Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. (...) Furthermore, we give a short and elementary proof of the fact that FAN is equivalent to each positive valued function with compact domain having positive infimum. (shrink)

The paper is a contribution to intuitionistic reverse mathematics. We introduce a formal system called Basic Intuitionistic MathematicsBIM, and then search for statements that are, over BIM, equivalent to Brouwer’s Fan Theorem or to its positive denial, Kleene’s Alternative to the Fan Theorem. The Fan Theorem is true under the intended intuitionistic interpretation and Kleene’s Alternative is true in the model of BIM consisting of the Turing-computable functions. The task of finding equivalents of Kleene’s Alternative is, intuitionistically, (...) a nontrivial extension of the task of finding equivalents of the Fan Theorem, although there is a certain symmetry in the arguments that we shall try to make transparent. We introduce closed-and-separable subsets of Baire space N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{N}}$$\end{document} and of the set R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}}$$\end{document} of the real numbers. Such sets may be compact and also positively noncompact. The Fan Theorem is the statement that Cantor space C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}, or, equivalently, the unit interval [0, 1], is compact and Kleene’s Alternative is the statement that C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}, or, equivalently, [0, 1], is positively noncompact. The class of the compact closed-and-separable sets and also the class of the closed-and-separable sets that are positively noncompact are characterized in many different ways and a host of equivalents of both the Fan Theorem and Kleene’s Alternative is found. (shrink)

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

Classically, weak König's lemma and Brouwer's fan theorem for detachable bars are equivalent. We give a direct constructive proof that the former implies the latter.

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space \ to the natural numbers \ which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction allows us to formulate several continuity principles on the Baire space stated in terms of bar recursion on continuous moduli which naturally characterise some variants of bar (...) induction. These principles state that a certain kind of continuous function from \ to \ admits a modulus of continuity with a bar recursor. The results for the Baire space are recast in the setting of the Cantor space \ using the notion of fan recursor, a bar recursor for binary trees. This yields characterisations of uniformly continuous functions on the Cantor space and fan theorem in terms of fan recursors. The results for the Cantor space hold over the extensional version of intuitionistic arithmetic in all finite types ), and those for the Baire space hold over \ extended with the type for Brouwer operations. Our work places Spector’s bar recursion in a proper context of Brouwer’s intuitionistic mathematics and clarifies the connection between bar recursion and bar induction. (shrink)

We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: fan theorem for decidable fans but arbitrary bars; continuity (...) principle and the axiom of choice both for arbitrary formulae; and $\Sigma _2$ induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO ; but that limited principle of omniscience LPO makes the situation completely classical. (shrink)

A weak constructive sequential compactness property of metric spaces is introduced. It is proved that for complete, totally bounded metric spaces this property is equivalent to Brouwer's fan theorem for detachable bars. Our results form a part of constructive reverse mathematics.

In this paper, we axiomatize the negatable consequences in dependence and independence logic by extending the systems of natural deduction of the logics given in [22] and [11]. We prove a characterization theorem for negatable formulas in independence logic and negatable sentences in dependence logic, and identify an interesting class of formulas that are negatable in independence logic. Dependence and independence atoms, first-order formulas belong to this class. We also demonstrate our extended system of independence logic by giving explicit (...) derivations for Armstrong's Axioms and the Geiger-Paz-Pearl axioms of dependence and independence atoms. (shrink)

It is shown that a principle, which can be seen as a constructivised version of sequential compactness, is equivalent to a form of Brouwer’s fan theorem. The complexity of the latter depends on the geometry of the spaces involved in the former.

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Earlier work showed all of these implications to be strict. Here we reprove one of the strictness results, using very different arguments. The technique used is a mixture of realizability, forcing in the guise of Heyting-valued models, and Kripke models.

Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the (...) fan theorem, weak König's lemma, in the original classical setting of reverse mathematics started by Friedman and Simpson. (shrink)

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension (...) greater than or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes. (shrink)

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional. The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+as defined there cannot be shown to have the required properties (...) inEL+ QF-AC, the reason being that a change in the definition ofW12alone does not suffice—if one wishes to establish closure under the operations of HAωthe definitions ofW1σfor other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.For a proper understanding, we should perhaps note already here thaton the assumption of the fan theorem, ECF+as defined in [T4] and the new model of this note coincide ; but inELit is impossible to prove this. (shrink)

On the intended interpretation of intuitionistic logic, Heyting's Proof Interpretation, a proof of a proposition of the form p -> q consists in a construction method that transforms any possible proof of p into a proof of q. This involves the notion of the totality of all proofs in an essential way, and this interpretation has therefore been objected to on grounds of impredicativity (e.g. Gödel 1933). In fact this hardly ever leads to problems as in proofs of implications usually (...) nothing more is assumed about a proof of the antecedent than that it indeed is one, and this assumption does not require a further grasp of the totality of proofs. The prime example of an intuitionistic theorem that goes beyond that assumption is Brouwer's proof of the 'bar theorem': For every tree x, if x contains a decidable subset of nodes such that every path through the tree meets it (a 'bar'), then there is a well-ordered subtree of x that contains a bar for the whole of x. Instantiated with an arbitrary tree t, this proposition takes the form P(t) -> Q(t). Brouwer's proof of the bar theorem mainly consists in an analysis of the inner structure that a proof of P(t) must have, where proofs are taken to be primarily mental objects. So here Brouwer engages in phenomenological reflection by considering the acts in which we think about bars. From that analysis he obtains the information from which to construct a proof of Q(t). In this talk I will argue that Brouwer circumvents the problem of impredicativity by resorting to a transcendental argument based on phenomenological description, and defend this application by showing how common objections to transcendental arguments do not apply here. Finally, I will indulge in some historical speculation by relating the foregoing considerations to the remarkable change that Gödel's view on the Proof Interpretation underwent between his Yale Lecture (1941) and the Dialectica paper (1958). (shrink)

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.

The uniform continuity theorem states that every pointwise continuous real‐valued function on the unit interval is uniformly continuous. In constructive mathematics, is strictly stronger than the decidable fan theorem, but Loeb [17] has shown that the two principles become equivalent by encoding continuous real‐valued functions as type‐one functions. However, the precise relation between such type‐one functions and continuous real‐valued functions (usually described as type‐two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for (...) a modulus of a continuous real‐valued function on [0, 1], and show that real‐valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that is equivalent to the statement that every pointwise continuous real‐valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that is equivalent to a similar principle for real‐valued functions on the Cantor space. These results extend Berger's [2] characterisation of for integer‐valued functions on and unify some characterisations of in terms of functions having continuous moduli. (shrink)

In the United States, final amendments to the Federal Policy for the Protection of Human Subjects (“the Common Rule”) were published on January 19, 2017, and they will take effect on January 21, 2019. One of the most widely discussed provisions is that for the first time, federal regulations governing research with humans authorize the use of broad consent for future, unspecified research on individually identifiable biospecimens and associated data. Many questions have been raised about broad consent, including what effect (...) it will have on research and whether it adequately protects the interests of research participants.There are lessons to be learned for the U.S. and other countries by looking to countries that already have experience with broad consent for biobank collection and with the storage and subsequent use of the biospecimens and data. This article describes how broad consent works in five countries—Canada (in Quebec), Israel, Nigeria, Taiwan, and the United Kingdom—and with different types of biobanks: national biobanks, federated biobanks, and regional biobanks. Evaluating the provisions and challenges of the broad consent approaches in these countries can inform policies for this increasingly used approach to biobank regulation. (shrink)

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. (shrink)

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

We provide a fine‐grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over a nearly intuitionistic system containing a weak countable choice only. This implies that the general decidable fan (...) theorem is not equivalent to the binary decidable fan theorem in the absence of a strong countable choice. In addition, we introduce a principle which fills the gap between König's lemma and weak König's lemma, and show that König's lemma is equivalent to weak König's lemma plus this principle. We also solve several problems arising from the investigation. (shrink)

Brouwer's theorem of 1927 on the equivalence between virtual and inextensible order is discussed. Several commentators considered the theorem at issue as problematic in various ways. Brouwer himself, at a certain time, believed to have found a very simple counter-example to his theorem. In some later publications, however, he stated the theorem in the original form again. It is argued that the source of all criticisms is Brouwer's overly elliptical formulation of the definition of (...) inextensible order, as well as a certain ambiguity in his terminology. Once these drawbacks are removed, his proof goes through. (shrink)

This article presents a weak system of intuitionistic second-order arithmetic, WKV, a subsystem of the one in S.C. Kleene, R.E. Vesley [The Foundations of Intuitionistic Mathematics: Especially in Relation to Recursive Functions, North-Holland Publishing Company, Amsterdam, 1965]. It is then shown that some statements of real analysis, like a version of the Heine–Borel Theorem, and some statements of logic, e.g. compactness of classical proposition calculus, are equivalent to the Fan Theorem in this system.

L.E.J. Brouwer made a mistake in the formulation of his famous bar theorem, as was pointed out by S.C. Kleene. By repeating this mistake several times, Brouwer has caused confusion. We consider the assumption underlying his bar theorem, calling it Brouwer’s Thesis. This assumption is not refuted by Kleene’s example and we use it to obtain a conclusion different from Brouwer’s. Thus we come to support a view first expressed and defended by E. Martino and P. Giaretta in (...) [Martino 1981]. We also indicate that Brouwer’s Thesis has many more applications than Brouwer dreamt of. (shrink)

L.E.J. Brouwer made a mistake in the formulation of his famous bar theorem, as was pointed out by S.C. Kleene. By repeating this mistake several times, Brouwer has caused confusion. We consider the assumption underlying his bar theorem, calling it Brouwer’s Thesis. This assumption is not refuted by Kleene’s example and we use it to obtain a conclusion different from Brouwer’s. Thus we come to support a view first expressed and defended by E. Martino and P. Giaretta in (...) [Martino 1981]. We also indicate that Brouwer’s Thesis has many more applications than Brouwer dreamt of. (shrink)

In intuitionistic analysis, "Brouwer's Continuity Principle" implies, together with an "Axiom of Countable Choice", that the positively Borel sets form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower level.

I survey Brouwer’s weak counterexamples to classical theorems, with a view to discovering what useful mathematical work is done by weak counterexamples; whether they are rigorous mathematical proofs or just plausibility arguments; the role of Brouwer’s notion of the creative subject in them, and whether the creative subject is really necessary for them; what axioms for the creative subject are needed; what relation there is between these arguments and Brouwer’s theory of choice sequences. I refute one of Brouwer’s claims with (...) a weak counterexample of my own. I also examine Brouwer’s 1927 proof of the negative continuity theorem, which appears to be a weak counterexample reliant on both the creative subject and the concept of choice sequence; I argue that it provides a good justification for the weak continuity principle, but it is not a weak counterexample and it does not depend essentially on the creative subject. (shrink)

When Kleene extended his recursive realizability interpretation from intuitionistic arithmetic to analysis, he was forced to use more than recursive functions to interpret sequences and conditional constructions. In fact, he used what classically appears to be the full continuum. We describe here a generalization to higher type of Kleene's realizability, one case of which, -realizability, uses general recursive functions throughout, both to realize theorems and to interpret choice sequences. -realizability validates a version of the bar theorem and the usual (...) continuity principles, while also providing naturally, as Kleene's 1965 realizability does not, for versions of lawless sequence axioms, as well as of Church's Thesis. (shrink)

The lives of few mathematicians offer the drama that is presented by the life of L. E. J. Brouwer, correctly identified on the cover of this book as a topologist, intuitionist, and philosopher, and before we go any further, it will be worth indicating why.It is not just that Brouwer would rank high among mathematicians for his work in topology alone: he set standards for rigour and created a theory of dimension for topological spaces, and his fixed-point theorem is (...) of great importance. Nor is it just that his philosophy of intuitionism created a new and vibrant branch of logic, that is, arguably, the only viable alternative to naïve, classical logic outside the enclave of professional logicians. Rather, it is that there is a tension, popularly taken to amount to a contradiction, in the fundamental ideas behind his topology and his intuitionism. Added to that, he took a highly principled stand on mathematical issues that led him into confrontations with major figures and a certain degre .. (shrink)

Is a mathematical theorem proved because provable, or provable because proved? If Brouwer’s intuitionism is accepted, we’re committed, it seems, to the latter, which is highly problematic. Or so I will argue. This and other consequences of Brouwer’s attempt to found mathematics on the intuition of a move of time have heretofore been insufficiently appreciated. Whereas the mathematical anomalies of intuitionism have received enormous attention, too little time, I’ll try to show, has been devoted to some of the temporal (...) anomalies that Brouwer has invited us to introduce into mathematics. (shrink)

Recent data indicate that under a specific posthypnotic suggestion to circumvent reading, highly suggestible subjects successfully eliminated the Stroop interference effect. The present study examined whether an optical explanation could account for this finding. Using cyclopentolate hydrochloride eye drops to pharmacologically prevent visual accommodation in all subjects, behavioral Stroop data were collected from six highly hypnotizables and six less suggestibles using an optical setup that guaranteed either sharply focused or blurred vision. The highly suggestibles performed the Stroop task when naturally (...) vigilant, under posthypnotic suggestion not to read, and while visually blurred; the less suggestibles ran naturally vigilant, while looking away, and while visually blurred. Although visual accommodation was precluded for all subjects, posthypnotic suggestion effectively eliminated Stroop interference and was comparable to looking away in controls. These data strengthen the view that Stroop interference is neither robust nor inevitable and support the hypothesis that posthypnotic suggestion may exert a top-down influence on neural processing. (shrink)

Working within Bishop-style constructive mathematics, we examine some of the consequences of the anti-Specker property, known to be equivalent to a version of Brouwer's fan theorem. The work is a contribution to constructive reverse mathematics.

Recent data indicate that under a specific posthypnotic suggestion to circumvent reading, highly suggestible subjects successfully eliminated the Stroop interference effect. The present study examined whether an optical explanation could account for this finding. Using cyclopentolate hydrochloride eye drops to pharmacologically prevent visual accommodation in all subjects, behavioral Stroop data were collected from six highly hypnotizables and six less suggestibles using an optical setup that guaranteed either sharply focused or blurred vision. The highly suggestibles performed the Stroop task when naturally (...) vigilant, under posthypnotic suggestion not to read, and while visually blurred; the less suggestibles ran naturally vigilant, while looking away, and while visually blurred. Although visual accommodation was precluded for all subjects, posthypnotic suggestion effectively eliminated Stroop interference and was comparable to looking away in controls. These data strengthen the view that Stroop interference is neither robust nor inevitable and support the hypothesis that posthypnotic suggestion may exert a top-down influence on neural processing. (shrink)

Governments and other food system actors from the private sector, civil society, research and education institutions are being called upon to work together to enhance the sustainability, resilience and inclusiveness of food systems. The analysis presented in this study provides an insight into the process and direction of food system transformation, and the key capabilities required. It portrays the interplay of different internal and external dynamics combined with the capacity of food system actors to connect, forge alliances and commit to (...) specific actions that has enabled countries to move towards a more sustainable food system. (shrink)