Separation and Weak König's Lemma

Humphreys, A. James; Simpson, Stephen G.

doi:10.2307/2586763

Cited by 4 publications

(3 citation statements)

References 11 publications

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“…Our constructions will use a result from the development of Banach spaces in [12], and draw on ideas from the corresponding developments in recursive mathematics [20] and constructive mathematics [1]. There are, however, subtleties and key differences, some of which are discussed at the end of this section.…”

Section: Bases and Independent Generating Sequencesmentioning

confidence: 99%

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Fundamental notions of analysis in subsystems of second-order arithmetic

Avigad

Simic

2006

Annals of Pure and Applied Logic

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We develop fundamental aspects of the theory of metric, Hilbert, and Banach spaces in the context of subsystems of second-order arithmetic. In particular, we explore issues having to do with distances, closed subsets and subspaces, closures, bases, norms, and projections. We pay close attention to variations that arise when formalizing definitions and theorems, and study the relationships between them. For example, we show that a natural formalization of the mean ergodic theorem can be proved in ACA 0 ; but even recognizing the theorem's "equivalent" existence assertions as such can also require the full strength of ACA 0 .

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Section: Bases and Independent Generating Sequencesmentioning

confidence: 99%

“…The following lemma is a consequence of Humphreys and Simpson [12]; see also the "independence criterion" in [20, page 143], or [17, Lemma 2.4-1].…”

Section: Proof Let σ Be a Finite Sequence Of Elements Of A That Spanmentioning

confidence: 99%

Fundamental notions of analysis in subsystems of second-order arithmetic

Avigad

Simic

2006

Annals of Pure and Applied Logic

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“…We exploit the ideas of this proof to show Sep c HB. The original proof by Brown and Simpson of the "forward direction" (showing that WKL 0 proves the separable Hahn-Banach Theorem) has been simplified first by Shioji and Tanaka ([17], this is essentially the proof contained in [19,§IV.9]) and then by Humphreys and Simpson [11]. No details of these or other proofs of the Hahn-Banach Theorem are needed for showing HB c Sep. Brattka noticed the possibility of avoiding these details in [2] and wrote, "Surprisingly, the proof of this theorem does not require a constructivization of the classical proof but just an 'external analysis'."…”

Section: Introductionmentioning

confidence: 99%

How Incomputable Is the Separable Hahn-Banach Theorem?

Gherardi¹,

Marcone²

2009

Notre Dame J. Formal Logic

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We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL 0 . We study analogies and differences between WKL 0 and the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

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How Incomputable is the Separable Hahn-Banach Theorem?

Gherardi

Marcone

2008

Electronic Notes in Theoretical Computer Science

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We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibility for multi-valued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL 0 . We study analogies and differences between WKL 0 and the class of Sep-computable multi-valued functions. Extending work of Brattka, we show that a natural multi-valued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

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Separation and Weak König's Lemma

Cited by 4 publications

References 11 publications

Fundamental notions of analysis in subsystems of second-order arithmetic

Fundamental notions of analysis in subsystems of second-order arithmetic

How Incomputable Is the Separable Hahn-Banach Theorem?

How Incomputable is the Separable Hahn-Banach Theorem?

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