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62 found
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  1. An Equivalent of the Axiom of Choice in Finite Models of the Powerset Axiom.Alexander Abian & Wael A. Amin - 1990 - Notre Dame Journal of Formal Logic 31 (3):371-374.
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  2. Some Consequences of the Axiom of Power-Set.Alexander Abian & Samuel Lamacchia - 1965 - Journal of Symbolic Logic 30 (3):293-294.
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  3. A Geometric Form of the Axiom of Choice.J. L. Bell - unknown
    Consider the following well-known result from the theory of normed linear spaces ([2], p. 80, 4(b)): (g) the unit ball of the (continuous) dual of a normed linear space over the reals has an extreme point. The standard proof of (~) uses the axiom of choice (AG); thus the implication AC~(w) can be proved in set theory. In this paper we show that this implication can be reversed, so that (*) is actually eq7I2valent to the axiom of choice. From this (...)
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  4. Review of G. H. Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence[REVIEW]J. L. Bell - 2014 - Philosophia Mathematica 22 (1):131-134.
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  5. The Axiom of Choice in the Foundations of Mathematics.John Bell - manuscript
    The principle of set theory known as the Axiom of Choice (AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”1 It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted to it, and it has long played a prominently role in discussions on the foundations of (...)
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  6. The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories.John Bell - 2008 - Mathematical Logic Quarterly 54 (2):194-201.
    A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.
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  7. ECH, T. J.: "The Axiom of Choice". [REVIEW]John Bell - 1976 - British Journal for the Philosophy of Science 27:187.
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  8. M L.John L. Bell - unknown
    A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that beefing up WST with moderate extensionality principles or quotient sets enables the derivation to go through.
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  9. The Axiom of Choice.John L. Bell - 2008 - Stanford Encyclopedia of Philosophy.
    The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...)
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  10. On the Computational Content of the Axiom of Choice.Stefano Berardi, Marc Bezem & Thierry Coquand - 1998 - Journal of Symbolic Logic 63 (2):600-622.
    We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation.
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  11. On Specker's Refutation of the Axiom of Choice.Maurice Boffa - 2000 - Logique Et Analyse 43.
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  12. Sequential Compactness and the Axiom of Choice.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (1):89-92.
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  13. The Axiom of Choice in Topology.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (3):305-317.
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  14. Investigation of the Equivalence of the Axiom of Choice and Zorn's Lemma From the Viewpoint of the Hierarchy of Types.J. Richard Buchi - 1953 - Journal of Symbolic Logic 18 (2):125 - 135.
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  15. The Axiom of Choice and Combinatory Logic.Andrea Cantini - 2003 - Journal of Symbolic Logic 68 (4):1091-1108.
    We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice.
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  16. Distributivity and an Axiom of Choice.George E. Collins - 1954 - Journal of Symbolic Logic 19 (4):275-277.
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  17. Typical Ambiguity and the Axiom of Choice.Marcel Crabbé - 1984 - Journal of Symbolic Logic 49 (4):1074-1078.
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  18. An Investigation Concerning the Hilbert-Sierpi'nski Logical Form of the Axiom of Choice.Ch C. Davis - 1975 - Notre Dame Journal of Formal Logic 16 (2):145-184.
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  19. A Note on the Axiom of Choice in Leśniewski's Ontology.Charles C. Davis - 1976 - Notre Dame Journal of Formal Logic 17 (1):35-43.
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  20. Definitions of Compactness and the Axiom of Choice.Omar De la Cruz, Eric Hall, Paul Howard, Jean E. Rubin & Adrienne Stanley - 2002 - Journal of Symbolic Logic 67 (1):143-161.
    We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.
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  21. Canonical Universes and Intuitions About Probabilities.Randall Dougherty & Jan Mycielski - 2006 - Dialectica 60 (4):357–368.
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  22. Inconsistency of the Axiom of Choice with the Positive Theory GPK+ ∞.Olivier Esser - 2000 - Journal of Symbolic Logic 65 (4):1911 - 1916.
    The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without "too much" negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory GPK + ∞.
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  23. The Status of the Axiom of Choice in Set Theory with a Universal Set.T. E. Forster - 1985 - Journal of Symbolic Logic 50 (3):701-707.
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  24. The Construction of a Steiner Triple System on Sets of the Power of the Continuum Without the Axiom of Choice.William J. Frascella - 1966 - Notre Dame Journal of Formal Logic 7 (2):196-202.
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  25. A Generalization of Sierpiński's Theorem on Steiner Triples and the Axiom of Choice.William J. Frascella - 1965 - Notre Dame Journal of Formal Logic 6 (3):163-179.
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  26. Corrigendum and Addendum To: ``A Generalization of Sierpiński's Theorem on Steiner Triples and the Axiom of Choice''.William J. Frascella - 1965 - Notre Dame Journal of Formal Logic 6 (4):323-324.
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  27. The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory.Kurt Gödel - 1940 - Princeton University Press;.
  28. Relations Between Some Cardinals in the Absence of the Axiom of Choice.Lorenz Halbeisen & Saharon Shelah - 2001 - Bulletin of Symbolic Logic 7 (2):237-261.
    If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...)
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  29. Is the Axiom of Choice a Logical or Set-Theoretical Principle?Jaako Hintikka - 1999 - Dialectica 53 (3-4):283–290.
    A generalization of the axioms of choice says that all the Skolem functions of a true first‐order sentence exist. This generalization can be implemented on the first‐order level by generalizing the rule of existential instantiation into a rule of functional instantiation. If this generalization is carried out in first‐order axiomatic set theory , it is seen that in any model of FAST, there are sentences S which are true but whose Skolem functions do not exist. Since this existence is what (...)
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  30. The Axiom of Choice for Countable Collections of Countable Sets Does Not Imply the Countable Union Theorem.Paul E. Howard - 1992 - Notre Dame Journal of Formal Logic 33 (2):236-243.
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  31. Subgroups of a Free Group and the Axiom of Choice.Paul E. Howard - 1985 - Journal of Symbolic Logic 50 (2):458-467.
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  32. Independence Results for Class Forms of the Axiom of Choice.Paul E. Howard, Arthur L. Rubin & Jean E. Rubin - 1978 - Journal of Symbolic Logic 43 (4):673-684.
    Let NBG be von Neumann-Bernays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.
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  33. Maximal $P$-Subgroups and the Axiom of Choice.Paul E. Howard & Mary Yorke - 1987 - Notre Dame Journal of Formal Logic 28 (2):276-283.
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  34. The Axiom of Choice for Well-Ordered Families and for Families of Well- Orderable Sets.Paul Howard & Jean E. Rubin - 1995 - Journal of Symbolic Logic 60 (4):1115-1117.
    We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets are both true, but the axiom of choice is false.
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  35. The Axiom of Choice.Thomas Jech - 1973 - North-Holland.
    Provability, Computability and Reflection.
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  36. Le'sniewski's Ontology Extended with the Axiom of Choice.James Kowalski - 1977 - Notre Dame Journal of Formal Logic 18 (1):1-78.
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  37. Equivalents of a Weak Axiom of Choice.Melven Krom - 1981 - Notre Dame Journal of Formal Logic 22 (3):283-285.
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  38. On Ramsey's Theorem and the Axiom of Choice.Gabriele Lolli - 1977 - Notre Dame Journal of Formal Logic 18 (4):599-601.
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  39. The Independence of a Weak Axiom of Choice.Elliott Mendelson - 1956 - Journal of Symbolic Logic 21 (4):350-366.
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  40. Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice.David W. Miller - 2007 - Logica Universalis 1 (1):183-199.
    . Dzik [2] gives a direct proof of the axiom of choice from the generalized Lindenbaum extension theorem LET. The converse is part of every decent logical education. Inspection of Dzik’s proof shows that its premise let attributes a very special version of the Lindenbaum extension property to a very special class of deductive systems, here called Dzik systems. The problem therefore arises of giving a direct proof, not using the axiom of choice, of the conditional . A partial solution (...)
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  41. Cut-Elimination for Simple Type Theory with an Axiom of Choice.G. Mints - 1999 - Journal of Symbolic Logic 64 (2):479-485.
    We present a cut-elimination proof for simple type theory with an axiom of choice formulated in the language with an epsilon-symbol. The proof is modeled after Takahashi's proof of cut-elimination for simple type theory with extensionality. The same proof works when types are restricted, for example for second-order classical logic with an axiom of choice.
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  42. On Generic Extensions Without the Axiom of Choice.G. P. Monro - 1983 - Journal of Symbolic Logic 48 (1):39-52.
    Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let $M$ be a countable transitive model of ZF. The method of forcing extends $M$ to another model $M\lbrack G\rbrack$ of ZF (a "generic extension"). If the axiom of choice holds in $M$ it also holds in $M\lbrack G\rbrack$, that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive (...)
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  43. Lebesgue's Measure Problem and Zermelo's Axiom of Choice.Gregory H. Moore - 1983 - In Joseph Warren Dauben & Virginia Staudt Sexton (eds.), History and Philosophy of Science: Selected Papers. New York Academy of Sciences.
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  44. Notions of Compactness for Special Subsets of ℝ I and Some Weak Forms of the Axiom of Choice.Marianne Morillon - 2010 - Journal of Symbolic Logic 75 (1):255-268.
    We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the axiom of Dependent (...)
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  45. Russell’s Method of Analysis and the Axioms of Mathematics.Lydia Patton - 2017 - In Sandra Lapointe Christopher Pincock (ed.), Innovations in the History of Analytical Philosophy. London: Palgrave-Macmillan. pp. 105-126.
    In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s (...)
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  46. Two Forms of the Axiom of Choice for an Elementary Topos.Anna Michaelides Penk - 1975 - Journal of Symbolic Logic 40 (2):197-212.
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  47. Support Structures for the Axiom of Choice.David Pincus - 1971 - Journal of Symbolic Logic 36 (1):28-38.
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  48. Plural Quantification and the Axiom of Choice.Stephen Pollard - 1988 - Philosophical Studies 54 (3):393 - 397.
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  49. Set Theory and its Philosophy: A Critical Introduction.Michael D. Potter - 2004 - 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 that bedevil set (...)
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  50. A Note on the Axiom of Choice and the Continuum Hypothesis.Rolf Schock - 1977 - Notre Dame Journal of Formal Logic 18 (3):409-414.
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