The Logical Strength of the Uniform Continuity Theorem

Berger, Josef

doi:10.1007/11780342_4

Cited by 27 publications

(34 citation statements)

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“…Now LPO implies LLPO, which is provably false in either of the non-classical standard models of Bishop's theory, in recursive and intuitionistic mathematics [9, Chapters 3 and 5]. 1) The non-constructive character of MIN notwithstanding, [9, Chapter 2, Theorem 4.5] gives a constructive proof that inf x∈S F (x) can be computed whenever F is uniformly continuous, and S is totally bounded. So the question remains under which circumstances our problem allows for a constructive solution.…”

Section: Wklmentioning

confidence: 99%

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Unique solutions

Schuster

2006

Mathematical Logic Qtrly

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It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform "at most one" condition follows from its non-uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges.

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Section: Wklmentioning

confidence: 99%

“…Given the antecedent of this implication, a natural first attempt to obtain its consequent is to choose -of course by countable choice -a sequence (x n ) in S with (1) ∀n(F (x n ) < 1/n).…”

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confidence: 99%

Unique solutions

Schuster

2006

Mathematical Logic Qtrly

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“…It is known that, within BISH, the anti-Specker property for the interval [0, 1] is equivalent to Brouwer's fan theorem FT c for 'c-bars'. 3) Although FT c is not adopted as a principle of BISH, since it holds in the intuitionistic model of BISH it (and therefore the anti-Specker property for compact metric spaces) can be regarded as more-or-less constructive, at least provided you are prepared to dispense with a recursive interpretation of your constructive mathematics. 1 , are of particular interest, since one can show that the proof-theoretic strength of the uniform continuity theorem for continuous functions on compact metric spaces lies between them; but whether that theorem is actually equivalent to either FTc or FT Π 0 1 remains an open question.…”

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confidence: 99%

“…As it is more convenient in constructive analysis to work with purely analytical principles, rather than logical ones, the anti-Specker property for the interval [0, 1] is an important principle to investigate. For more on these matters, see [3,4].…”

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confidence: 99%

The anti‐Specker property, positivity, and total boundedness

Bridges

Diener

2010

Mathematical Logic Qtrly

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Working within Bishop-style constructive mathematics, we examine some of the consequences of the antiSpecker property, known to be equivalent to a version of Brouwer's fan theorem. The work is a contribution to constructive reverse mathematics.In this paper we continue the discussion [4,8], within Bishop's constructive mathematics (BISH) 1) , of generalisations of the anti-Specker property for [0, 1] (relative to R) -that is, If (z n ) n 1 is a sequence in R that is eventually bounded away from each point of [0, 1] , then (z n ) n 1 is eventually bounded away from the entire interval [0, 1]. 2) Specifically, we are interested in -the connection between generalised anti-Specker properties and the positivity of the infimum of a positivevalued function on a compact -that is, complete, totally bounded -metric space, and -the question whether every space with a generalised anti-Specker property is totally bounded. This work lies within the programme of constructive reverse mathematics, in which, on the one hand, we examine the constructive equivalence of classical statements (see, for example, [11,17]), and on the other, we seek, for example, to classify theorems according to the version of the fan theorem to which they are equivalent [14,15,16,26]. It is known that, within BISH, the anti-Specker property for the interval [0, 1] is equivalent to Brouwer's fan theorem FT c for 'c-bars'. 3) Although FT c is not adopted as a principle of BISH, since it holds in the intuitionistic model of BISH it (and therefore the anti-Specker property for compact metric spaces) can be regarded as more-or-less constructive, at least provided you are prepared to dispense with a recursive interpretation of your constructive mathematics. 1) This is simply mathematics carried out with intuitionistic logic and within some suitable set-theoretic framework such as that found in [1]. We shall also allow the use of countable and dependent choice.2) The anti-Specker property is in direct opposition to Specker's theorem, a fundamental result in recursive analysis [25]. For more about the anti-Specker property, see [7,8].3) There are currently four versions of Brouwer's fan theorem that have been investigated in the scope of constructive reverse mathematics. All of them enable one to conclude that a given bar is uniform; the difference between them lies in the required complexity of the bar. This ranges from the very strongest requirement -decidable -to no restriction on the bar at all. Between these two extremes lie fan theorems for bars that are c-sets and Π 0 1 -sets, respectively. These two fan theorems, FTc and FT Π 0 1 , are of particular interest, since one can show that the proof-theoretic strength of the uniform continuity theorem for continuous functions on compact metric spaces lies between them; but whether that theorem is actually equivalent to either FTc or FT Π 0 1 remains an open question. Many other analytical theorems have, however, been shown to be equivalent to versions of the fan theorem. As it is more convenient in constructiv...

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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%

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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The Logical Strength of the Uniform Continuity Theorem

Cited by 27 publications

References 4 publications

Unique solutions

Unique solutions

The anti‐Specker property, positivity, and total boundedness

10.4115/jla.v3i0.114

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