8 research outputs found
Nikolai Luzin and the problem of existence in mathematics
This paper presents excerpts of the discussion about the problem of existence for functions in each one of Baire’ s classes. First, it sets up the historical problem, introducing four existential categories, and then it shows Luzin’ s position and reframes it in terms of Cavaillès and Gardies’ theory of thematization.
MSC2010: 00A30, 26A21, 03E15
Set theory and the analyst
This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure - category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: "what every young analyst should know"
The Creating Subject, the Brouwer-Kripke Schema, and infinite proofs
Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's 'Creating Subject arguments'; other applications have been found since. I first look in detail at a representative choice of Brouwer's arguments. Then I discuss the original use of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer's own writings. Finally, I defend the Schema and the Theory against a number of objections that have been made