The interdependence of certain consequences of the axiom of choice
A. Lévy (1964)
Fundamenta Mathematicae
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A. Lévy (1964)
Fundamenta Mathematicae
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K. Wiśniewski (1972)
Fundamenta Mathematicae
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Omar De la Cruz, Damir D. Dzhafarov, Eric J. Hall (2006)
Fundamenta Mathematicae
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A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (AC) is assumed, is equivalent to stating that the set is finite; several such definitions have been studied over the years. In this article we introduce a framework for generating definitions of finiteness in a systematical way: basic definitions are obtained from properties of certain classes of binary relations, and further definitions are obtained from the basic ones by closing them...
M. Jelić (1990)
Matematički Vesnik
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Horst Herrlich, Paul Howard, Eleftherios Tachtsis (2015)
Bulletin of the Polish Academy of Sciences. Mathematics
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We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice ( AC ), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y → X.
Alas, Ofelia T. (1969)
Portugaliae mathematica
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Nancy Moler, Patrick Suppes (1968)
Compositio Mathematica
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P. Andrews (1963)
Fundamenta Mathematicae
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M. Bleicher (1964)
Fundamenta Mathematicae
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A. Szepietowski (2010)
RAIRO - Theoretical Informatics and Applications
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In this paper we show that neither the set of all valid equations between shuffle expressions nor the set of schemas of valid equations is recursively enumerable. Thus, neither of the sets can be recursively generated by any axiom system.
A. Lévy (1962)
Fundamenta Mathematicae
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Perry Smith (1982)
Publications de l'Institut Mathématique
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Rolando Chuaqui
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CONTENTSIntroduction............................................................................................................ 5I. Axiom system and elementary consequences........................................... 61. Axioms........................................................................................................................ 62. Definitions and elementary consequences........................................................ 9II. Principles of definitions by recursion..............................................................