Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees. 1.

Abstract. Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.?1. Introduction. In 1963, Paul Cohen introduced the method of forcing to prove the independence of both the axiom of choice and the continuum hypothesis from Zermelo-Fraenkel set theory. It was not long before Saul Kripke noted a connection between forcing and his semantics for modal and

Undecidability of various properties of first order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2-complete. The particular problem of local confluence turns out to be Π 0 2-complete for ground terms, but only Σ 0 1-complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1-complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.

Dedicated to Grigori Mints in honor of his seventieth birthday. Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical first-order arithmetic, and reflects on some of the relationships between them. Variants of the Gödel-Gentzen doublenegation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory. 1

Abstract. Undecidability of various properties of first order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy, where also quantification over function variables is allowed. In this paper we consider properties of first order term rewriting systems and classify them in this hierarchy. Weak and strong normalization for single terms turn out to be Σ 0 1-complete, while their uniform versions as well as dependency pair problems with minimality flag are Π 0 2-complete. We find that confluence is Π 0 2-complete both for single terms and uniform. Unexpectedly weak confluence for ground terms turns out to be harder than weak confluence for open terms. The former property is Π 0 2-complete while the latter is Σ 0 1-complete (and thereby recursively enumerable). The most surprising result is on dependency pair problems without minimality flag: we prove this to be Π 1 1-complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic. 1