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Constructive Topology and Combinatorics
 In proceeding of the conference Constructivity in Computer Science, San Antonio, LNCS 613
, 1991
"... We present a method to extract constructive proofs from classical arguments proved by topogical means. Typically, this method will apply to the nonconstructive use of compactness in combinatorics, often in the form of the use of König's lemma (which says that a finitely branching tree that is infini ..."
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We present a method to extract constructive proofs from classical arguments proved by topogical means. Typically, this method will apply to the nonconstructive use of compactness in combinatorics, often in the form of the use of König's lemma (which says that a finitely branching tree that is infinite has an infinite branch.) The method consists roughly of working with the corresponding pointfree version of the topological argument, which can be proven constructively using only as primitive the notion of inductive definition. We illustrate here this method on the classical "minimal bad sequence" argument used by NashWilliams in his proof of Kruskal's theorem. The proofs we get by this method are wellsuited for mechanisation in interactive proof systems that allow the user to introduce inductively defined notions, such as NuPrl, or MartinLof set theory.
The Borel hierarchy and the projective hierarchy in intuitionistic mathematics
 University of Nijmegen Department of Mathematics
, 2001
"... this paper. Once we agree to accept and use it we enter a new world and discover many facts for which there does not exist a classical counterpart. The principle entails for instance that the union of the two closed sets [0, 1] and [1, 2] is not a countable intersection of open subsets of R. One ma ..."
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Cited by 4 (2 self)
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this paper. Once we agree to accept and use it we enter a new world and discover many facts for which there does not exist a classical counterpart. The principle entails for instance that the union of the two closed sets [0, 1] and [1, 2] is not a countable intersection of open subsets of R. One may also infer that there are unions of three closed sets di#erent from every union of two closed sets. These observations are the tip of an iceberg. The intuitionistic Borel Hierarchy shows o# an exquisite fine structure
Individual Choice Sequences in the Work of L.E.J. Brouwer
, 2002
"... Choice sequences are sequences not completely determined by a law. We state that the introduction of particular choice sequences by Brouwer in the late twenties was not recognised as such. We claim that their later use in the method of the creative subject was not traced back to this original use of ..."
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Choice sequences are sequences not completely determined by a law. We state that the introduction of particular choice sequences by Brouwer in the late twenties was not recognised as such. We claim that their later use in the method of the creative subject was not traced back to this original use of them and has been misinterpreted. We show where these particular choice sequences appear in the work of Brouwer and we show how they should be handled.
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. ..."
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This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question
A NOVEL PROOF OF THE HEINEBOREL THEOREM
, 808
"... Abstract. Every beginning real analysis student learns the classic HeineBorel theorem, that the interval [0, 1] is compact. In this article, we present a proof of this result that doesn’t involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. ..."
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Abstract. Every beginning real analysis student learns the classic HeineBorel theorem, that the interval [0, 1] is compact. In this article, we present a proof of this result that doesn’t involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The HeineBorel theorem is an immediate corollary. 1. The HeineBorel theorem Think back to your first real analysis class. In the beginning, most of the definitions were fairly straightforward. Open and closed sets made sense, because of the common usage of open and closed intervals in previous math classes. It was a bit odd that open sets could also be closed, or that sets could be neither open nor closed. But this was “higher math,” so you could let that one slide. Then came the definition of “compact. ” Completely out of nowhere. Why anyone would ever find themselves with an open cover, let alone try to extract a finite subcover, was beyond your wildest dreams. As you sat there in class trying to figure out what this really meant, the professor wrote the following two sentences on the board, with “HeineBorel ” preceding one of them. 1 • The interval [0, 1] is compact. • A subset of R n is compact iff it is closed and bounded. You might remember what came next. From an arbitrary infinite sequence contained in [0, 1], a divideandconquer technique to construct a particular sequence of nested intervals, from this a subsequence of real numbers, and then a summon to the completeness of the reals, one of those blatantly obvious analysis facts that you had no idea how to prove (and likely still don’t know). It felt a little unsatisfying, and almost seemed like cheating. About this time, it dawned on you that your roommate was right: mathematicians make a living saying the simplest things in the most difficult roundabout way. By now, you understand in ways you never could have imagined back then, how wise your old roommate was. But you also remember what attracted you to mathematics in the first place, those mysterious qualities that, like the silver bell in the Polar Express, could
A short note on Spector’s proof of consistency of analysis
, 2012
"... In 1962, Clifford Spector gave a consistency proof of analysis using socalled bar recursors. His paper extends an interpretation of arithmetic given by Kurt Gödel in 1958. Spector’s proof relies crucially on the interpretation of the socalled (numerical) double negation shift principle. The argume ..."
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In 1962, Clifford Spector gave a consistency proof of analysis using socalled bar recursors. His paper extends an interpretation of arithmetic given by Kurt Gödel in 1958. Spector’s proof relies crucially on the interpretation of the socalled (numerical) double negation shift principle. The argument for the interpretation is ad hoc. On the other hand, William Howard gave in 1968 a very natural interpretation of bar induction by bar recursion. We show directly that, within the framework of Gödel’s interpretation, (numerical) double negation shift is a consequence of bar induction. The 1958 paper [4] of Kurt Gödel presented an interpretation (now known as the dialectica interpretation) of Heyting arithmetic HA into a quantifierfree calculus T of finitetype functionals. The terms of T denote certain computable functionals of finite type (a primitive notion in Gödel’s paper, as it were): the socalled primitive recursive functionals in the sense of Gödel. These terms can be rigorously defined and they include as primitives the combinators (a burocracy of terms for dealing with the “logical ” part of the calculus) and the arithmetical constants: 0, the successor constant and, importantly, the recursors. 1 The dialectica interpretation assigns to each formula A of the language of firstorder arithmetic a (quantifierfree) formula AD(x, y) of the language of T, and Gödel showed that if HA ⊢ A, then there is a term t (in which y does not occur free) of the language of T such that T ⊢ AD(t, y). 2 The combinators play a central role in showing the preservation of the interpretation under (intuitionistic) logic and, unsurprisingly, the recursors play an essential role in interpreting the induction axioms. It is convenient to extend the dialectica interpretation to Heyting arithmetic in all finite types HA ω. 3 Within the language of this theory, one can formulate the characteristic principles of the interpretation: 1 The reader can consult [11], [1] or [7] for a precise description of the calculus T and of its terms in particular. These are good references for details concerning the dialectica interpretation. 2 We are taking some liberties here (and will take in the sequel). Rigorously, either one should speak of tuple of variables x: = x1,..., xn and y: = y1,..., ym or allow convenient product types.
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric