the forcing adding “string” objects and thus replacing classical points in spacetime. Turning to strings would be something fundamental and connected with set theoretical forcing. The field theory/strings dualities originate at the set (...)
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  16. Hakwan Lau (2008). A Higher Order Bayesian Decision Theory of Consciousness. In Rahul Banerjee & B. K. Chakrabarti (eds.), Models of Brain and Mind: Physical, Computational, and Psychological Approaches. Elsevier
    It is usually taken as given that consciousness involves superior or more elaborate forms of information processing. Contemporary models equate consciousness with global processing, system complexity, or depth or stability of computation. This is in stark contrast with the powerful philosophical intuition that being conscious is more than just having the ability to compute. I argue that it is also incompatible with current empirical findings. I present a model that is free from the strong assumption that consciousness predicts superior performance. (...)
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  17.  19
    Bart Jacobs (1989). The Inconsistency of Higher Order Extensions of Martin-Löf's Type Theory. Journal of Philosophical Logic 18 (4):399 - 422.
    Martin-Löf's constructive type theory forms the basis of this paper. His central notions of category and set, and their relations with Russell's type theories, are discussed. It is shown that addition of an axiom - treating the category of propositions as a set and thereby enabling higher order quantification - leads to inconsistency. This theorem is a variant of Girard's paradox, which is a translation into type theory of Mirimanoff's paradox (concerning the set of all well-founded sets). (...)
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  18. Jaakko Hintikka, Reforming Logic (and Set Theory).
    1. Frege’s mistake Frege is justifiably considered the most important thinker in the development of our contemporary “modern” logic. One corollary to this historical role of Frege’s is that his mistakes are found in a magnified form in the subsequent development of logic. This paper examines one such mistake and its later history. Diagnosing this history also reveals ways of overcoming some of the limitations that Frege’s mistake has unwittingly imposed on current forms of modern logic. Frege’s mistake concerns the (...)
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  19.  2
    Lucius T. Schoenbaum (2010). On the Syntax of Logic and Set Theory. Review of Symbolic Logic 3 (4):568-599.
    We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set-theoretic (...)
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  20.  39
    Michael D. Potter (2004). Set Theory and its Philosophy: A Critical Introduction. Oxford University Press.
    Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes (...)
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  21.  54
    A. A. Fraenkel, Y. Bar-Hillel & A. Levy (1973). Foundations of Set Theory. North Holland.
    HISTORICAL INTRODUCTION In Abstract Set Theory) the elements of the theory of sets were presented in a chiefly generic way: the fundamental concepts were ...
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  22.  15
    Willard V. Quine (1963). Set Theory and its Logic. Harvard University Press.
    This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject.
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  23. Penelope Maddy (2011). Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford University Press.
    Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account (...)
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  24.  88
    Salvatore Florio & Stewart Shapiro (2014). Set Theory, Type Theory, and Absolute Generality. Mind 123 (489):157-174.
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or (...)
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  25.  47
    Zach Weber (2010). Extensionality and Restriction in Naive Set Theory. Studia Logica 94 (1):87 - 104.
    The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set (...), leads to triviality. (shrink)
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  26.  10
    Jonas R. Becker Arenhart (2012). Finite Cardinals in Quasi-Set Theory. Studia Logica 100 (3):437-452.
    Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a theory is to deal properly with collections of items like particles in non-relativistic quantum mechanics when these are understood as being non-individuals in the sense that they may be indistinguishable although identity does not apply to (...)
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  27.  53
    M. Rathjen (2001). Kripke-Platek Set Theory and the Anti-Foundation Axiom. Mathematical Logic Quarterly 47 (4):435-440.
    The paper investigates the strength of the Anti-Foundation Axiom, AFA, on the basis of Kripke-Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength.
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  28.  18
    Peter Verdée (2013). Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics. Foundations of Science 18 (4):655-680.
    In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also (...)
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  29.  40
    Shunsuke Yatabe (2007). Distinguishing Non-Standard Natural Numbers in a Set Theory Within Łukasiewicz Logic. Archive for Mathematical Logic 46 (3-4):281-287.
    In ${\mathbf{H}}$ , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of ${\mathbf{H}}$ , and we prove an analogy of Hájek’s theorem with a very simple procedure.
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  30.  36
    Jonas Rafael Becker Arenhart (2011). A Discussion on Finite Quasi-Cardinals in Quasi-Set Theory. Foundations of Physics 41 (8):1338-1354.
    Quasi-set theory Q is an alternative set-theory designed to deal mathematically with collections of indistinguishable objects. The intended interpretation for those objects is the indistinguishable particles of non-relativistic quantum mechanics, under one specific interpretation of that theory. The notion of cardinal of a collection in Q is treated by the concept of quasi-cardinal, which in the usual formulations of the theory is introduced as a primitive symbol, since the usual means of cardinal definition fail for collections (...)
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  31.  11
    Richard Kaye & Tin Lok Wong (2007). On Interpretations of Arithmetic and Set Theory. Notre Dame Journal of Formal Logic 48 (4):497-510.
    This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. (...)
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  32.  25
    Ross T. Brady (2014). The Simple Consistency of Naive Set Theory Using Metavaluations. Journal of Philosophical Logic 43 (2-3):261-281.
    The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of (...)
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  33.  61
    Jeffrey W. Roland (2010). Concept Grounding and Knowledge of Set Theory. Philosophia 38 (1):179-193.
    C. S. Jenkins has recently proposed an account of arithmetical knowledge designed to be realist, empiricist, and apriorist: realist in that what’s the case in arithmetic doesn’t rely on us being any particular way; empiricist in that arithmetic knowledge crucially depends on the senses; and apriorist in that it accommodates the time-honored judgment that there is something special about arithmetical knowledge, something we have historically labeled with ‘a priori’. I’m here concerned with the prospects for extending Jenkins’s account beyond arithmetic—in (...)
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  34.  8
    Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (2015). Multiverse Conceptions in Set Theory. Synthese 192 (8):2463-2488.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
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  35.  10
    Moshe Machover (1996). Set Theory, Logic and Their Limitations. Cambridge University Press.
    This is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations.
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  36. Richard Pettigrew (2010). The Foundations of Arithmetic in Finite Bounded Zermelo Set Theory. Cahiers du Centre de Logique 17:99-118.
    In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of (...)
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  37.  3
    Laurence Kirby (2010). Substandard Models of Finite Set Theory. Mathematical Logic Quarterly 56 (6):631-642.
    A survey of the isomorphic submodels of Vω, the set of hereditarily finite sets. In the usual language of set theory, Vω has 2ℵ0 isomorphic submodels. But other set-theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of Vω.
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  38.  10
    Ray-Ming Chen & Michael Rathjen (2012). Lifschitz Realizability for Intuitionistic Zermelo–Fraenkel Set Theory. Archive for Mathematical Logic 51 (7-8):789-818.
    A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition, but not the general form of Church’s thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101–106, 1979). A Lifschitz counterpart to Kleene’s realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805–821, 1990). In that paper he also extended Lifschitz’ realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo–Fraenkel set theory, IZF. The (...)
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  39.  5
    Maria Emilia Maietti & Silvio Valentini (1999). Can You Add Power‐Sets to Martin‐Lof's Intuitionistic Set Theory? Mathematical Logic Quarterly 45 (4):521-532.
    In this paper we analyze an extension of Martin-Löf s intensional set theory by means of a set contructor P such that the elements of P are the subsets of the set S. Since it seems natural to require some kind of extensionality on the equality among subsets, it turns out that such an extension cannot be constructive. In fact we will prove that this extension is classic, that is “ true holds for any proposition A.
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  40.  85
    Han Geurdes, Heisenberg Quantum Mechanics, Numeral Set-Theory And.
    In the paper we will employ set theory to study the formal aspects of quantum mechanics without explicitly making use of space-time. It is demonstrated that von Neuman and Zermelo numeral sets, previously efectively used in the explanation of Hardy’s paradox, follow a Heisenberg quantum form. Here monadic union plays the role of time derivative. The logical counterpart of monadic union plays the part of the Hamiltonian in the commutator. The use of numerals and monadic union in the classical (...)
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  41.  8
    Frode Bjørdal (2011). The Inadequacy of a Proposed Paraconsistent Set Theory. Review of Symbolic Logic 4 (1):106-108.
    We show that a paraconsistent set theory proposed in Weber (2010) is strong enough to provide a quite classical nonprimitive notion of identity, so that the relation is an equivalence relation and also obeys full substitutivity: a = b -> F(b)). With this as background it is shown that the proposed theory also proves the negation of x=x. While not by itself showing that the proposed system is trivial in the sense of proving all statements, it is argued (...)
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  42. Richard Pettigrew (2009). On Interpretations of Bounded Arithmetic and Bounded Set Theory. Notre Dame Journal of Formal Logic 50 (2):141-152.
    In 'On interpretations of arithmetic and set theory', Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.

    THEOREM 1 The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.

    In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot (...)
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  43.  1
    Michael Rathjen (2008). The Natural Numbers in Constructive Set Theory. Mathematical Logic Quarterly 54 (1):83-97.
    Constructive set theory started with Myhill's seminal 1975 article [8]. This paper will be concerned with axiomatizations of the natural numbers in constructive set theory discerned in [3], clarifying the deductive relationships between these axiomatizations and the strength of various weak constructive set theories.
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  44.  8
    A. R. D. Mathias (2001). Slim Models of Zermelo Set Theory. Journal of Symbolic Logic 66 (2):487-496.
    Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence $\langle A_{\lambda} \mid \lambda \text{a limit ordinal} \rangle$ where for each $\lambda, A_{\lambda} \subseteq ^{\lambda}2$ , there is a supertransitive inner model of Zermelo containing all ordinals in which for every λ (...)
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  45.  84
    Mary Tiles (1989/2004). The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. Dover Publications.
    David Hilbert famously remarked, “No one will drive us from the paradise that Cantor has created.” This volume offers a guided tour of modern mathematics’ Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor’s transfinite paradise; axiomatic set theory; logical objects and logical types; independence results and the universe of sets; and the constructs and reality of mathematical structure. Philosophers (...)
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  46.  3
    Michael Rathjen (2006). A Note on Bar Induction in Constructive Set Theory. Mathematical Logic Quarterly 52 (3):253-258.
    Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive Zermelo-Fraenkel Set Theory, CZF. It is shown that CZF augmented by decidable Bar Induction proves the 1-consistency of CZF. This answers a question of P. Aczel who used Bar Induction to give a proof of the Lusin Separation Theorem in the constructive set theory CZF.
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  47.  28
    Rodrigo A. Freire (2012). On Existence in Set Theory. Notre Dame Journal of Formal Logic 53 (4):525-547.
    The aim of the present paper is to provide a robust classification of valid sentences in set theory by means of existence and related notions and, in this way, to capture similarities and dissimilarities among the axioms of set theory. In order to achieve this, precise definitions for the notions of productive and nonproductive assertions, constructive and nonconstructive productive assertions, and conditional and unconditional productive assertions, among others, will be presented. These definitions constitute the result of a semantical (...)
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  48.  16
    Petr Andreev & Karel Hrbacek (2004). Standard Sets in Nonstandard Set Theory. Journal of Symbolic Logic 69 (1):165-182.
    We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.
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  49.  22
    Alexander Paseau (2001). Should the Logic of Set Theory Be Intuitionistic? Proceedings of the Aristotelian Society 101 (3):369–378.
    It is commonly assumed that classical logic is the embodiment of a realist ontology. In “Sets and Semantics”, however, Jonathan Lear challenged this assumption in the particular case of set theory, arguing that even if one is a set-theoretic Platonist, due attention to a special feature of set theory leads to the conclusion that the correct logic for it is intuitionistic. The feature of set theory Lear appeals to is the open-endedness of the concept of set. This (...)
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  50.  45
    P. T. Johnstone (1987). Notes on Logic and Set Theory. Cambridge University Press.
    A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included (...)
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