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 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

For a formal theory T, the diagonalizable algebra (a.k.a. Magari algebra) of T, denoted DT, is the Lindenbaum sentence algebra of T endowed with the unary operator T arising from the provability predicate of T: (the equivalence class of) a sentence ' is sent by T to (the equivalence class of) the T-sentence expressing that T proves '. It was shown in Shavrukov [6] that the diagonalizable algebras of PA and ZF, as well as the diagonalizable algebras of similarly related pairs of 1-sound theories, are not isomorphic. Neither are these algebras rst-order equivalent (Shavrukov [7, Theorem 2.11]). In the present paper we establish a su cient condition, which we name B ( ) coherence, for the diagonalizable algebras of two theories to be isomorphic. It is then immediately seen that DZF = DGB, which answers a question of Smorynski [11]. We also construct non-identity automorphisms of diagonalizable algebras of all theories under consideration. The techniques we useareacombination of those developed in the