Formal systems for some branches of intuitionistic analysis

Kreisel, G.; Troelstra, A. S.

doi:10.1016/0003-4843(70)90001-x

Cited by 136 publications

(72 citation statements)

References 14 publications

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“…Inspired by Kreisel's and Troelstra's [14] definition of realizability for higher order Heyting arithmetic, realizability was first applied to systems of set theory by Myhill [18] and Friedman [9]. More recently, realizability models of set theory were investigated by Beeson [3,5] (for non-extensional set theories) and McCarty [16,17] (directly for extensional set theories).…”

Section: Realizability For Set Theoriesmentioning

confidence: 99%

Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory

Chen

Rathjen

2012

Arch. Math. Logic

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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 V. Lifschitz in [15]. A Lifschitz counterpart to Kleene's realizability for functions (in Baire space) was developed by van Oosten [19]. 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 would also work for extensions of IZF with large set axioms. In addition to separating Church's thesis with uniqueness condition from its general form in intuitionistic set theory, we also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, AC ω,ω , asserting that a countable family of inhabited sets of natural numbers has a choice function. AC ω,ω is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of AC ω,ω , namely that the sets of Cauchy reals and Dedekind reals are isomorphic, remains valid in this interpretation.Another interesting aspect of this realizability is that it validates the lesser limited principle of omniscience.MSC:03E25,03E35,03E70,03F25, 03F35,03F50,03F55,03F60

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Section: Realizability For Set Theoriesmentioning

confidence: 99%

Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory

Chen

Rathjen

2012

Arch. Math. Logic

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“…One interesting result regarding choice sequences is the so-called "elimination of choice sequences" theorem [46, Sec.2; 44, Ch.7; 69, Ch.3; 31, pp.221-222; 30] that eliminates quantifications over choice sequences. This theorem relies on a mapping from the formulae of the CS formal system [44] to formulae of the IDB 1 formal system [44] that do not contain choice sequence variables. It is left to future work to study whether a similar result could be used to prove that BI is consistent with Nuprl without using choice sequences.…”

Section: ∀X(∀y(xρy → Q(y)) → Q(x)) → ∀Xq(x)mentioning

confidence: 99%

Bar induction: The good, the bad, and the ugly

Rahli

Bickford

Constable

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Abstract-We present an extension of the computation system and logic of the Nuprl proof assistant with intuitionistic principles, namely versions of Brouwer's bar induction principle, which is equivalent to transfinite induction. We have substantially extended the formalization of Nuprl's type theory within the Coq proof assistant to show that two such bar induction principles are valid w.r.t. Nuprl's semantics (the Good): one for sequences of numbers that involved only minor changes to the system, and a more general one for sequences of name-free (the Ugly) closed terms that involved adding a limit constructor to Nuprl's term syntax in our model of Nuprl's logic. We have proved that these additions preserve Nuprl's key metatheoretical properties such as consistency. Finally, we show some new insights regarding bar induction, such as the nontruncated version of bar induction on monotone bars is intuitionistically false (the Bad).

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“…The constructions of these models and the proofs of their properties can be performed in an intuitionistic system like IDB (see Kreisel & Troelstra(1970)). This means that the theories of choice sequences that we provide models for are all consistent with Church's thesis ("all lawlike sequences are recursive") and countable choice.…”

Section: Va(a(a) ->-3f(a E Im(f) 1\ Vb E Im(f)a\b»)mentioning

confidence: 99%

“…More accurately, we will construct a sequence of models, starting with a model for lawless sequences of neighbourhood functions, and arriving by subsequent modifications at a model for the theory CS of Kreisel & Troelstra(1970).…”

Section: Introductionmentioning

confidence: 99%