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.

At first sight, the argument which F. P. Ramsey gave for (the infinite case of) his famous theorem from 1927, is hopelessly unconstructive. If suitably reformulated, the theorem is true intuitionistically as well as classically: we offer a proof which should convince both the classical and the intuitionistic reader. 1.

This paper essentially contains material from chapter 8 of the author's Habilitationschrift. Some of the results were presented at the Logic Colloquium 94 at Clermont--Ferrand (see [7]). 1 As is well-known (cf. the discussion at the end of x3 of [10]), the use of classical logic (on which the systems GnA ! are based) has the consequence that the extractability of an effective (and for n = 2 polynomial) bound from a proof of an 89A--sentence is (in general) guaranteed only if A is quantifier-- free (or purely existential). In the present paper we study proofs which may use mathematically strong non--constructive analytical principles as e.g. 1) Attainment of the maximum of f 2 C([0; 1] d ; IR) 2) Mean value theorem for integrals 3) Cauchy--Peano existence theorem 4) Brouwer's fixed point theorem for continuous functions f : [0; 1] d ! [0; 1] d 5) A generalization WKL 2 seq of the binary Konig's lemma WKL 6) Comprehension for negated formulas: CA ae : : 9\Phi 0ae x ae :1 0 8y ae \Gamma \Phiy = 0 0 $ :A(y) \Delta ; where A is arbitrary: as well as the non-intuitionistic logical principles 7) The `double negation shift' DNS : 8x ae ::A ! ::8x ae A for arbitrary types ae and formulas A 8) The `lesser limited principle of omniscience' LLPO : 8x 1 ; y 1 9k 0 1([k = 0 ! x IR y] [k = 1 ! y IR x]) 9) The independence of premise principle for negated formulas IP: : (:A ! 9y ae B) ! 9y ae (:A ! B); where y is not free in A, plus the schema AC of full choice but apply these principles only in the context of the intuitionistic versions (E)--GnA ! i of the theories (E)--G n A ! . The restriction to intuitionistic logic guarantees the extractability of (uniform) effective bounds for arbitrary 89A--sentences (see theorem 4.1 below). Indeed w...

We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue-measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0,1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T on a separable Hilbert space, and any element f, it is possible to compute a bound on the rate of convergence of (Anf) from T, f, and the norm ‖f ∗ ‖ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L2(X), then there is a computable bound on the rate of convergence of the sequence (Anf). The mean ergodic theorem is equivalent to the assertion that for every function K(n) and every ε> 0, there is an n with the property that the ergodic averages Amf are stable to within ε on the interval [n, K(n)]. Even in situations where the sequence (Anf) does not have a computable limit, one can give explicit bounds on such n in terms of K and ‖f‖/ε. This tells us how far one has to search to find an n so that the ergodic averages are “locally stable ” on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of “proof mining.” 1

Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1

Abstract. In this paper we show how to extend a constructive type theory with a principle that captures the spirit of Markov’s principle from constructive recursive mathematics. Markov’s principle is especially useful for proving termination of specific computations. Allowing a limited form of classical reasoning we get more powerful resulting system which remains constructive and valid in the standard constructive semantics of a type theory. We also show that this principle can be formulated and used in a propositional fragment of a type theory.

We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and Hahn-Banach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions needed in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and Helly-Hahn-Banach 1 theorems. Earlier pointfree formulations of the Hahn-Banach theorem, in a topostheoretic setting, were presented by Mulvey and Pelletier (1987,1991) and by Vermeulen (1986). A constructive proof based on points was given by Bishop (1967). In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As usual in pointfree topology, our guideline is to deøne the objects under analysis as formal points of a suitable formal space. After this has...

MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive LCF-style tactic-based proof assistant, a logical framework, a logical programming environment, and a formal methods programming toolkit. MetaPRL is distributed under an open-source license and can be downloaded from http://metaprl.org/. This paper provides an overview of the system focusing on the features that did not exist in the previous generations of PRL systems.

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in all finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a non-constructive -operator and his well-known results on the strength of this operator from the 70's. In this note we consider a weaker form LNOS (lesser numerical omniscience schema) of NOS which su#ces to derive the strong form of binary Konig's lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semi-classical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper. # Basic Research in Computer Science, Centre...

ms that hold only in computational models. For example, Brouwer proved that every totally defined function on the real line is continuous. A theory where this is provable cannot refer to the classical universe, so we have to consider whether we like its models better than the classical model. But any theorem in constructive mathematics is a theorem in classical mathematics, so in this case it is not a question of choosing between models, but of deciding whether it is worthwhile to talk about computational models in addition to the classical model. By comparing mathematical realism with intuitionism from an informal axiomatic point of view, we can steer clear of most of the metaphysical problems involved in analyzing these notions from the ground up, and concentrate on what may be termed the purely mathematical aspects. In particular we won't have to consider whether intuitionists hold that a theorem "isn't true until it's known to be true", as Blais 1 <F