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Alternative axiomatic set theories

Stanford Encyclopedia of Philosophy (2008)

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  1. Admissible sets and structures: an approach to definability theory.Jon Barwise - 1975 - New York: Springer Verlag.
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  • Slim Models of Zermelo Set Theory.A. R. D. Mathias - 2001 - 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 $\Phi$ 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 $\lambda A_{\lambda} = \{\alpha (...)
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  • Consistency of strictly impredicative NF and a little more ….Sergei Tupailo - 2010 - Journal of Symbolic Logic 75 (4):1326-1338.
    An instance of Stratified Comprehension ∀x₁ … ∀x n ∃y∀x (x ∈ y ↔ φ(x, x₁, …, x n )) is called strictly impredicative iff, under minimal stratification, the type of x is 0. Using the technology of forcing, we prove that the fragment of NF based on strictly impredicative Stratified Comprehension is consistent. A crucial part in this proof, namely showing genericity of a certain symmetric filter, is due to Robert Solovay. As a bonus, our interpretation also satisfies some (...)
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  • Logic for mathematicians.J. Barkley Rosser - 1978 - Mineola, N.Y.: Dover Publications.
    Hailed by the Bulletin of the American Mathematical Society as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields. Starting with an introduction to symbolic logic, the first eight chapters develop logic through the restricted predicate calculus. Topics include the statement calculus, the use of names, an axiomatic treatment of the statement (...)
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  • Ackermann's set theory equals ZF.William N. Reinhardt - 1970 - Annals of Mathematical Logic 2 (2):189.
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  • On ordered pairs.W. V. Quine - 1945 - Journal of Symbolic Logic 10 (3):95-96.
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  • Failure of cartesian closedness in NF.Colin McLarty - 1992 - Journal of Symbolic Logic 57 (2):555-556.
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  • The strength of Mac Lane set theory.A. R. D. Mathias - 2001 - Annals of Pure and Applied Logic 110 (1-3):107-234.
    Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is (...)
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  • On Ackermann's set theory.Azriel Lévy - 1959 - Journal of Symbolic Logic 24 (2):154-166.
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  • A very strong set theory?Andrzej Kisielewicz - 1998 - Studia Logica 61 (2):171-178.
    Using two distinct membership symbols makes possible to base set theory on one general axiom schema of comprehension. Is the resulting system consistent? Can set theory and mathematics be based on a single axiom schema of comprehension?
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  • On the consistency of a slight (?) Modification of quine'smew foundations.Ronald Björn Jensen - 1968 - Synthese 19 (1-2):250 - 264.
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  • The Usual Model Construction for NFU Preserves Information.M. Randall Holmes - 2012 - Notre Dame Journal of Formal Logic 53 (4):571-580.
    The usual construction of models of NFU (New Foundations with urelements, introduced by Jensen) is due to Maurice Boffa. A Boffa model is obtained from a model of (a fragment of) Zermelo–Fraenkel with Choice (ZFC) with an automorphism which moves a rank: the domain of the Boffa model is a rank that is moved. “Most” elements of the domain of the Boffa model are urelements in terms of the interpreted NFU. The main result of this paper is that the restriction (...)
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  • The set-theoretic multiverse.Joel David Hamkins - 2012 - Review of Symbolic Logic 5 (3):416-449.
    The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...)
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  • On the Consistency of a Positive Theory.Olivier Esser - 1999 - Mathematical Logic Quarterly 45 (1):105-116.
    In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is a “strong” theory since “On (...)
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  • On the consistency of an impredicative subsystem of Quine's NF.Marcel Crabbé - 1982 - Journal of Symbolic Logic 47 (1):131-136.
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  • Frege's double correlation thesis and Quine's set theories NF and ML.Nino B. Cocchiarella - 1985 - Journal of Philosophical Logic 14 (1):1 - 39.
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  • Non-Well-Founded Sets.Peter Aczel - 1988 - Palo Alto, CA, USA: Csli Lecture Notes.
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  • Principles of Mathematics.Bertrand Russell - 1903 - New York,: Routledge.
    First published in 1903, _Principles of Mathematics_ was Bertrand Russell’s first major work in print. It was this title which saw him begin his ascent towards eminence. In this groundbreaking and important work, Bertrand Russell argues that mathematics and logic are, in fact, identical and what is commonly called mathematics is simply later deductions from logical premises. Highly influential and engaging, this important work led to Russell’s dominance of analytical logic on western philosophy in the twentieth century.
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  • Die grundlagen der arithmetik.Gottlob Frege - 1934 - Breslau,: M. & H. Marcus.
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  • Mathematics in Alternative Set Theory.Petr Vopĕnka - 1979 - Leipzig, Germany: Teubner.
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  • Logic, computers, and sets.Hao Wang - 1962 - New York,: Chelsea Pub. Co..
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  • The Ethics.Benedict de Spinoza - unknown
    Definitions Axioms Prop. I. Substance is by nature prior to its modifications Prop. II. Two substances, whose attributes are different, have nothing in common Prop III. Things, which have nothing in common, cannot be one the cause of the other Prop. IV. Two or more distinct things are distinguished one from the other either by the difference of the attributes of the substance, or by the differences of their modifications Prop. V. There cannot exist in the universe two or more (...)
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  • Enriched stratified systems for the foundations of category theory.Solomon Feferman - unknown
    Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including the ones constructed under (R1); (R3) In addition, S should allow us to establish the (...)
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  • Mathematics: Form and Function.Saunders Mac Lane - 1990 - Studia Logica 49 (3):424-426.
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  • Cantorian Set Theory and Limitation of Size.Michael Hallett - 1990 - Studia Logica 49 (2):283-284.
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