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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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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.
SOME AXIOMS FOR CONSTRUCTIVE ANALYSIS
"... This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and numbertheoretic sequences, we propose some modified cho ..."
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This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and numbertheoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis.
ANOTHER UNIQUE WEAK KÖNIG’S LEMMA WKL!!
"... In [2] J. Berger and Ishihara proved, via a circle of informal implications involving countable choice, that Brouwer’s Fan Theorem for detachable bars on the binary fan is equivalent in Bishop’s sense to various principles including a version WKL! of Weak König’s Lemma with a strong effective uniq ..."
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In [2] J. Berger and Ishihara proved, via a circle of informal implications involving countable choice, that Brouwer’s Fan Theorem for detachable bars on the binary fan is equivalent in Bishop’s sense to various principles including a version WKL! of Weak König’s Lemma with a strong effective uniqueness hypothesis. Schwichtenberg [9] proved the equivalence directly and formalized his proof in Minlog. We verify that his result does not require countable choice, and derive a separation principle SP from the Fan Theorem, in a minimal intuitionistic system M of analysis with function comprehension. In contrast, WKL!! comes from Weak König’s lemma WKL by adding the hypothesis that any two infinite paths must agree. WKL!! is interderivable over M with the conjunction of a consequence of Markov’s Principle and the double negation of WKL. This decomposition is in the spirit of Ishihara’s [4] and J. Berger’s [1]. Kleene’s function realizability and the author’s modified realizability establish that WKL!! is strictly weaker than WKL and strictly stronger than WKL!.
Note on Π 0 n+1LEM, Σ0 n+1LEM and Σ0 n+1DNE⋆
"... Abstract. In [1] Akama, Berardi, Hayashi and Kohlenbach used a monotone modified realizability interpretation to establish the relative independence of Σ 0 n+1DNE from Π 0 n+1LEM over HA, and hence the independence of Σ 0 n+1LEM from Π 0 n+1LEM over HA, for all n ≥ 0. We show that the same relat ..."
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Abstract. In [1] Akama, Berardi, Hayashi and Kohlenbach used a monotone modified realizability interpretation to establish the relative independence of Σ 0 n+1DNE from Π 0 n+1LEM over HA, and hence the independence of Σ 0 n+1LEM from Π 0 n+1LEM over HA, for all n ≥ 0. We show that the same relative independence results hold for these arithmetical principles over Kleene and Vesley’s system FIM of intuitionistic analysis [3], which extends HA and is consistent with PA but not with classical analysis. 1 The double negations of the closures of Σ 0 n+1LEM, Σ 0 n+1DNE and Π 0 n+1LEM are also considered, and shown to behave differently with respect to HA and FIM. Various elementary questions remain to be answered.