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21
Analyzing Proofs in Analysis
 LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE
, 1993
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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 34 (21 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
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 31 (18 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
Pointwise hereditary majorization and some applications
 Arch. Math. Logic
, 1992
"... A pointwise version of the Howard–Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel’s T as well as Kleene/Feferma ..."
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Cited by 14 (11 self)
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A pointwise version of the Howard–Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel’s T as well as Kleene/Feferman’s P R) is applied to systems of intuitionistic and classical arithmetic (H and Hc) in all finite types with full induction as well as to the corresponding systems with restricted induction ˆ H  \ and ˆ H  \c 1) H and ˆ H  \ are closed under a generalized fan–rule. For a restricted class of formulae this also holds for H c and ˆ H  \c 2) We give a new and very perspicuous proof that for each Φ 2 ∈ T ( P R) one can construct a functional ˜ Φ 2 ∈ T ( P R) such that ˜ Φα is a modulus of uniform continuity for Φ on {β 1 ∀n(βn ≤ αn)}. Such a modulus can also be obtained by majorizing any modulus of pointwise continuity for Φ. 3) The type structure M of all pointwise majorizable set–theoretical functionals of finite type is used to give a short proof that quantifier–free “choice ” with uniqueness (AC!) 1,0 –qf. is not provable within classical arithmetic in all finite types plus comprehension (given by the schema (C) ρ: ∃y0ρ∀xρ (yx = 0 ↔ A(x)) for arbitrary A), dependent ω–choice and bounded choice. Furthermore M separates several µ–operators. 1
Elimination of Skolem functions for monotone formulas in analysis
 ARCHIVE FOR MATHEMATICAL LOGIC
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Strongly Uniform Bounds from SemiConstructive Proofs
, 2004
"... In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and nor ..."
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Cited by 10 (6 self)
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In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semiintuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain nonconstructive principles. Contrary to
On the Uniform Weak König's Lemma
, 1999
"... The socalled 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 higherorder arithmetic which allow to carry out very substantial parts of classical mathematics b ..."
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Cited by 10 (5 self)
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The socalled 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 higherorder 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 erfree 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 erfree rule of extensionality QFER 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.
On the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness
 PURE AND APPLIED LOGIC
, 1997
"... 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 ..."
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Cited by 9 (4 self)
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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