Results 1  10
of
16
Analyzing Proofs in Analysis
 LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE
, 1993
"... ..."
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 ..."
Abstract

Cited by 39 (23 self)
 Add to MetaCart
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
"... ..."
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 ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 15 (6 self)
 Add to MetaCart
(Show Context)
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 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 ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
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
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 fo ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
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.
Synthesis of moduli of uniform continuity by the Monotone Dialectica Interpretation
 in the proofsystem MINLOG. Electronic Notes in Theoretical Computer Science
, 2007
"... We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel’s T of type (N→N)→(N→N). The generic solution may then be quickly inferred by the human. The automated synthesis of such moduli proceeds from a proof of the heredi ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel’s T of type (N→N)→(N→N). The generic solution may then be quickly inferred by the human. The automated synthesis of such moduli proceeds from a proof of the hereditarily extensional equality (≈) of t to itself, hence a proof in a weakly extensional variant of BergerBuchholzSchwichtenberg’s system Z of t ≈(N→N)→(N→N) t. We use an implementation on the machine, in Schwichtenberg’s MinLog proofsystem, of a nonliteral adaptation to Natural Deduction of Kohlenbach’s monotone functional interpretation. This new version of the Monotone Dialectica produces terms in NbEnormal form by means of a recurrent partial NbEnormalization. Such partial evaluation is strictly necessary.
PROOF INTERPRETATIONS AND MAJORIZABILITY
"... Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpret ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents