[HM] Understanding the infinite

Carlos Cesar Araujo (carlos.cesar@taskmail.com.br)
14 May 99 22:38:51 -0400 (EDT)

Dear friends,

I don't know Lavine's book but Kutler's comments on it may create the
impression that Lavine invaded a virgin territory. The idea that
Cantor's concept of set is paradox-free is not new and, as far as I
can see, go back at least to Goedel in his widely quoted "What is
Cantor's Continuum Problem?". For this and much more, one can consult
the anthology "Philosophy of Mathematics" edited by Paul Benacerraf
and Hilary Putnam (Part IV). Another eminent figure in this direction
(despite his acrimonious style) is Kreisel. For instance:

"Formalization, by Frege, perhaps helped one derive paradoxes (not to
correct them). Even this was not needed since, back in 1885, in a
review [Review of Frege's Grundalagen der Arithmetik, Deutsche
Literaturzeitung 6 (1885), 728-729], Cantor explained very clearly why
Frege's formulation could not be expected to hold for the concept of
set described and used by Cantor."
[Georg Kreisel, "What have we learnt from Hilbert's second problem" in
Mathematical Developments Arising from Hilbert Problems, Proceedings
of Symposia in Pure Mathematics, 1976, V. XXVIII, Part I, p. 95.]

"It is perhaps natural that the antinomies are often used – in effect
if not by intention – to introduce a bit of drama into foundations, a
subject by and large devoted to the undramatic business of "analysing"
what (we believe) we know anyway. But it is simply historically false
to think that the antinomies provide evidence for any failure of the
"logical intuitions" of Cantor, let alone of his contemporaries ..."
[Kreisel in his review of L. E. Brouwer collected works, Bull. A. M.
S., 1977.]

Carlos Cesar Araujo