Brouwer's fan theorem and unique existence in constructive analysis

Berger, Josef; Ishihara, Hajime

doi:10.1002/malq.200410038

Cited by 50 publications

(44 citation statements)

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“…in [2] and [9] evidently does not need countable choice, so can be formalized in M. The reverse entailment also holds constructively, as we now verify. in [2] and [9] evidently does not need countable choice, so can be formalized in M. The reverse entailment also holds constructively, as we now verify.…”

Section: Verification That Ft D Constructively Entails Wkl!supporting

confidence: 69%

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Another Unique Weak König’s Lemma WKL!!

Moschovakis¹

2012

Logic, Construction, Computation

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For Helmut Schwichtenberg, with respect and appreciation for his encouragement and friendship.In [2] J. Berger and Ishihara proved, via a circle of informal implications involving countable choice, that Brouwer's Fan Theorem for detachable bars on the binary fan is equivalent in Bishop's sense to various principles including a version WKL! of Weak König's Lemma with a strong effective uniqueness hypothesis. Schwichtenberg [9] proved the equivalence directly and formalized his proof in Minlog. We verify that his result does not require countable choice, and derive a separation principle SP from the Fan Theorem, in a minimal intuitionistic system M of analysis with function comprehension.In contrast, WKL!! comes from Weak König's lemma WKL by adding the hypothesis that any two infinite paths must agree. WKL!! is interderivable over M with the conjunction of a consequence of Markov's Principle and the double negation of WKL. This decomposition is in the spirit of Ishihara's [4] and J. Berger's [1]. Kleene's function realizability and the author's modified realizability establish that WKL!! is strictly weaker than WKL and strictly stronger than WKL!.

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Section: Verification That Ft D Constructively Entails Wkl!supporting

confidence: 69%

“…2 J. Berger's and Ishihara's round-robin proof in [2] of the equivalence of WKL! 2 J. Berger's and Ishihara's round-robin proof in [2] of the equivalence of WKL!…”

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confidence: 99%

Another Unique Weak König’s Lemma WKL!!

Moschovakis¹

2012

Logic, Construction, Computation

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“…It is tempting to believe that the 'at most one solution implies there is a solution' version of Peano's existence theorem is equivalent, over BISH, to FT D . However, a proof of this equivalence is elusive, perhaps because the Peano existence problem deals with a very specific compact space, namely S , whereas equivalents of FT D normally deal with statements that apply to all compact metric spaces (see [9]). …”

Section: Discussionmentioning

confidence: 98%

Constructive Solutions of Ordinary Differential Equations

Bridges¹

2012

Logic, Construction, Computation

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For Helmut Schwichtenberg, with thanks for his friendship and for his hospitality in Munich on numerous occasions over the past 15 years.A Bishop-style constructive analysis is given for the Peano existence theorem for solutions of the differential equation y = f (x, y) with specified initial condition. In particular, it is shown that the existence of a solution in the general case implies the omniscience principle LLPO, but that introducing an a priori hypothesis of uniqueness of a solution can enable the construction of the solution.

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“…The final constructive problem arises in the application of Baire's category theorem: although the intersection of a sequence of dense open sets in a complete metric space is, as classically, dense, the classical contrapositive version of Baire's theorem -the version applied above -does not hold in BISH without some quite strong extra hypotheses; see [8,Chapter 2] and also the paper [11]. 3) We conclude that the constructive problems in proving Theorem 1 are nontrivial. As we now show, the trick is to set things up for an application of Baire's theorem in the "dense open sets" version.…”

Section: Llpomentioning

confidence: 99%

Continuous homomorphisms of R onto a compact group

Bridges¹,

Hendtlass

2010

Math. Log. Quart.

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It is shown within Bishop's constructive mathematics that, under one extra, classically automatic, hypothesis, a continuous homomorphism from R onto a compact metric abelian group is periodic, but that the existence of the minimum value of the period is not derivable.In this paper, which is written within the framework of Bishop's constructive mathematics (BISH), 1) we consider a partial abstraction of the well-known classical result COPEvery compact orbit of a dynamical system is periodic [12].The abstraction is the following: Theorem 1 Let θ be a continuous homomorphism of R onto a compact (metric) abelian group G, such thatis open. Then θ is periodic, in the sense that there exists τ > 0 such that θ(τ) = 0.Here we use standard additive notation for abelian group operations. By a metric abelian group we mean an abelian group G equipped with a metric ρ, such that the mapping (x, y)y − x is pointwise continuous at (0, 0) ∈ G × G, and uniformly continuous on compact subsets of G × G. The mappings x −x and (x, y)x + y are then pointwise continuous throughout their domains, and uniformly continuous on compact subsets of their domains. Moreover, for each positive integer n, the mapping x nx is continuous at 0. Note that if G is locally compact -that is, every bounded subset of G is contained in a compact subset of G -then the pointwise continuity of the mapping (x, y)y − x is a consequence of its uniform continuity on compact sets. We say that a homomorphism θ of the abelian group R into a metric abelian group G is continuous if it is uniformly continuous on each compact (or, equivalently, on each bounded) subset of R.Although we know of no reference for Theorem 1 in the literature, we believe that the following argument, based on the standard classical one used to prove COP, would be the natural one for the classical mathematician to use. Under the hypotheses of Theorem 1, first observe that for each positive integer n, since θ is continuous, the closed subsetof G is compact. Since C 1 ⊃ C 2 ⊃ · · · , the set n 1 C n is nonempty; from which it is not hard to deduce that each C n is dense in G. On the other hand, by the continuity of θ, the sets θ[−n, n] are compact and hence closed. Since G is complete and equals the union of the sets θ[−n, n] (n 1), it follows from the Baire category theorem that the interior of θ[−N, N] is nonempty for some N. We can therefore find t 1 , t 2 such that |t 1 | N, t 2 N + 1, and θ(t 1 ) = θ(t 2 ). Setting τ = t 2 − t 1 , with very little more work we see that θ(τ) = 0.

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Brouwer's fan theorem and unique existence in constructive analysis

Cited by 50 publications

References 4 publications

Another Unique Weak König’s Lemma WKL!!

Another Unique Weak König’s Lemma WKL!!

Constructive Solutions of Ordinary Differential Equations

Continuous homomorphisms of R onto a compact group

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