Abstract not found

Abstract not found

A pointwise version of the Howard--Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Godel's T as well as Kleene/Feferman's PR) is applied to systems of intuitionistic and classical arithmetic (H and H in all finite types with full induction as well as to the corresponding systems with restricted induction . 1) H and H| \ are closed under a generalized fan--rule. For a restricted class of formulae this also holds for H . 2) We give a new and very perspicuous proof that for each # PR) one can construct a functional PR) such that ## is a modulus of uniform continuity for # on {# |#n(#n # #n)}. Such a modulus can also be obtained by majorizing any modulus of pointwise continuity for #. 3) The type structure of all pointwise majorizable set--theoretical functionals of finite type is used to give a short proof that quantifier--free "choice" with uniqueness --qf. is not provable within classical arithmetic in all finite types plus comprehension (given by the schema (C) (yx = 0 A(x)) for arbitrary A), dependent #--choice and bounded choice. Furthermore separates several --operators. . 1

This paper is part of a sequence of papers ([9],[10],[11],[12]) resulting from our Habilitation thesis [8] addressing the following question: What is the impact on the growth of extractable uniform bounds the use of various analytical principles \Gamma in a given proof of an 89--sentence might have? In particular we are interested in analyzing proofs of sentences having the form (1) 8u

In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

this paper and discuss now only (PCM) in order to motivate the results of the present paper which is the second one in a sequence of papers resulting from the authors Habilitationsschrift [12]. All undefined notions are used in the sense of [14] on which this paper relies. A 0 , B 0 , C 0 , . . . always denote quantifier-free formulas. Using a convenient representation of real numbers, (PCM) can be formalized as follows: (PCM) : 0 hk(|a m )). (PCM) immediately follows from its arithmetical weakening (PCM - ) : # #k 0 n(|a m by an application of AC ar to n(|a m k + 1 1 (# IR 1 follows from the fact that real numbers are given as Cauchy sequences of rationals with fixed rate of convergence in our theories). It is well--known that a constructive functional interpretation of the negative translation of AC ar requires so--called bar-recursion and cannot be caried out e.g. in Godel's term calculus T (see [21] and [15] ). AC ar is (using classical logic) equivalent to CA ar +AC --qf, where CA ar : (g(x) = 0 0 A(x)) with A (and AC --qf is the restriction of AC ar to quantifier-free formulas) and therefore causes an immense rate of growth (when added to e.g. G 2 A ). From the work in the context of `reverse mathematics' (see e.g. [3],[20]) it is known that 1)--5) imply CA ar relatively to (a second-order version of) \ +AC --qf (see [1] for the definition of \ ). In [12] it is shown that this holds even relatively to G 2 A . In contrast to these general facts we prove in this paper a meta--theorem which in particular implies that if (PCM) is applied in a proof only to sequences (a n ) which are given explicitely in the parameters of the proposition (which is proved) then this pr...

this paper is to show that the arguments given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistic point of view and that the later argument given by Nash-Williams is not. The paper consists of the following 11 Sections. 1. Dickson's Lemma 2. Almost full relations 3. Brouwer's Thesis 4. Ramsey's Theorem 5. The Finite Sequence Theorem 6. Vazsonyi's Conjecture for binary trees 7. Higman's Theorem 8. Vazsonyi's Conjecture and the Tree Theorem 9. Minimal-Bad-Sequence Arguments 10. The Principle of Open Induction 11. Concluding Remarks Except for Section 9, we will argue intuitionistically. 1 1 Dickson's Lemma

In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in T n -proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As

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...

The so-called 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 higher-order 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 er-free 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 er-free rule of extensionality QF-ER 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.