A Palatable Substitute for Kripke's Schema

Vesley, R. E.

doi:10.1016/s0049-237x(08)70750-x

Cited by 9 publications

(7 citation statements)

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“…A strong version ∀α[¬∀x¬A(α, x) ⊃ ∃xA(α, x)] of Markov's Principle, while realizable classically and hence consistent with FIM, is unprovable (Kleene [74] p. 131, using a modification of function-realizability inspired by Kreisel [79]). A weak form ∀α¬¬GR(α) of Church's Thesis is also consistent with and independent of FIM, where GR(α) is a formula of the form ∃e∀x∃y[T(e, x, y) ∧ ∀z(U(y, z) ⊃ α(x) = z)] expressing "α is general recursive" (J. R. Moschovakis [91], using a modified function-realizability interpretation inspired by Vesley [129]). The strong form of Church's Thesis conflicts with Brouwer's Principle, and in the presence of Markov's Principle the weak form conflicts with the Bar Theorem (cf.…”

Section: Kleene and Vesley's Formalism For Intuitionistic Mathematicsmentioning

confidence: 92%

The Logic of Brouwer and Heyting

Moschovakis¹

2009

Handbook of the History of Logic

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Intuitionistic logic consists of the principles of reasoning which were used informally by L. E. J. Brouwer, formalized by A. Heyting (also partially by V. Glivenko), interpreted by A. Kolmogorov, and studied by G. Gentzen and K. Gödel during the first third of the twentieth century. Formally, intuitionistic first-order predicate logic is a proper subsystem of classical logic, obtained by replacing the law of excluded middle A ∨ ¬A by ¬A ⊃ (A ⊃ B); it has infinitely many distinct consistent axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete). However, intuitionistic second-order logic is inconsistent with classical second-order logic. In terms of expressibility, the intuitionistic logic and language are richer than the classical; as Gödel and Gentzen showed, classical first-order arithmetic can be faithfully translated into the negative fragment of intuitionistic arithmetic, establishing proof-theoretical equivalence and clarifying the distinction between the classical and constructive consequences of mathematical axioms.The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises. Any proof of a disjunction of two statements can be effectively transformed into a proof of one of the disjuncts, while any proof of an existential statement contains an effective prescription for finding a witness. The negation of a statement is interpreted as asserting that the statement is not merely false but absurd, i.e., leads to a contradiction. Brouwer objected to the general law of excluded middle as claiming a priori that every mathematical problem has a solution, and to the general law ¬¬A ⊃ A of double negation as asserting that every consistent mathematical statement holds.Brouwer called "the First Act of Intuitionism" the separation of mathematics (an exclusively mental activity) from language. For him, all mathematical objects (including proofs) were mental constructions carried out by individual mathematicians. This insight together with "the intuition of the bare two-oneness" guided the creation of intuitionistic logic and arithmetic, with full induction and a theory of the rational numbers. It was limited, by Brouwer's early insistence on predicative definability, to the construction of entities which were finitely specifiable and thus at most denumerably infinite in number; this seemed to exclude the possibility of a good intuitionistic theory of the continuum.A later insight led Brouwer to "the Second Act of Intuitionism," which accepted infinitely proceeding sequences of independent choices of objects already constructed (choice sequences of natural numbers, for example) as legitimate mathematical entities. He succeeded in extracting positive information about the structure of the continuum from the fact that an individual choice sequence may be known only by its finite approximations. His "Bar Theorem" (actually a new mathematical axiom) essentially accepted induction up ...

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Section: Kleene and Vesley's Formalism For Intuitionistic Mathematicsmentioning

confidence: 92%

The Logic of Brouwer and Heyting

Moschovakis¹

2009

Handbook of the History of Logic

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“…instead, from which follows Vesley’s [55] alternative formalization of creative subject mentioned in f.n 14…”

Section: Final Remarksmentioning

confidence: 99%

A Marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic

Nemoto¹,

Sato²

2021

J. symb. log.

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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 (i.e., which part is equiconsistent with $\textbf {PRA}$ or consistent provably in $\textbf {PRA}$ ) 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: (i) fan theorem for decidable fans but arbitrary bars; (ii) continuity principle and the axiom of choice both for arbitrary formulae; and (iii) $\Sigma _2$ induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO (except the choice along functions); but that limited principle of omniscience LPO makes the situation completely classical.

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“…Vesley [189] proposed a schema that is implied by BKS − but does not imply it [189, p.199], yet allows alternative arguments for Brouwerian counterexamples without appealing to the Creating Subject. It is also, unlike BKS − , consistent with FIM and with ∀α ∃β-continuity.…”

Section: A Palatable Substitute For Bks −mentioning

confidence: 99%

“…[Note MvA: In that volume[78], BKS is discussed in the contributions by Kreisel[89], Myhill[122], Van Rootselaar[179], Vesley[189], and Scott[134].] 108 [Note MvA: Kreisel's reference '[6]' is to[89]; see item 2 in this list.…”

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confidence: 99%

The Creating Subject, the Brouwer–Kripke Schema, and infinite proofs

Atten¹

2018

Indagationes Mathematicae

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Forthcoming in Indagationes Mathematicae. AbstractKripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's 'Creating Subject arguments'; other applications have been found since. I first look in detail at a representative choice of Brouwer's arguments. Then I discuss the original use of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer's own writings. Finally, I defend the Schema and the Theory against a number of objections that have been made.

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A Palatable Substitute for Kripke's Schema

Cited by 9 publications

References 1 publication

The Logic of Brouwer and Heyting

The Logic of Brouwer and Heyting

A Marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic

The Creating Subject, the Brouwer–Kripke Schema, and infinite proofs

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