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Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
"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 show how ..."
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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.
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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Cited by 4 (3 self)
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS
, 2010
"... We study the computabilitytheoretic complexity and prooftheoretic strength of the following statements: (1) “If X is a wellordering, then so is εX”, and (2) “If X is a wellordering, then so is ϕ(α, X)”, where α is a fixed computable ordinal and ϕ represents the twoplaced Veblen function. For th ..."
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We study the computabilitytheoretic complexity and prooftheoretic strength of the following statements: (1) “If X is a wellordering, then so is εX”, and (2) “If X is a wellordering, then so is ϕ(α, X)”, where α is a fixed computable ordinal and ϕ represents the twoplaced Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA + 0 over RCA0. To prove the latter statement we need to use ωα iterations of the Turing jump, and we show that the statement is equivalent to Π0 ωαCA0. Our proofs are purely computabilitytheoretic. We also give a new proof of a result of Friedman: the statement “if X is a wellordering, then so is ϕ(X, 0)” is equivalent to ATR0 over RCA0.
Contents
, 2013
"... • A brief history of proof theory • Sequent calculi for classical and intuitionistic logic, Gentzen’s Hauptsatz: Cut elimination ..."
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• A brief history of proof theory • Sequent calculi for classical and intuitionistic logic, Gentzen’s Hauptsatz: Cut elimination
SubComputabilities
"... Abstract. Every recursively enumerable set of integers (r.e. set) is enumerable by a primitive recursive function. But if the enumeration is required to be oneone, only a proper subset of all r.e. sets qualify. Starting from a collection of total recursive functions containing the primitive recursi ..."
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Abstract. Every recursively enumerable set of integers (r.e. set) is enumerable by a primitive recursive function. But if the enumeration is required to be oneone, only a proper subset of all r.e. sets qualify. Starting from a collection of total recursive functions containing the primitive recursive functions, we thus define a subcomputability as an enumeration of the r.e. sets that are themselves oneone enumerable by total functions of the given collection. Notions similar to the classical computability ones are introduced and variants of the classical theorems are shown. We also introduce subreducibilities and study the related completeness notions. One of the striking results is the existence of natural (recursive) sets which play the role of low (nonrecursive) solutions to Post’s problem for these subreducibilities. The similarity between subcomputabilities and (complete) computability is surprising, since there are so many missing r.e. sets in subcomputabilities. They can be seen as toy models of computability.