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"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.
An Interpolation Theorem
 Bull. Symbolic Logic
, 2000
"... Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of rstorder logic, there is an interpolant in which each relation symbol appears only in those polarities in which it appears in both the antecedent and the succedent of the given i ..."
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Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of rstorder logic, there is an interpolant in which each relation symbol appears only in those polarities in which it appears in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some xed tuple U of unary predicates U , all formulae under consideration have all quantiers explicitly relativised to one of the U . Under this stipulation, existential (universal) quantication over U contributes a positive (negative) occurrence of U . It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a manysorted interpolation theorem focusing o...
A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' resul ..."
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Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a selfcontained proof requiring only basic knowledge of mathematical logic.
Which Quantifiers are Logical? A
"... combined semantical and inferential criterion ..."
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Harmonious Logic Harmonious Logic: Craig’s Interpolation Theorem and its Descendants
"... For Bill Craig, with great appreciation for his fundamental contributions to our subject, and for his perennially open, welcoming attitude and fine personality that enhances every encounter. Abstract: Though deceptively simple and plausible on the face of it, Craig’s interpolation theorem (published ..."
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For Bill Craig, with great appreciation for his fundamental contributions to our subject, and for his perennially open, welcoming attitude and fine personality that enhances every encounter. Abstract: Though deceptively simple and plausible on the face of it, Craig’s interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig’s theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of manysorted interpolation theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by prooftheoretic arguments while most of the applications are modeltheoretic in nature. The article concludes with the role of the interpolation property in the quest for “reasonable ” logics extending firstorder logic within the framework of abstract model theory.
Ah, Chu!
"... A theorem obtained by van Benthem for preservation of formulas under Chu transforms between Chu spaces is strengthened and derived from a general manysorted interpolation theorem. The latter has been established both by prooftheoretic and modeltheoretic methods; there is some discussion as to how ..."
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A theorem obtained by van Benthem for preservation of formulas under Chu transforms between Chu spaces is strengthened and derived from a general manysorted interpolation theorem. The latter has been established both by prooftheoretic and modeltheoretic methods; there is some discussion as to how these methods compare and what languages they apply to. In the conclusion, several further questions are raised. A theorem obtained by van Benthem for preservation of formulas under Chu transforms between Chu spaces is strengthened and derived from a general manysorted interpolation theorem. The latter has been established both by prooftheoretic and modeltheoretic methods; there is some discussion as to how these methods compare and what languages they apply to. In the conclusion, several further questions are raised. Dear Johan,
FIRST ORDER PRESERVATION THEOREMS IN TWOSORTED LANGUAGES
"... The purpose of this note is to extend to twosorted first order languages several wellknown onesorted preservation theorems, and to give some preservation theorems which are new even in the onesorted case. For reasons of uniformity and economy we adopt the presentation of Lindstrom [5]. However t ..."
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The purpose of this note is to extend to twosorted first order languages several wellknown onesorted preservation theorems, and to give some preservation theorems which are new even in the onesorted case. For reasons of uniformity and economy we adopt the presentation of Lindstrom [5]. However the techniques of Keisler [2], [3] were used to find the theorems. It also seems likely that the methods of Makkai [7] could be used here and, in addition, to extend these results to various infinitary languages. The generalization of the work here to manysorted languages is clear. Feferman [1] obtained such a result for sentences preserved by extensions. In section 1 we give the generalization of Lindstrom's main theorem to twosorted languages. In section 2 we apply this to get the extensions of some known preservation theorems in Theorem 2.1, and other preservation theorems in Theorems 2.2, 2.3 and 2.4. As applications of these, let S be the twosorted theory of modules (see [8], [9]). A set A x of sentences is syntactically described such that a sentence 0 is preserved by homomorphisms of modules, where the homomorphism is an isomorphism of the base rings, if and only if there is a 9 in Ax such that S h (j> <>0 (Theorem 2.1). Let nx < n2 < «3 <... be an arbitrary but fixed sequence of positive integers. Let 0WM denote the countable weak direct power, and M &quot; the direct power, of the module M. Then F is a restricted isomorphism from ®WM onto ®aN if it is an isomorphism and, for all /, F(M &quot; 1) — N&quot; ' (F being an isomorphism of the base rings). A set A2 of sentences is syntactically described such that a sentence $ is preserved, from M to JV, by restricted isomorphisms from ©w M onto ®a N if and only if there is a 9 in A2 such that S h 4> <> 9 (Theorem 2.2). This theorem in the onesorted case would give a corresponding result with, for example, S the theory of groups. Other consequences, involving modules being direct factors of other modules, are noted in the final paragraph of this paper. 1. We shall consider a twosorted first order language L, with connectives