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12
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
Elimination of Skolem functions for monotone formulas in analysis
 ARCHIVE FOR MATHEMATICAL LOGIC
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Foundational and mathematical uses of higher types
 REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN
, 1999
"... In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles ..."
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Cited by 11 (4 self)
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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles which generalize (and for n = 0 coincide with) the socalled `weak' König'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)
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
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 9 (1 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
A Note on Spector's QuantifierFree Rule of Extensionality
 Arch. Math. Logic
, 1999
"... In this note we show that the socalled weakly extensional arithmetic in all nite types, which is based on a quanti erfree rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for add ..."
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Cited by 6 (4 self)
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In this note we show that the socalled weakly extensional arithmetic in all nite types, which is based on a quanti erfree rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for 1  axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept xed) was known.
The Computational Strength of Extensions of Weak König's Lemma
, 1998
"... The weak Konig's lemma WKL is of crucial signi cance in the study of on the other hand have a low prooftheoretic 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 erfree. Whereas in g ..."
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Cited by 1 (0 self)
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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 prooftheoretic 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 erfree. Whereas in general the computational and prooftheoretic 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 1WKL. 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 erfree 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 bAC In the absence of functional parameters (so in particular in a second order context), the corresponding versions of 1WKL and 1 bAC 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 1WKL and 1 bAC to a nonstandard axiom F (introduced in a previous paper) and a new elimination result for F relative to various fragment of arithmetic in...