A fan-theoretic equivalent of the antithesis of Specker's theorem

Berger, Josef; Bridges, Douglas S.

doi:10.1016/s0019-3577(07)00012-2

Cited by 23 publications

(29 citation statements)

References 4 publications

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“…Various antitheses of Specker's theorem have been studied as constructive substitutes for sequential compactness; see [2,7,8,12,14,15]. One of the weakest of those notions is the limited anti-Specker property (relative to one-point extensions), AS X L : If X ∪ {ω} is a one-point extension of X, and (x n ) n 1 is a sequence in X ∪ {ω} that is eventually bounded away from each point of X, then there exists n such that x n = ω.…”

Section: Anti-specker Properties and Z-stabilitymentioning

confidence: 99%

Z-stability in Constructive Analysis

Bridges¹,

Dent²,

McKubre-Jordens

2017

Logical Methods in Computer Science

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Abstract. We introduce Z-stability, a notion capturing the intuition that if a function f maps a metric space into a normed space and if f(x) is small, then x is close to a zero of f. Working in Bishop's constructive setting, we first study pointwise versions of Z-stability and the related notion of good behaviour for functions. We then present a recursive counterexample to the classical argument for passing from pointwise Z-stability to a uniform version on compact metric spaces. In order to effect this passage constructively, we bring into play the positivity principle, equivalent to Brouwer's fan theorem for detachable bars, and the limited anti-Specker property, an intuitionistic counterpart to sequential compactness. The final section deals with connections between the limited anti-Specker property, positivity properties, and (potentially) Brouwer's fan theorem for detachable bars.

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Section: Anti-specker Properties and Z-stabilitymentioning

confidence: 99%

Z-stability in Constructive Analysis

Bridges¹,

Dent²,

McKubre-Jordens

2017

Logical Methods in Computer Science

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“…(For background, see [2,3,4].) Douglas Bridges showed such closure under BD-N [8], speculating that they were equivalent.…”

Section: Introductionmentioning

confidence: 99%

Principles Weaker than BD-N

Lubarsky¹,

Diener²

2013

J. symb. log.

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“…In [6], this principle has been shown to be equivalent to a version of Brouwer's Fan theorem, which itself is weaker than UCT [5], but stronger than the Fan theorem for decidable bars. 3 We will show that BUCT [0,1] is enough to show that AS [0,1] holds.…”

Section: H Diener and I Loebmentioning

confidence: 99%

“…Assume there is n M such that x kn ∈ [0, 1]. Equation (6) shows that there is a point y ∈ [0, 1] such that…”

Section: Journal Ofmentioning

confidence: 99%

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10.4115/jla.v3i0.114

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Abstract:We study Picard's Theorem and Peano's Theorem from a constructive reverse perspective. This means that we have to change our focus from global properties to local properties. We also extend the theory of pointwise continuously differentiable functions to include Rolle's Theorem, the Mean Value Theorem, and the full Fundamental Theorem of Calculus.2000 Mathematics Subject Classification 03F60, 26E40 (primary); 03F55 (secondary)

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A fan-theoretic equivalent of the antithesis of Specker's theorem

Cited by 23 publications

References 4 publications

Z-stability in Constructive Analysis

Z-stability in Constructive Analysis

Principles Weaker than BD-N

10.4115/jla.v3i0.114

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