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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 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.
Things that can and things that can't be done in PRA
, 1998
"... It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoW ..."
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Cited by 3 (1 self)
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It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoWeierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper
On Tao’s “finitary” infinite pigeonhole principle
 The Journal of Symbolic Logic
, 2010
"... In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasifinitization of the infinite pigeonhole principle IPP, arriving at the “finitary ” infinite pigeonho ..."
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Cited by 2 (2 self)
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In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasifinitization of the infinite pigeonhole principle IPP, arriving at the “finitary ” infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay. We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP2 and IPP ↔ FIPP3 in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao’s notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e. in terms of the “big five ” subsystems of second order arithmetic. 1
On the Steiner Tree 3/2Approximation For QuasiBipartite Graphs
, 1999
"... IR + be a nonnegative weighting of the edges of G. Assume V is partitioned as R X . A Steiner tree is any tree T of G such that every node in R is incident with at least one edge of T . The metric Steiner tree problem asks for a Steiner tree of minimum weight, given that w is a metric. When ..."
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IR + be a nonnegative weighting of the edges of G. Assume V is partitioned as R X . A Steiner tree is any tree T of G such that every node in R is incident with at least one edge of T . The metric Steiner tree problem asks for a Steiner tree of minimum weight, given that w is a metric. When X is a stable set of G, then (G, R, X) is called quasibipartite.
PROOF INTERPRETATIONS AND MAJORIZABILITY
"... Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpret ..."
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Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents
Classical provability of uniform versions and intuitionistic provability
, 2013
"... Along the line of HirstMummert [9] and Dorais [4], we analyze the relationship between the classical provability of uniform versions Uni(S) of Π2statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for e ..."
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Along the line of HirstMummert [9] and Dorais [4], we analyze the relationship between the classical provability of uniform versions Uni(S) of Π2statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for every Π2statement S of some syntactical form, if its uniform version derives the uniform variant of ACA over a classical system of arithmetic in all finite types with weak extensionality, then S is not provable in strong semiintuitionistic systems including bar induction BI in all finite types but also nonconstructive principles such as König’s lemma KL and uniform weak König’s lemma UWKL. Our result is applicable to many mathematical principles whose sequential versions imply ACA. 1