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Intuitionism As Generalization
 Philosophia Math
, 1990
"... ms that hold only in computational models. For example, Brouwer proved that every totally defined function on the real line is continuous. A theory where this is provable cannot refer to the classical universe, so we have to consider whether we like its models better than the classical model. But an ..."
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ms that hold only in computational models. For example, Brouwer proved that every totally defined function on the real line is continuous. A theory where this is provable cannot refer to the classical universe, so we have to consider whether we like its models better than the classical model. But any theorem in constructive mathematics is a theorem in classical mathematics, so in this case it is not a question of choosing between models, but of deciding whether it is worthwhile to talk about computational models in addition to the classical model. By comparing mathematical realism with intuitionism from an informal axiomatic point of view, we can steer clear of most of the metaphysical problems involved in analyzing these notions from the ground up, and concentrate on what may be termed the purely mathematical aspects. In particular we won't have to consider whether intuitionists hold that a theorem "isn't true until it's known to be true", as Blais 1 <F
An Application of Constructive Completeness.
 In Proceedings of the Workshop TYPES '95
, 1995
"... this paper, we explore one possible effective version of this theorem, that uses topological models in a pointfree setting, following Sambin [11]. The truthvalues, instead of being simply booleans, can be arbitrary open of a given topological space. There are two advantages with considering this m ..."
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this paper, we explore one possible effective version of this theorem, that uses topological models in a pointfree setting, following Sambin [11]. The truthvalues, instead of being simply booleans, can be arbitrary open of a given topological space. There are two advantages with considering this more abstract notion of model. The first is that, by using formal topology, we get a remarkably simple completeness proof; it seems indeed simpler than the usual classical completeness proof. The second is that this completeness proof is now constructive and elementary. In particular, it does not use any impredicativity and can be formalized in intuitionistic type theory; this is of importance for us, since we want to develop model theory in a computer system for type theory. Formal topology has been developed in the type theory implementation ALF [1] by Cederquist [2] and the completeness proof we use has been checked in ALF by Persson [9]. In view of the extreme simplicity of this proof, it might be feared that it has no interesting applications. We show that this is not the case by analysing a conservativity theorem due to Dragalin [4] concerning a nonstandard extension of Heyting arithmetic. We can transpose directly the usual model theoretic conservativity argument, that we sketched above, in this framework. It seems likely that a direct syntactical proof of this result would have to be more involved. The first part of this paper presents a definition of topological models, Sambin's completeness proof, and an alternative completeness proof; we also discuss how Beth models relate to our approach. The second part shows how to use this in order to give a proof of Dragalin's conservativity result; our proof is different from his and, we believe, simpler. In [8] a stronger ...
Arguments for the Continuity Principle
, 2000
"... Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences ..."
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Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences . . . . . . . . . 15 3.2 Kripke's Schema and full PEM . . . . . . . . . . . . . . . . . 15 3.3 The KLST theorem . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion 19 1 The continuity principle There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the rst time in print in [Brouwer 1918]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fa
A Negationless Interpretation of Intuitionistic Axiomatic Theories: Higherorder Arithmetic
, 1998
"... This work is a sequel to our [14]. It is shown how Theorem 6 of [14], dealing with the translatability of HA (Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types. Introduction The main purpose of the present paper is to exte ..."
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This work is a sequel to our [14]. It is shown how Theorem 6 of [14], dealing with the translatability of HA (Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types. Introduction The main purpose of the present paper is to extend Theorem 6 of [14], asserting the translatability of HA into negationless arithmetic NA, to the case of intuitionistic higherorder arithmetic HAH and a formalization (denoted by NAH) of negationless higherorder arithmetic to be described in Section 1. In Section 2 we consider the connection of NAH with a version of intuitionistic higherorder arithmetic, the system HAH. An extension of the translatability result to these systems turns out easy to perform, since the only essential distinction between HAH and NAH is that the former is based on HA (instead of NA). In Section 3 we study the relationship between HAH and HAH and then prove the main result on the translatability of HAH into N...
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providi ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...
The History of Mathematical Logic (vastly abbreviated and horribly simplified)
, 1997
"... F11.95> manipulating such forms in order to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate "building blocks". In ..."
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F11.95> manipulating such forms in order to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate "building blocks". In particular, one has to ask the questions about the meaning of such building blocks, of various terms and categories of terms and, furthermore, of their combinations. 3. Finally, there is the question of how to represent these patterns. Although apparently of secondary importance, it is the answer to this question which can be, to a high degree, considered the beginning of modern mathematical logic. The first three sections sketch the development along the respective lines until Renessance. In section 4, we indicate the development in modern era, with particular emphasis on the last two centuries. Section 5 indicates some basic aspect
M4M 2007 Continuous Functions on Final Coalgebras
"... It can be traced back to Brouwer that continuous functions of type StrA → B, where StrA is the type of infinite streams over elements of A, can be represented by well founded, Abranching trees whose leafs are elements of B. This paper generalises the above correspondence to functions defined on fin ..."
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It can be traced back to Brouwer that continuous functions of type StrA → B, where StrA is the type of infinite streams over elements of A, can be represented by well founded, Abranching trees whose leafs are elements of B. This paper generalises the above correspondence to functions defined on final coalgebras for powerseries functors on the category of sets and functions. While our main technical contribution is the characterisation of all continuous functions, defined on a final coalgebra and taking values in a discrete space by means of inductive types, a methodological point is that these inductive types are most conveniently formulated in a framework of dependent type theory.
ITERATED DEFINABILITY, LAWLESS SEQUENCES AND BROUWER’S CONTINUUM
"... Abstract. The research on which this article is based was motivated by the wish to find a model of Kreisel’s lawless sequence axioms in which the lawlike and lawless sequences form disjoint, inhabited, welldefined classes within Brouwer’s continuum. The original results, reported as they developed ..."
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Abstract. The research on which this article is based was motivated by the wish to find a model of Kreisel’s lawless sequence axioms in which the lawlike and lawless sequences form disjoint, inhabited, welldefined classes within Brouwer’s continuum. The original results, reported as they developed in four papers over a period of ten years from 1986 to 1996, have so far lacked a readerfriendly presentation. Since the question of absolute definability is related to the subject of these Bristol Workshops, I offer here a straightforward exposition of the final model and formal system with axioms for numbers, lawlike sequences, and arbitrary choice sequences. A choice sequence is defined to be lawless if it satisfies an extensional (un)predictability condition from which extensional versions of Kreisel’s axioms of open data and strong continuous choice follow. The law of excluded middle can be assumed for properties of lawlike and independent lawless sequences only, while Brouwer’s continuity principle applies to properties of all choice sequences. Iterating definability, quantifying over numbers and over lawlike and independent lawless sequences, yields a classical model of the lawlike sequences with a natural wellordering. Under the (classically consistent and intuitionistically plausible) assumption that the closure ordinal of the iteration is countable, a realizability interpretation establishes the consistency of a common extension FIRM(≺) of classical analysis R and Kleene’s intuitionistic analysis FIM. Lawlike sequences behave classically, while the lawless sequences form a disjoint, Baire comeager collection of choice sequences, of classical measure zero. Thus Brouwer’s continuum can be understood as a relatively chaotic expansion of a completely determined, wellordered classical continuum. 1.