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- W2057631483 abstract "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 number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CFd asserting that every decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC00 of countable choice can be decomposed into a monotone choice schema AC 00 m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC00. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms $${AB_{1/2,0}^{2^{mathbb{N}}}}$$ . We also introduce a version WKL!! of Weak König’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FTd." @default.
- W2057631483 created "2016-06-24" @default.
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- W2057631483 date "2012-03-21" @default.
- W2057631483 modified "2023-09-27" @default.
- W2057631483 title "Some axioms for constructive analysis" @default.
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- W2057631483 doi "https://doi.org/10.1007/s00153-012-0273-z" @default.
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