This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifier--free choice AC--qf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non-- standard' axiom F - which does not hold in the full set--theoretic model but in the strongly majorizable functionals): From a proof GnA # +AC--qf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full set--theoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +AC--qf+F - proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +AC--qf+# for suitable #. 1

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

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.

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The weak Konig's lemma WKL is of crucial signi cance in the study of on the other hand have a low proof-theoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKL is also `weak' in that the tree predicate is quanti er-free. Whereas in general the computational and proof-theoretic strength increases when logically more complex trees are allowed, we show that this is not the case for trees which are given by formulas in a class 1 where we allow an arbitrary function quanti er pre x over bounded functions in front of a 1 -formula. This results in a schema 1-WKL. Another way of looking at WKL is via its equivalence to the principle 8x9y 18z A 0 (x; y; z) ! 9f x:18x; z A 0 (x; fx; z); where A 0 is a quanti er-free formula (x; y; z are natural number variables). We generalize this to 1 -formulas as well and allow function quanti ers `9g s' instead of `9y 1', where g s is de ned pointwise. The resulting schema is Basic Research in Computer Science, Centre of the Danish National Research Foundation. called 1 -b-AC In the absence of functional parameters (so in particular in a second order context), the corresponding versions of 1-WKL and 1 -b-AC turn out to be equivalent to WKL. This changes completely in the presence of functional variables of type 2 where we get proper hierarchies of principles n -WKL and . Variables of type 2 however are necessary for a direct representation of analytical objects and { sometimes { for a faithful representation of such objects at all as we will show in a subsequent paper. By a reduction of 1-WKL and 1 -b-AC to a non-standard axiom F (introduced in a previous paper) and a new elimination result for F relative to various fragment of arithmetic in...

Abstract. We generalize to abstract many-sorted algebras the classical proof-theoretic result due to Parsons, Mints and Takeuti that an assertion ∀x ∃y P(x,y) (where P is Σ0 1), provable in Peano arithmetic with Σ0 1 induction, has a primitive recursive selection function. This involves a corresponding generalization to such algebras of the notion of primitive recursiveness. The main difficulty encountered in carrying out this generalization turns out to be the fact that equality over these algebras may not be computable, and hence atomic formulae in their signatures may not be decidable. The solution given here is to develop an appropriate concept of realizability of existential assertions over such algebras, generalized to realizability of sequents of existential assertions. In this way, the results can be seen to hold for classical proof systems. This investigation may give some insight into the relationship between specifiability and computability for data types such as the reals, where the atomic formulae, i.e., equations between terms of type real, are not computable. Key words and phrases: generalized computability, realizability, selection function2 1

Contents 1 Introduction 1 2 General Background 2 2.1 Godel's System T . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Intuitionistic Arithmetic for Functionals . . . . . . . . . . . . . . 6 2.3 Program Extraction from Constructive Proofs . . . . . . . . . . . 7 2.4 Example: Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . 13 3 Computational Content of Classical Proofs 14 3.1 Definite and Goal Formulas . . . . . . . . . . . . . . . . . . . . . 14 3.2 Computational Content . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Example: Fibonacci Numbers Again . . . . . . . . . . . . . . . . 23 3.4 Example: Integer Square Roots . . . . . . . . . . . . . . . . . . . 26 3.5 Example: The Greatest Common Divisor . . . . . . . . . . . . . 28 3.6 Example: Dickson's Lemma . . . . . . . . . . . . . . . . . . . . . 35 3.7 Towards More Interesting Examples . . . . . . . . . . . . . . . . 38 1 Introduction It is