Introduction to the constructive point of view in the foundations of mathematics, in particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation and formalization of logic is described. For constructive (intuitionistic) arithmetic, Kleene’s realizability interpretation is given; this provides an example of the possibility of a constructive mathematical practice which diverges from classical mathematics. The crucial notion in intuitionistic analysis, choice sequence, is briefly described and some principles which are valid for choice sequences are discussed. The second half of the article deals with some aspects of proof theory, i.e., the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions are outlined: his introduction of the so-called Gentzen systems which use sequents instead of formulas and his result on first-order arithmetic showing that (suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in first-order arithmetic.

We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.

U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show that, using ideas from the theory of intersection types [2, 6, 7, 21] and Martin-Löf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for Martin-Löf dependent type theory extended with bar recursion, and with some form of proof-irrelevance. 1

We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the well-known abbreviation

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.

A domain model characterising strong normalisation

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These interpretations use strong computation principles with difficult termination proofs. 2 The problem Given a strongly normalizing type theory T and a system of rewrite rules R, show that the combined system T + R is strongly normalizing. We are mainly interested in higher-order rewrite rules that go beyond structural recursion. Example T = simple types over the booleans and the natural numbers R = Gödel primitive recursion + modified bar recursion: