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The Undecidability of kProvability
 Annals of Pure and Applied Logic
, 1989
"... The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X... ..."
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The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X...
"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.
Does Reductive Proof Theory Have A Viable Rationale?
 Erkenntnis
, 2000
"... The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or foundational sch ..."
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The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistencyproof rationale and the foundational reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper, various reducti...
A HigherOrder Extension of Constraint Programming in Discourse Analysis
 First Workshop on Principles and Practice of Constraint Programming (Rhode Island
, 1993
"... Variables on which constraints are imposed incrementally can be said to carry "existential" force in the following sense. Under a translation, commonly used in analyzing natural language discourse, of firstorder formulas into programs from quantified dynamic logic, such a variable is introduced (at ..."
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Variables on which constraints are imposed incrementally can be said to carry "existential" force in the following sense. Under a translation, commonly used in analyzing natural language discourse, of firstorder formulas into programs from quantified dynamic logic, such a variable is introduced (at the level of formulas) by an existential quantifier. The present paper extends that translation to support constraints on variables introduced, as it were, by universal quantification. The extension rests on a certain program construct ) that can be interpreted (following Kleene's realizability analysis of universalexistential clauses) by closing the collection of states on which the programs act under partial functions. An alternative "reduced" interpretation of ) can also be given over sets of states from dynamic logic (not unlike Concurrent Dynamic Logic). These interpretations can be related by what is essentially a reduction to disjunctive normal form, involving socalled andor comput...
Term Induction
, 2001
"... In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid firstorder inference rule (A is quantifierfree). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called ter ..."
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In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid firstorder inference rule (A is quantifierfree). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called term induction, it derives a restricted term built from successor s and the constant 0. We call such terms numerals. To characterise the difference between T (tind) and pure logic, we employ proof theoretic methods. Firstly we establish a variant of Herbrand’s Theorem for T (tind). Let ∃¯xF (¯x) be a Σ1 formula; provable by Π. Then there exists a disjunction � N i1 · · · � N il M1(s i1 (0),..., s il(0)) ∨ · · · ∨ Mm(s i1 (0),..., s il(0)), denoted by H that is valid for some N ∈ IN, furthermore the Mi are instances of F (ā). In T (tind) it is not possible to bound the length of Herbrand disjunctions in terms of proof length and logical complexity of the endformula as usual. The main result is that we can bound the length of the {s, 0}matrix of the above disjunctions in this way.