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Spreads or choice sequences? (English) Zbl 0769.03004

Author’s summary: “Intuitionistically, a set has to be given by a finite construction or by a construction-project generating the elements of the set in the course of time. Quantification is only meaningful if the range of each quantifier is a well-circumscribed set. Thinking upon the meaning of quantification, one is led to insights — in particular, the so- called continuity principles — which are surprising from a classical point of view. We believe that such considerations lie at the basis of Brouwer’s reconstruction of mathematics. The predicate ‘\(\alpha\) is lawless’ is not acceptable, the lawless sequences do not form a well- circumscribed intuitionistic set, and quantification over lawless sequences does not make sense.”
The terms of this paper are wholly outside a contemporary mathematical, scientific or, for that matter, professional philosophical education; for example, on p. 211, such jargon as ‘philosophically sound principles’. On p. 212 there is reference to a rule \(\alpha: \omega\mapsto\{0,1\}\) s.t. \(\alpha(n)=1\) if the author has a proof or refutation of Fermat’s (so- called last) theorem \(F\) at time \(n\), and \(\alpha(n)=0\) otherwise. Then it is asked whether \(\alpha\) is lawlike or lawless without a moment’s thought that something, here \(\alpha\), may be good for a joke, but not rewarding as an object of — here, logical — theory (and, of course, vice versa). Depending on their temperament readers may nevertheless find the paper stimulating, for example, the rule \(\alpha\) above: What, if anything, would the author do if, at this moment, he had Wiles’ proof of \(F\) (in his hands, which is perhaps not the same thing as having it in one’s head)?
Reviewer: G.Kreisel (Oxford)

MSC:

03-03 History of mathematical logic and foundations
03A05 Philosophical and critical aspects of logic and foundations
03F50 Metamathematics of constructive systems
01A60 History of mathematics in the 20th century
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References:

[1] Brouwer, L.E.J. (1918) ’Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil: Allgemeine Mengenlehre1, in 1975, 150-190. · JFM 46.0310.06
[2] Brouwer, L.E.J., 1925 ’Zur Begründung der intuitionistischen Mathematik I’ in 1975. 301-314.
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[13] DOI: 10.1007/BF00247189 · Zbl 0518.03023 · doi:10.1007/BF00247189
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