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On the NoCounterexample Interpretation
 J. SYMBOLIC LOGIC
, 1997
"... In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functi ..."
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Cited by 18 (10 self)
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In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder 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 t ..."
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Cited by 17 (5 self)
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder 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
On the computational content of the Krasnoselski and Ishikawa fixed point theorems
, 2000
"... This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general ..."
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Cited by 11 (10 self)
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This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general socalled KrasnoselskiMann iterations. These iterations converge to fixed points of f under certain compactness conditions. But, as we show, already for uniformly convex spaces in general no bound on the rate of convergence can be computed uniformly in f . This is related to the nonuniqueness of fixed points. However, the iterations yield even without any compactness assumption and for arbitrary normed spaces approximate fixed points of arbitrary quality for bounded C (asymptotic regularity, Ishikawa 1976). We apply proof theoretic techniques (developed in previous papers of us) to noneffective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 9 (1 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
A most artistic package of a jumble of ideas
"... In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defen ..."
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Cited by 2 (2 self)
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In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel’s recasting of intuitionistic arithmetic via the “Dialectica ” interpretation, discuss the extra principles that the interpretation validates, and comment on extensionality and higher order equality. The latter sections focus on the role of majorizability considerations within the “Dialectica ” and related interpretations for extracting computational information from ordinary proofs in mathematics. I Kurt Gödel’s realism, a stance “against the current ” of his time, is now wellknown
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
2 1 General introduction
"... This paper (and its companion [29]) is another case study in the project of ‘proof mining ’ 1 in analysis by which we mean the logical analysis of mathematical proofs (typically using noneffective analytical tools) with the aim of extracting new numerically relevant information (e.g. effective unif ..."
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This paper (and its companion [29]) is another case study in the project of ‘proof mining ’ 1 in analysis by which we mean the logical analysis of mathematical proofs (typically using noneffective analytical tools) with the aim of extracting new numerically relevant information (e.g. effective uniform bounds or algorithms etc.) hidden in the proofs. 2 Many problems in numerical (functional) analysis can be seen as instances of the following general task: construct a solution x of an equation A(x): ≡ (F(x) = 0), where x is an element of some Polish (i.e. complete separable metric) space (typically with additional structure) and F: X → IR (usually F will depend on certain parameters a which again belong to Polish spaces). Quite often the construction of such a solution is obtained in two steps: 1) One shows how to construct (uniformly in the parameters of A) approximate solutions (sometimes called ‘εsolutions’) xε ∈ X for an εversion of the original equation Aε(x): ≡ ( F(x)  < ε). 2) Exploiting compactness conditions on X one concludes that either (x 1 n)n∈IN itself or some subsequence of it converges to a solution of A(x). The first step usually is constructive. However, the noneffectivity of the second step in many cases prevents one from being able to compute a solution x of A effectively within a prescribed error 1 k, i.e. to compute a function n(k) such that dX(xn(k), x) < 1 k. In many cases X: = K is compact and x is uniquely determined. Then (xn) itself converges to x so that no subsequence needs to be selected. However, the problem of how to get apriori bounds (in particular not depending on x itself) on the rate of convergence of that sequence remains. In numerical analysis, often such rates are not provided (due to the ineffectivity of the proof of the uniqueness of x). 3 In a series of papers we have demonstrated the applicability of proof theoretic techniques to extract socalled uniform moduli of uniqueness (which generalize 1 The term ‘proof mining ’ (instead of G. Kreisel’s ‘unwinding of proofs’) for the activity of extracting additional information hidden in given proofs using proof theoretic tools was suggested to the author by Professor Dana Scott. 2 For a different case study in analysis in the context of best approximation theory see [21],[22]. For other kinds of logical analyses of specific proofs see [33] and [36].