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Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... 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 ..."
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Cited by 31 (20 self)
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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 quantifierfree choice ACqf 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 settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #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 settheoretic 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 # +ACqf+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 # +ACqf+# for suitable #. 1
A note on the Π 0 2induction rule
 Arch. Math. Logic
, 1995
"... It is well–known (due to C. Parsons) that the extension of primitive recursive arithmetic PRA by first–order predicate logic and the rule of Π 0 2 –induction Π0 2 –IR is Π02 –conservative over PRA. We show that this is no longer true in the presence of function quantifiers and quantifier–free choice ..."
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Cited by 3 (3 self)
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It is well–known (due to C. Parsons) that the extension of primitive recursive arithmetic PRA by first–order predicate logic and the rule of Π 0 2 –induction Π0 2 –IR is Π02 –conservative over PRA. We show that this is no longer true in the presence of function quantifiers and quantifier–free choice for numbers AC0,0 – qf. More precisely we show that T:=PRA2 + Π0 2 –IR+AC0,0–qf proves the totality of the Ackermann function, where PRA2 is the extension of PRA by number and function quantifiers and Π0 2 –IR may contain function parameters. This is true even for PRA2 + Σ0 1 –IR+Π02 –IR−+AC0,0–qf, where Π0 2 –IR − is the restriction of Π0 2 –IR without function parameters. 1 Let (PRA) denote the extension of primitive recursive arithmetic obtained by adding first–order predicate logic. By the rule IR of induction we mean