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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 proof-theoretic 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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Cited by 4 (2 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's proof-theoretic 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.
A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1-formulas. 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 ..."
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Cited by 2 (1 self)
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Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1-formulas. 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 self-contained proof requiring only basic knowledge of mathematical logic.
An Interpolation Theorem
- Bull. Symbolic Logic
, 2000
"... Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of rst-order 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 implic ..."
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Cited by 1 (0 self)
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Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of rst-order 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 many-sorted interpolation theorem focusing o...
Combination Results for Many Sorted Theories with Overlapping Signatures
, 2004
"... We present a combination result for many-sorted rst-order theories whose signatures may share common symbols (i.e. overlapping or non-disjoint signatures), extending the recent results by Ghilardi for the unsorted case. Furthermore, we give practical conditions under which the combination method ..."
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We present a combination result for many-sorted rst-order theories whose signatures may share common symbols (i.e. overlapping or non-disjoint signatures), extending the recent results by Ghilardi for the unsorted case. Furthermore, we give practical conditions under which the combination method becomes a semi-decision procedure, and additional sucient conditions which turn it into a decision procedure.
