In the course of ten short sections, we comment on Gödel's seminal dialectica paper of fifty years ago and its aftermath. We start by suggesting that Gödel's use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics, in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the dialectica interpretation, discuss the extra principles that (...) the interpretation validates and comment on extensionality and higher order equality. The latter sections focus on the role of majorizability considerations within the dialectica and related interpretations for extracting computational information from ordinary proofs in mathematics. (shrink)

Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of certain set-valued mappings between function spaces. This paper deals with the computational properties of these accretive and (generalized) monotone set-valued operators. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie “non-computational” proofs from the mainstream literature. To this end, we establish logical metatheorems (...) that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On the one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches. (shrink)

In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems Tnω in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive functions of low growth. We reduce the use of instances of these principles in Tnω-proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical (...) content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper.As corollaries we obtain new conservation results for fragments of analysis over fragments of arithmetic which strengthen known purely first-order conservation results.We also characterize the provably recursive functions of type 2 of the extensions of Tnω based on these fragments of arithmetical comprehension, choice and uniform boundedness. (shrink)

In this article we study applications of the bounded functional interpretation to theories of feasible arithmetic and analysis. The main results show that the novel interpretation is sound for considerable generalizations of weak König’s Lemma, even in the presence of very weak induction. Moreover, when this is combined with Cook and Urquhart’s variant of the functional interpretation, one obtains effective versions of conservation results regarding weak König’s Lemma which have been so far only obtained non-constructively.

We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finite-type arithmetic. As a consequence, some uniform boundedness principles are interpreted while maintaining unmoved the -sentences of arithmetic. We explain why this interpretation is tailored to yield conservation results.

We will apply the methods developed in the field of ‘proof mining’ to the Bolzano-Weierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation as well as the monotone functional interpretation of BW for the product space Πi ∈ℕ[–ki, ki] . This results in optimal program and bound extraction theorems for proofs based on fixed instances of BW, i.e. for BW applied to fixed (...) sequences in Πi ∈ℕ[–ki, ki]. (shrink)

We present a refinement ofthe bounded modified realizability which provides both upper and lower bounds for witnesses. Our interpretation is based on a generalisation of Howard/Bezem's notion of strong majorizability. We show how the bounded modified realizability coincides with our interpretation in the case when least elements exist . The new interpretation, however, permits the extraction of more accurate bounds, and provides an ideal setting for dealing directly with data types whose natural ordering is not well-founded.

We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including the FAN theorem, weak König’s lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end up discussing some applications of the (...) new interpretation to theories of classical arithmetic and analysis. (shrink)

The main result of this paper is the equivalence of several definition schemas of bar recursion occurring in the literature on functionals of finite type. We present the theory of functionals of finite type, in [T] denoted byqf-WE-HA ω, which is necessary for giving the equivalence proofs. Moreover we prove two results on this theory that cannot be found in the literature, namely the deduction theorem and a derivation of Spector's rule of extensionality from [S]: ifP→T 1=T 2 and Q[X∶≡T1], (...) then P→Q[X∶≡ T2], from the at first sight weaker rule obtained by omitting “P→”. (shrink)

We adapt a continuous logic axiomatization of tracial von Neumann algebras due to Farah, Hart and Sherman in order to prove a metatheorem for this class of structures in the style of proof mining, a research programme that aims to obtain the hidden computational content of ordinary mathematical proofs using tools from proof theory.

Several extensions of Gödel's system TT with new forms of recursion have been designed for the purpose of giving a computational interpretation to classical analysis. One can organise many of these extensions into two groups: those based on bar recursion , which include Spector's original bar recursion, modified bar recursion and the more recent products of selections functions, or those based on open recursion which in particular include the symmetric Berardi–Bezem–Coquand functional. We relate these two groups by showing that both (...) open recursion and the BBC functional are primitive recursively equivalent to a variant of modified bar recursion. Our results, in combination with existing research, essentially complete the classification up to primitive recursive equivalence of those extensions of system TT used to give a direct computational interpretation to choice principles. (shrink)

In the present paper we give a functional interpretation of Aczel's constructive set theories CZF − and CZF in systems T ∈ and T ∈ + of constructive set functionals of finite types. This interpretation is obtained by a translation × , a refinement of the ∧ -translation introduced by Diller and Nahm 49–66) which again is an extension of Gödel's Dialectica translation. The interpretation theorem gives characterizations of the definable set functions of CZF − and CZF in terms of (...) constructive set functionals. In a second part we introduce constructive set theories in all finite types. We expand the interpretation to these theories and give a characterization of the translation × . We further show that the simplest non-trivial axiom of extensionality is not interpretable by functionals of T ∈ and T ∈ + . We obtain this result by adapting Howard's notion of hereditarily majorizable functionals to set functionals. Subject of the last section is the translation ∨ that is defined in Burr for an interpretation of Kripke-Platek set theory with infinity. (shrink)

This article presents a parametrized functional interpretation. Depending on the choice of two parameters one obtains well-known functional interpretations such as Gödel's Dialectica interpretation, Diller-Nahm's variant of the Dialectica interpretation, Kohlenbach's monotone interpretations, Kreisel's modified realizability, and Stein's family of functional interpretations. A functional interpretation consists of a formula interpretation and a soundness proof. I show that all these interpretations differ only on two design choices: first, on the number of counterexamples for A which became witnesses for ¬A when defining (...) the formula interpretation and, second, the inductive information about the witnesses of A which is considered in the proof of soundness. Sufficient conditions on the parameters are also given which ensure the soundness of the resulting functional interpretation. The relation between the parametrized interpretation and the recent bounded functional interpretation is also discussed. (shrink)

Along the line of Hirst‐Mummert and Dorais, we analyze the relationship between the classical provability of uniform versions Uni(S) of Π2‐statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for every Π2‐statement S of some syntactical form, if its uniform version derives the uniform variant of over a classical system of arithmetic in all finite types with weak extensionality, then S is not provable in strong semi‐intuitionistic systems (...) including bar induction in all finite types but also nonconstructive principles such as Kőnig's lemma and uniform weak Kőnig's lemma. Our result is applicable to many mathematical principles whose sequential versions imply. (shrink)

The technical contribution of this paper is threefold.First we show how to encode functionals in a ‘flat’ applicative structure by adding oracles to untyped λ-calculus and mimicking the applicative behaviour of the functionals with an impredicatively defined reduction relation. The main achievement here is a Church-Rosser result for the extended reduction relation.Second, by combining the previous result with the model construction based on partial equivalence relations, we show how to extend a λ-closed simple type structure to a model of the (...) polymorphic λ-calculus.Third, we specialize the previous result to a counter model against a simple minimization. This minimization is realized by a bar recursive functional, which contrasts the results of Spector and Girard which imply that the bar recursive functions are exactly those that are definable in the polymorphic λ-calculus. As a spin-off, we obtain directly the non-conservativity of the extensions of Gödel's T with bar recursion, fan functional, and Luckhardt's minimization functional, respectively. For the latter two extensions these results are new. (shrink)

We show that Spector's “restricted” form of bar recursion is sufficient (over system T) to define Spector's search functional. This new result is then used to show that Spector's restricted form of bar recursion is in fact as general as the supposedly more general form of bar recursion. Given that these two forms of bar recursion correspond to the (explicitly controlled) iterated products of selection function and quantifiers, it follows that this iterated product of selection functions is T‐equivalent to the (...) corresponding iterated product of quantifiers. (shrink)

In recent years, proof theoretic transformations that are based on extensions of monotone forms of Gödel’s famous functional interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for bounded closed and (...) convex subsets of an abstract Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω∗ definable from bar recursion of lowest type. While a case study on the analysis of strong convergence results that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an explicit bound on a metastable version of this theorem that is primitive recursive relative to Ω∗.In the current paper we for the first time give the construction of Ω∗ . As a corollary to the fine analysis of the use of bar recursion in this construction we obtain that Ω∗ elevates arguments in Tn at most to resulting functionals in Tn+2 . In particular, one can conclude from this that our bound on Baillon’s theorem is at least definable in T4. (shrink)

Let the chain antichain principle (CAC) be the statement that each partial order on $\mathbb{N}$ possesses an infinite chain or an infinite antichain. Chong, Slaman, and Yang recently proved using forcing over nonstandard models of arithmetic that CAC is $\Pi^1_1$-conservative over $\text{RCA}_0+\Pi^0_1\text{-CP}$ and so in particular that CAC does not imply $\Sigma^0_2$-induction. We provide here a different purely syntactical and constructive proof of the statement that CAC (even together with WKL) does not imply $\Sigma^0_2$-induction. In detail we show using a (...) refinement of Howard's ordinal analysis of bar recursion that $\text{WKL}_0^\omega+\text{CAC}$ is $\Pi^0_2$-conservative over PRA and that one can extract primitive recursive realizers for such statements. Moreover, our proof is finitary in the sense of Hilbert's program. CAC implies that every sequence of $\mathbb{R}$ has a monotone subsequence. This Bolzano-Weierstraß}-like principle is commonly used in proofs. Our result makes it possible to extract primitive recursive terms from such proofs. We also discuss the Erdős-Moser principle, which—taken together with CAC—is equivalent to $\text{RT}^2_2$. (shrink)

We prove that the (non-intuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application. we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case.

In [U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 89–128], the second author obtained metatheorems for the extraction of effective bounds from classical, prima facie non-constructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT and normed linear spaces and guarantee the independence of the bounds from parameters ranging over metrically bounded spaces. Recently ]), the authors obtained generalizations of these metatheorems which (...) allow one to prove similar uniformities even for unbounded spaces as long as certain local boundedness conditions are satisfied. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to the classical case, there are practically no restrictions on the logical complexity of theorems for which bounds can be extracted. Again, our metatheorems guarantee very general uniformities, even in cases where the existence of uniform bounds is not obtainable by straightforward functional analytic means. Already in the purely intuitionistic case, where the existence of effective bounds is implicit, the metatheorems allow one to derive uniformities that may not be obvious at all from a given constructive proof. Finally, we illustrate our main metatheorem by an example from metric fixed point theory. (shrink)

In this paper we apply methods of proof mining to obtain a highly uniform effective rate of asymptotic regularity for the Ishikawa iteration associated with nonexpansive self-mappings of convex subsets of a class of uniformly convex geodesic spaces. Moreover, we show that these results are guaranteed by a combination of logical metatheorems for classical and semi-intuitionistic systems.

A pointwise version of the Howard-Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel'sT as well as Kleene/Feferman's ) is applied to systems of intuitionistic and classical arithmetic (H andH c) in all finite types with full induction as well as to the corresponding systems with restricted inductionĤ↾ andĤ↾c.H and Ĥ↾ are closed under a generalized (...) fan-rule. For a restricted class of formulae this also holds forH c andĤ↾c.We give a new and very perspicuous proof that for each one can construct a functional such that $\tilde \Phi \alpha $ is a modulus of uniform continuity for Φ on {β1∣∧n(βn≦αn)}. Such a modulus can also be obtained by majorizing any modulus of pointwise continuity for Φ.The type structure ℳ of all pointwise majorizable set-theoretical functionals of finite type is used to give a short proof that quantifier-free “choice” with uniqueness (AC!)1,0-qf. is not provable within classical arithmetic in all finite types plus comprehension [given by the schema (C)ϱ:∨y 0ϱ∧x ϱ(yx=0⇔A(x)) for arbitraryA], dependent ω-choice and bounded choice. Furthermore ℳ separates several μ-operators. (shrink)

In finite type arithmetic, the real numbers are represented by rapidly converging Cauchy sequences of rational numbers. Ulrich Kohlenbach introduced abstract types for certain structures such as metric spaces, normed spaces, Hilbert spaces, etc. With these types, the elements of the spaces are given directly, not through the mediation of a representation. However, these abstract spaces presuppose the real numbers. In this paper, we show how to set up an abstract type for the real numbers. The appropriateness of our construction (...) works in tandem with the bounded functional interpretation. (shrink)

We show how to interpret weak Kőnig's lemma in some recently defined theories of nonstandard arithmetic in all finite types. Two types of interpretations are described, with very different verifications. The celebrated conservation result of Friedman's about weak Kőnig's lemma can be proved using these interpretations. We also address some issues concerning the collecting of witnesses in herbrandized functional interpretations.