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  1. Epistemology Versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf.Peter Dybjer, Sten Lindström, Erik Palmgren & Göran Sundholm (eds.) - 2012 - Dordrecht, Netherland: Springer.
    This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...)
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  • Extensional realizability.Jaap van Oosten - 1997 - Annals of Pure and Applied Logic 84 (3):317-349.
    Two straightforward “extensionalisations” of Kleene's realizability are considered; denoted re and e. It is shown that these realizabilities are not equivalent. While the re-notion is a subset of Kleene's realizability, the e-notion is not. The problem of an axiomatization of e-realizability is attacked and one arrives at an axiomatization over a conservative extension of arithmetic, in a language with variables for finite sets. A derived rule for arithmetic is obtained by the use of a q-variant of e-realizability; this rule subsumes (...)
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  • A new model construction by making a detour via intuitionistic theories III: Ultrafinitistic proofs of conservations of Σ 1 1 collection. [REVIEW]Kentaro Sato - 2023 - Annals of Pure and Applied Logic 174 (3):103207.
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  • Bounded inductive dichotomy: separation of open and clopen determinacies with finite alternatives in constructive contexts.Kentaro Sato - 2022 - Archive for Mathematical Logic 61 (3):399-435.
    In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary ) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded ) positive operators, the present article investigates inductive dichotomy for these operators, and applies it to (...)
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  • From the weak to the strong existence property.Michael Rathjen - 2012 - Annals of Pure and Applied Logic 163 (10):1400-1418.
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  • A marriage of brouwer’s intuitionism and hilbert’s finitism I: Arithmetic.Takako Nemoto & Sato Kentaro - 2022 - Journal of Symbolic Logic 87 (2):437-497.
    We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: fan theorem for decidable fans but arbitrary bars; continuity principle (...)
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  • Some axioms for constructive analysis.Joan Rand Moschovakis & Garyfallia Vafeiadou - 2012 - Archive for Mathematical Logic 51 (5-6):443-459.
    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 (...)
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  • Analyzing realizability by Troelstra's methods.Joan Rand Moschovakis - 2002 - Annals of Pure and Applied Logic 114 (1-3):203-225.
    Realizabilities are powerful tools for establishing consistency and independence results for theories based on intuitionistic logic. Troelstra discovered principles ECT 0 and GC 1 which precisely characterize formal number and function realizability for intuitionistic arithmetic and analysis, respectively. Building on Troelstra's results and using his methods, we introduce the notions of Church domain and domain of continuity in order to demonstrate the optimality of “almost negativity” in ECT 0 and GC 1 ; strengthen “double negation shift” DNS 0 to DNS (...)
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  • Classical Consequences of Continuous Choice Principles from Intuitionistic Analysis.François G. Dorais - 2014 - Notre Dame Journal of Formal Logic 55 (1):25-39.
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  • Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory.Ray-Ming Chen & Michael Rathjen - 2012 - Archive for Mathematical Logic 51 (7-8):789-818.
    A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition, but not the general form of Church’s thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101–106, 1979). A Lifschitz counterpart to Kleene’s realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805–821, 1990). In that paper he also extended Lifschitz’ realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo–Fraenkel set theory, IZF. The machinery (...)
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