We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of classical minimal logic into minimal logic, which we refer to as the Peirce translation, and which we apply to interpret both a strengthening of the double-negation shift and the axioms of countable and dependent choice, via infinite products of selection functions.

Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domain-theoretic semantics. The theorem applies to extensions of G"odel's system T, but also to various forms of bar recursion for which strong normalization was hitherto unknown.

In this note we show that the so-called weakly extensional arithmetic in all nite types, which is based on a quanti er-free rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for 1 - axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept xed) was known.

In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. 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

In a previous paper we have given a representation of continuous functions on streams, both discrete-valued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We gave also a combinator on the representations of stream processing functions that reflects composition. Streams are the simplest example of a non-trivial final coalgebras, playing in the coalgebraic realm the same role as do the natural numbers in the algebraic realm. Here we extend our previous results to cover the case of final coalgebras for a broad class of functors generalising (×A). The functors we deal with are those that arise from countable signatures of finiteplace untyped operators. These have many applications. The topology we put on the final coalgebra for such a functor is that induced by taking for basic neighbourhoods the set of infinite objects which share a common prefix, according to the usual definition of the final coalgebra as the limit of a certain inverse chain starting at �. 1

Abstract. We show how different functional interpretations can be combined via a multi-modal linear logic. A concrete hybrid of Kreisel’s modified realizability and Gödel’s Dialectica is presented, and several small applications are given. We also discuss how the hybrid interpretation relates to variants of Dialectica and modified realizability with non-computational quantifiers. 1

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Abstract. The purpose of this article is to present a parametrised functional interpretation. Depending on the choice of the parameter relations 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 translation and a proof translation. We show that all these interpretation only differ on two choices: firstly, on “how much ” of the counterexamples for A became witnesses for ¬A when defining the formula translation, and, secondly, “how much ” of the witnesses of A one is interested in when defining the proof translation.

Abstract. The last section of “Lecture at Zilsel’s ” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Gödel’s reformulation, and discuss the relation between the two proofs. 1. Let me begin with a description of Gentzen’s consistency proof. As had already been noted in [5], we may express it in terms of a game. 1 To simplify things, we can assume that the logical constants of the classical system of number theory, P A, are ∧, ∨, ∀ and ∃ and that negations are applied only to atomic formulas. ¬φ in general is represented by the complement φ of φ, obtained by interchanging ∧ with ∨, ∀ with ∃, and atomic sentences with their negations. The components of the sentences φ ∨ ψ and φ ∧ ψ are φ and ψ. The components of the sentences ∃xφ(x) and ∀xφ(x) are the sentences φ(¯n) for each numeral ¯n. A ∧- or ∀-sentence, called a �-sentence, is thus expressed by the conjunction of its components and a ∨- or ∃-sentence, called a �-sentence, is expressed by the disjunction of them; and so it follows that every sentence can be represented as an infinitary propositional formula built up from prime sentences— atomic or negated atomic sentences. Disjunctive and conjunctive sentences with the components φn (where the range of n is 1, 2 or ω) will be denoted respectively by

Abstract. 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. §1. Introduction and background. In 1962 [14], Clifford Spector gave a remarkable characterization of the provably recursive functionals of full secondorder arithmetic (a.k.a. analysis). The central result of his paper is an extension, from arithmetic to analysis, of the (then quite recent) dialectica interpretation of Gödel of 1958 [7]. Spector’s extension relies on a form of well-founded recursion