Intuitionistic Fixed Point Theories for Strictly Positive Operators

Rüede, Christian; Strahm, Thomas

doi:10.1002/1521-3870(200202)48:2<195::aid-malq195>3.3.co;2-j

Cited by 4 publications

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“…In arithmetic, the absence of the subscript 0 indicates that induction for arbitrary formulas along both α and ω is available. See also [5, Subsection 2.4], [2, p. 64] and [6]. Note however, that the theory presented in [6] is parameter free.…”

Section: Definitionmentioning

confidence: 99%

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From hierarchies to well-foundedness

Flumini

Sato

2014

Arch. Math. Logic

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We highlight that the connection of well-foundedness and recursive definitions is more than just convenience. While the consequences of making well-foundedness a sufficient condition for the existence of hierarchies (of various complexity) have been extensively studied, we point out that (if parameters are allowed) well-foundedness is a necessary condition for the existence of hierarchies e.g. that even in an intuitionistic setting (Π 0 1 -CA 0 ) α wf(α) where (Π 0 1 -CA 0 ) α stands for the iteration of Π 0 1 comprehension (with parameters) along some ordinal α and wf(α) stands for the well-foundedness of α.

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Section: Definitionmentioning

confidence: 99%

“…See also [5, Subsection 2.4], [2, p. 64] and [6]. Note however, that the theory presented in [6] is parameter free. Also note that the references use slightly different notations; Π 0 α -CA 0 in [5] and ACA − α in [6, §4].…”

Section: Definitionmentioning

confidence: 99%

From hierarchies to well-foundedness

Flumini

Sato

2014

Arch. Math. Logic

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“…Then they conclude that ID i (strict) is conservative over the intuitionistic arithmetic HA with respect to negative formulas. Let us try to prove the full conservation result in [5] along the line in [18]. The intuitionistic version of Lemma 1.1.2 is easy to see, which says that ID i (acc) is a conservative extension of HA.…”

Section: The Classical Theory Id(acc) Is Interpretable In the Classicmentioning

confidence: 99%

“…In a language of arithmetic strongly positive formulae with respect to X are generated from arithmetic formulae and atomic ones X(t) by means of positive connectives ∨, ∧, ∃, ∀. Then Rüede and Strahm [18] extends the result to the intuitionistic fixed point theory FiX i (HA) for strictly positive formulae Q(X, x), in which the predicate symbol X does not occur in the antecedent ϕ of implications ϕ → ψ nor in the scope of negations ¬. Indeed as shown in [5] FiX i (HA) is a conservative extension of HA.…”

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Intuitionistic fixed point theories over set theories

Arai

2015

Arch. Math. Logic

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In this paper we show that the intuitionistic fixed point theory FiX i (T ) over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.1 Intuitionistic fixed point theory over set theories T For a theory T in a laguage L, let Q(X, x) be an X-positive formula in the language L ∪ {X} with an extra unary predicate symbol X. Introduce a fresh unary predicate symbol Q together with the axiom stating that Q is a fixed point of Q(X, x):By the completeness theorem, it is obvious that the resulting extension of T is conservative over T , though it has a non-elementary speed-up over T when T is a recursive theory containing the elementary recursive arithmetic EA, cf.[3]. When T has an axiom schema, e.g., T = PA, the Peano arithmetic with the complete induction schema, let us define the fixed point extension FiX(PA) to have the induction schema for any formula with the fixed point predicate Q. Then FiX(PA) is stronger than PA, e.g., FiX(PA) proves the consistency of PA. For the proof-theretic strength of the fixed point theory FiX(PA), see [2,10,16].On the other side, W. Buchholz [15] shows that an intuitionistic fixed point theory over the intuitionistic (Heyting) arithmetic HA for strongly positive formulae Q(X, x) is proof-theoretically reducible to HA. In a language of arithmetic strongly positive formulae with respect to X are generated from arithmetic formulae and atomic ones X(t) by means of positive connectives ∨, ∧, ∃, ∀. Then Rüede and Strahm [18] extends the result to the intuitionistic fixed point theory FiX i (HA) for strictly positive formulae Q(X, x), in which the predicate symbol X does not occur in the antecedent ϕ of implications ϕ → ψ nor in the scope 1

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Quick cut-elimination for strictly positive cuts

Arai

2011

Annals of Pure and Applied Logic

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In this paper we show that the intuitionistic theory ID i <ω (SP ) for finitely many iterations of strictly positive operators is a conservative extension of the Heyting arithmetic. The proof is inspired by the quick cut-elimination due to G. Mints. This technique is also applied to fragments of Heyting arithmetic. * The paper has been finished when I visited München. I would like to thank to Prof. W. Buchholz for his interests, the valuable comments and the hospitality in my visit.

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Intuitionistic Fixed Point Theories for Strictly Positive Operators

Cited by 4 publications

References 0 publications

From hierarchies to well-foundedness

From hierarchies to well-foundedness

Intuitionistic fixed point theories over set theories

Quick cut-elimination for strictly positive cuts

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