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Ramsey's theorem and the pigeonhole principle in intuitionistic mathematics
, 1992
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Classical And Constructive Hierarchies In Extended Intuitionistic Analysis
"... This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire sp ..."
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Cited by 8 (3 self)
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This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire space with the property that every constructive partial functional defined on {# : R(#)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(#) is equivalent in #) for some stable A(#, #) (which belongs to the classical analytical hierarchy). The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems.
König’s Lemma and Kleene Tree
, 2006
"... I present a basic result about Cantor space in the context of computability theory: the computable Cantor space is computably noncompact. This is in sharp contrast with the classical theorem that Cantor space is compact. The note is written for mathematicians with classical training in topology an ..."
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I present a basic result about Cantor space in the context of computability theory: the computable Cantor space is computably noncompact. This is in sharp contrast with the classical theorem that Cantor space is compact. The note is written for mathematicians with classical training in topology and analysis. I assume nothing from computability theory, except the basic intuition about how computers work by executing instructions given by a finite program. 1
1 Unique paths as formal points
"... SCHUSTER Abstract:A pointfree formulation of the König Lemma for trees with uniformly at most one infinite path allows for a constructive proof without unique choice. ..."
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SCHUSTER Abstract:A pointfree formulation of the König Lemma for trees with uniformly at most one infinite path allows for a constructive proof without unique choice.
EMIL POST
, 1897
"... Augustów, a town at that time within the Russian empire, but after 1918 in the province of Bialystok in Eastern Poland. In 1897 his father Arnold emigrated to join his brother in America. Seven years later, in May 1904, with the success of the family clothing and fur business in New York, his wife ..."
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Augustów, a town at that time within the Russian empire, but after 1918 in the province of Bialystok in Eastern Poland. In 1897 his father Arnold emigrated to join his brother in America. Seven years later, in May 1904, with the success of the family clothing and fur business in New York, his wife Pearl, together with Emil and his sisters Anna and Ethel, joined him. The family lived in a comfortable home in Harlem. Figure 1. Emil Post, June 1924 As a child, Post was particularly interested in astronomy, but an accident at the age of twelve foreclosed that choice of career. As he reached for a lost ball under a parked car, a second car crashed into it, and he lost his left arm below the shoulder. As a high school senior, Post wrote to several observatories inquiring whether his handicap would prevent him pursuing a career as an astronomer. The responses he received, though not uniformly negative, were sufficient to discourage him from following his childhood ambition; instead, he turned towards mathematics.