Results 1  10
of
30
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 ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
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
Elimination of Skolem functions for monotone formulas in analysis
 ARCHIVE FOR MATHEMATICAL LOGIC
"... ..."
Foundational and mathematical uses of higher types
 REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN
, 1999
"... In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles which generalize (and for n = 0 coincide with) the socalled `weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context)
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 ..."
Abstract

Cited by 11 (10 self)
 Add to MetaCart
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...
Groundwork for weak analysis
 the Journal of Symbolic Logic
, 2002
"... Abstract. This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variabl ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
Abstract. This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we show how to interpret the theory BTFA in Robinson’s theory of arithmetic Q. This fact entails that the elementary theory of the real closed ordered fields is interpretable in Q. §1. Introduction. The formalization of mathematics within secondorder arithmetic has a long and distinguished history. We may say that it goes back to Richard Dedekind, and that it has been pursued by, among others, Hermann Weyl, David Hilbert, Paul Bernays, Harvey Friedman, and Stephen Simpson and his students (we may also mention the insights of Georg Kreisel, Solomon Feferman,
Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
"... ..."
On the Uniform Weak König's Lemma
, 1999
"... The socalled weak König's lemma WKL asserts the existence of an in nite path b in any in nite binary tree (given by a representing function f ). Based on this principle one can formulate subsystems of higherorder arithmetic which allow to carry out very substantial parts of classical mathematics b ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
The socalled weak König's lemma WKL asserts the existence of an in nite path b in any in nite binary tree (given by a representing function f ). Based on this principle one can formulate subsystems of higherorder arithmetic which allow to carry out very substantial parts of classical mathematics but are 2  conservative over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In [10] we established such conservation results relative to nite type extensions PRA of PRA (together with a quanti erfree axiom of choice schema). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional which selects uniformly in a given in nite binary tree f an in nite path f of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA only has a quanti erfree rule of extensionality QFER instead of the full axioms (E) of extensionality for all nite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be 2 conservative over PRA, PRA + (E)+UWKL contains (and is conservative over) full Peano arithmetic PA.