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27
Lectures on proof theory
 in Proc. Summer School in Logic, Leeds 67
, 1968
"... This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite " in mathematics from proofs of ..."
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This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite " in mathematics from proofs of purely nitary statements. One of the main approaches that turned out to be the most useful in pursuit of this program was that due to Gerhard Gentzen, in the 1930s, via his calculi of \sequents" and his CutElimination Theorem for them. Following that we trace how and why prima facie in nitary concepts, such as ordinals, and in nitary methods, such as the use of in nitely long proofs, gradually came to dominate prooftheoretical developments. In this rst lecture I will give anoverview of the developments in proof theory since Hilbert's initiative in establishing the subject in the 1920s. For this purpose I am following the rst part of a series of expository lectures that I gave for the Logic Colloquium `94 held in ClermontFerrand 2123 July 1994, but haven't published. The theme of my lectures there was that although Hilbert established his theory of proofs as a part of his foundational program and, for philosophical reasons whichwe shall get into, aimed to have it developed in a completely nitistic way, the actual work in proof theory This is the rst of three lectures that I delivered at the conference, Proof Theory: History
Syntactic cutelimination for common knowledge
 In Methods for Modalities
"... We first look at an existing infinitary sequent system for common knowledge for which there is no known syntactic cutelimination procedure and also no known nontrivial bound on the proofdepth. We then present another infinitary sequent system based on nested sequents that are essentially trees an ..."
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We first look at an existing infinitary sequent system for common knowledge for which there is no known syntactic cutelimination procedure and also no known nontrivial bound on the proofdepth. We then present another infinitary sequent system based on nested sequents that are essentially trees and with inference rules that apply deeply inside of these trees. Thus we call this system “deep ” while we call the former system “shallow”. In contrast to the shallow system, the deep system allows to give a straightforward syntactic cutelimination procedure. Since both systems can be embedded into each other, this also yields a syntactic cutelimination procedure for the shallow system. For both systems we thus obtain an upper bound of ϕ20 onthe depth of proofs, where ϕ is the Veblen function. Key words: cut elimination, infinitary sequent system, nested sequents, common knowledge 1.
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
"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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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 founda ..."
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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...
Dialgebraic Specification and Modeling
"... corecursive functions COALGEBRA state model constructors destructors data model recursive functions reachable hidden abstraction observable hidden restriction congruences invariants visible abstraction ALGEBRA visible restriction!e Swinging Cube ..."
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corecursive functions COALGEBRA state model constructors destructors data model recursive functions reachable hidden abstraction observable hidden restriction congruences invariants visible abstraction ALGEBRA visible restriction!e Swinging Cube
Prooftheoretic analysis by iterated reflection
 Arch. Math. Logic
"... Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omegarule. We compare the information obtained by this kind of analysis with the results obtained by the more usual prooftheoretic techni ..."
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Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omegarule. We compare the information obtained by this kind of analysis with the results obtained by the more usual prooftheoretic techniques. In some cases the techniques of iterated reflection principles allows to obtain sharper results, e.g., to define prooftheoretic ordinals relevant to logical complexity Π 0 1. We provide a more general version of the fine structure formulas for iterated reflection principles (due to U. Schmerl [24]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣn, IΣ − n, IΠ − n and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1reflection principle for T is Σ2conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl’s theorem. 1
Classifying the provably total functions of PA
 BSL
, 2006
"... We give a selfcontained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as good as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for ..."
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We give a selfcontained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as good as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for teaching purposes and just requires basic familiarity with PA and the ordinals below ε0. (Familiarity with a cut elimination theorem for a Gentzen or Tait calculus is helpful but not presupposed). 1