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Modal logics for topological spaces
 Thesis (Ph.D.)–City University of New York
, 1993
"... In this thesis we shall present two logical systems, MP and MP ∗ , for the purpose of reasoning about knowledge and effort. These logical systems will be ..."
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In this thesis we shall present two logical systems, MP and MP ∗ , for the purpose of reasoning about knowledge and effort. These logical systems will be
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
Hermann Weyl’s Intuitionistic Mathematics. Dirk
"... It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to t ..."
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It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, how come Weyl was so wellinformed about Brouwer’s new intuitionism, in what respect did Weyl’s intuitionism differ from Brouwer’s intuitionism, what did Brouwer think of Weyl’s views,........? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl’s career. Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics ” 1. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [Weyl 1918] ; this contained in addition to a technical logical – mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary — and hence also impredicative — Dedekind cuts. This book did not cause much of a stir in mathematics, that is to say, it was ritually quoted in the literature but, probably, little understood. It had to wait for a proper appreciation until the phenomenon of impredicativity was better understood 2. The paper “On the new foundational crisis in mathematics ” had a totally different effect, it was the proverbial stone thrown into the quiet pond of mathematics. Weyl characterised it in retrospect with the somewhat apologetic words: Only with some hesitation I acknowledge these lectures, which reflect in their style, which was here and there really bombastic, the mood of excited times — the times immediately following the First World War. 3 Indeed, Weyl’s “New crisis ” reads as a manifesto to the mathematical community, it uses an evocative language with a good many explicit references to the political
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 ...