Abstract not found

Abstract not found

Abstract not found

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

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. 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 second-order 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,

In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are proof-theoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on non-collapsing hierarchies ( n -WKL+ ; n -WKL+ ) of principles which generalize (and for n = 0 coincide with) the so-called `weak' Konig'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) Basic Research in Computer Science, Centre of the Danish National Research Foundation.

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...

A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a proof-theoretic method, Herbrand analysis, yielding uniform model-theoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the model-theoretic version. 1

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