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33
Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
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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
Saturated models of universal theories
 Annals of Pure and Applied Logic
"... A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a ..."
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Cited by 7 (3 self)
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A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the modeltheoretic version. 1
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we sho ..."
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Cited by 5 (2 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
Things that can and things that can't be done in PRA
, 1998
"... It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoW ..."
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Cited by 3 (1 self)
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It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoWeierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper
Injecting uniformities into Peano arithmetic
, 2008
"... We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic. As a consequence, some uniform boundedness principles (not necessarily settheoretically true) are interpre ..."
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Cited by 2 (1 self)
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We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic. As a consequence, some uniform boundedness principles (not necessarily settheoretically true) are interpreted while maintaining unmoved the Π0 2sentences of arithmetic. We explain why this interpretation is taylored to yield conservation results.
A most artistic package of a jumble of ideas
"... In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defen ..."
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Cited by 2 (2 self)
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In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel’s recasting of intuitionistic arithmetic via the “Dialectica ” interpretation, discuss the extra principles that the interpretation validates, and comment on extensionality and higher order equality. The latter sections focus on the role of majorizability considerations within the “Dialectica ” and related interpretations for extracting computational information from ordinary proofs in mathematics. I Kurt Gödel’s realism, a stance “against the current ” of his time, is now wellknown
RAMSEY’S THEOREM FOR PAIRS AND PROVABLY RECURSIVE FUNCTIONS
"... Abstract. This paper addresses the strength of Ramsey’s theorem for pairs (RT2 2) over a weak base theory from the perspective of ‘proof mining’. Let RT 2− 2 denote Ramsey’s theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a ..."
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Cited by 2 (1 self)
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Abstract. This paper addresses the strength of Ramsey’s theorem for pairs (RT2 2) over a weak base theory from the perspective of ‘proof mining’. Let RT 2− 2 denote Ramsey’s theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König’s lemma and a substantial amount of Σ0 1induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). In the resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of ∀∃theorems. There are two components this work. The first component is a general prooftheoretic result, due to the second author ([13, 14]), that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey’s theorem due to Erdős and Rado can be formalized using these restricted principles. So from the perspective of proof unwinding the computational content of concrete proofs based on RT2 2 the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of RT2 2 that are primitive recursive in f. 1.