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.

The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to safely add new decision procedures to the Nuprl prover. It relies on a reflection mechanism and is applicable to any tactic-oriented prover with sufficient reflection. The lectures explain reflection in the setting of constructive type theory, the core logic of Nuprl.

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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.

Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is well-known that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14]. There is a link between the presence of cut formulas with nested quantifiers and the non-elementary expansion needed to prove a theorem without the help of such formulas. If one considers the graph defined by tracing the flow of occurrences of formulas (in the sense of [2]) for proofs allowing a non-elementary compression, one Preprint submitted to Elsevier Preprint 7 November 1997 finds that such graphs contain cycles [5] or almost cyclic structures[6]. These cycles codify in a small space (i.e. a proof with a small number of lines) all the information which is present in the proof once cuts on formulas wit

Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs. © 2001 Academic Press Key Words: formal proofs; logical flow graphs; cut elimination; bridge groups; Baumslag–Solitar groups; Gersten groups.

The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be non-elementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with O(k n+1 ) lines. In particular, there is a quadratic time algorithm which eliminates a single cycle from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.

ABSTRACT. Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen’s cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures. Many mathematicians and philosophers of mathematics think that it’s somehow valuable for a proof to be “pure”, that is, for it not to use notions extraneous to what is being proved. Not worrying about why that is for now, we would like to make better sense of the suggestion. What does it mean for a notion used in a proof to be extraneous to the theorem being proved? One way of making this sharper would be to develop a syntactic way of evaluating extraneousness. I want to consider such a proposal, using Gerhard Gentzen’s cut-elimination theorem. I will argue that there are serious obstacles to making this proposal work. 1. FOUR CLAIMS CONCERNING EXTRANEOUSNESS Bertrand’s postulate states that for every natural number n ≥ 1, there is a prime number between n and 2n. It is so named because it was verified by computation for

We extend arithmetic with a new predicate, Pr, giving axioms for Pr based on first-order versions of Lob's derivability conditions. We hoped that the addition of a reflection schema mentioning Pr would then give a non-conservative extension of the original arithmetic theory. The paper investigates this possibility. It is shown that, under special conditions, the extension is indeed non-conservative. However, in general such extensions turn out to be conservative. 1 Introduction In any recursively axiomatized theory of arithmetic, T , one can follow Godel's construction to obtain a `provability predicate', a \Sigma 1 -formula Bew T (x) such that Bew T (pAq) is true if and only if T ` A, where pAq is the Godel number of the formula A. Moreover, if T is sufficiently strong then Bew T satisfies the following predicate (or `uniform') versions of Lob's derivability conditions [7]: (D1) if T ` 8xA then T ` 8xBew T (pAhxiq); (D2) T ` 8x(Bew T (p(A ! B)hxiq) ! (Bew T (pAhxiq) ! Bew T (pBhxiq)...

Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for actual points to move around inside them, perhaps to parameterize them in some way. The concept of formal proofs has the interesting feature that it provides opportunities for both perspectives. The inner perspective has been largely overlooked, but in fact lengths of proofs lead to new ways to measure the information content of mathematical objects. The disparity between minimal lengths of proofs with and without "lemmas" provides an indication of internal symmetry of mathematical objects and their descriptions.