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General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 31 (18 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 9 (1 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
On the logical analysis of proofs based on nonseparable Hilbert space theory
, 2010
"... Starting in [15] and then continued in [9, 17, 24] and [18], general logical metatheorems were developed that guarantee the extractability of highly uniform effective bounds from proofs of theorems that hold for general classes of structures such as ..."
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Cited by 4 (4 self)
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Starting in [15] and then continued in [9, 17, 24] and [18], general logical metatheorems were developed that guarantee the extractability of highly uniform effective bounds from proofs of theorems that hold for general classes of structures such as
Injecting uniformities into Peano arithmetic
, 2008
"... We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic. As a consequence, some uniform boundedness principles (not necessarily settheoretically true) are interpre ..."
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Cited by 2 (1 self)
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We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic. As a consequence, some uniform boundedness principles (not necessarily settheoretically true) are interpreted while maintaining unmoved the Π0 2sentences of arithmetic. We explain why this interpretation is taylored to yield conservation results.
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 1 (1 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
URL: www.emis.de/journals/AFA/ UNIFORM BOUNDEDNESS PRINCIPLES FOR ORDERED TOPOLOGICAL VECTOR SPACES
"... Abstract. We obtain uniform boundedness principles for a new class of families of mappings from topological vector spaces to ordered topological vector spaces. The new class of families of mappings includes the family of linear mappings and many other families which consist of nonlinear mappings. 1. ..."
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Abstract. We obtain uniform boundedness principles for a new class of families of mappings from topological vector spaces to ordered topological vector spaces. The new class of families of mappings includes the family of linear mappings and many other families which consist of nonlinear mappings. 1. Introduction and