Strongly majorizable functionals of finite type: A model for barrecursion containing discontinuous functionals

Abstract. We show how two iterated products of selection functions can both be used in conjunction with system T to interpret, via the dialectica interpretation and modified realizability, full classical analysis. We also show that one iterated product is equivalent over system T to Spector's bar recursion, whereas the other is Tequivalent to modified bar recursion. Modified bar recursion itself is shown to arise directly from the iteration of a different binary product of 'skewed' selection functions. Iterations of the dependent binary products are also considered but in all cases are shown to be T-equivalent to the iteration of the simple products. §1. Introduction. Gödel's [13] so-called dialectica interpretation reduces the consistency of Peano arithmetic to the consistency of the quantifier-free calculus of functionals T. In order to extend Gödel's interpretation to full classical analysis PA ω + CA, Spector [18] made use of the fact that PA ω + CA can be embedded, via the negative translation, into HA ω + AC N + DNS. Here PA ω and HA ω denote Peano and Heyting arithmetic, respectively, formulated in the language of finite types, andis countable choice, and DNS : @n N B(n) Ñ @nB(n), is the double negation shift, with A(n) and A(n, x) standing for arbitrary formulas, and B(n) " Dx A(n, x). Since HA ω + AC N , excluding the double negation shift, has a straightforward (modified) realizability interpretation [20], as well as a dialectica interpretation [1,13], the remaining challenge is to give a computational interpretation to DNS.A computational interpretation of DNS was first given by Spector [18], via the dialectica interpretation. Spector devised a form of recursion on well-founded trees, nowadays known as Spector bar recursion, and showed that the dialectica interpretation of DNS can be witnessed by such kind of recursion. A computational interpretation of DNS via realizability only came recently, first in [2], via a non-standard form of realizability, and then in [4,5], via Kreisel's modified realizability. The realizability interpretation of DNS makes use of a new form of bar recursion, termed modified bar recursion.It has been shown in [5] that Spector's bar recursion is definable in system T extended with modified bar recursion, but not conversely, since Spector's bar recursion Received by the editors PREPRINT, MARCH 1, 2018.

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“…We note that SPEC follows from CONT, but it also holds in the model of strongly majorizable functionals [7].…”

Section: Definition 22 (Discrete and Compact Types)mentioning

confidence: 78%

Bar Recursion and Products of Selection Functions

Escardó

Oliva

2015

J. symb. log.

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“…In [9] we introduced the following 'non-standard' axiom F which is not valid in the full set-theoretic type structure S ω of all set-theoretic functionals but is true in the type structure of all strongly majorizable functionals M ω which was introduced in [1] for different purposes (see [9] for details; a special case of F -called F 0 in [9]-was already studied in [7]). In this section we review some of the results on F from [9].…”

Section: The Axiom F and The Principle Of Uniform Boundednessmentioning

confidence: 99%

“…In [9] also we introduced new axioms F and F − which both (essentially) have the form (4) and are true in the type structure of all strongly majorizable functionals (see [1]) but are false in the full set-theoretic model (weaker versions of these axioms were studied already in [7]). Thus, whereas F, F − do not contribute to the construction of bounds extracted from a proof, the verification of these bounds so long uses these axioms.…”

Section: Introductionmentioning

confidence: 99%

“…Since we deal with theories which do not contain CA 1 ar it is consistent to assume as an axiom that all functions G : [0, 1] d → IR, which are given explicitly by a functional Φ 1 (1) , are uniformly continuous. 2 This is achieved by the axiom F : From the work on the program of so-called 'reverse mathematics' (see [3], [16]) it is known that e.g.…”

Section: Introductionmentioning

confidence: 99%

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The Use of a Logical Principle of Uniform Boundedness in Analysis

Kohlenbach

1999

Logic and Foundations of Mathematics

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“…Here denotes Bezem's [1] strong majorization relation (which is a variant of Howard's [4] majorization). 1 Obviously, (2) implies (1) (take y := ϕ(x)) but the other direction in general does not hold for the simple reason that (1) only claims something for majorizable functionals x while (2) talks about all x. E.g.…”

Section: Introductionmentioning

confidence: 99%

A note on the monotone functional interpretation

Kohlenbach

2011

Mathematical Logic Quarterly

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We prove a result relating the author's monotone functional interpretation to the bounded functional interpretation due to Ferreira and Oliva. More precisely we show that (over model of majorizable functionals) largely a solution for the bounded interpretation also is a solution for monotone functional interpretation although the latter uses the existence of an underlying precise witness. This makes it possible to focus on the extraction of bounds (as in the bounded interpretation) while using the conceptual benefit of having precise realizers at the same time without having to construct them.

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Strongly majorizable functionals of finite type: A model for barrecursion containing discontinuous functionals

Abstract: In this paper a model for barrecursion is presented. It has as a novelty that it contains discontinuous functionals. The model is based on a concept called strong majorizability. This concept is a modification of Howard's majorizability notion; see [T, p. 456].

Cited by 86 publications

References 2 publications

Bar Recursion and Products of Selection Functions

Bar Recursion and Products of Selection Functions

The Use of a Logical Principle of Uniform Boundedness in Analysis

A note on the monotone functional interpretation

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