In [15],[16] Kreisel introduced the no-counterexample 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 first-order 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

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...

It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano-Weierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper

Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1-formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a self-contained proof requiring only basic knowledge of mathematical logic.