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Herman Jurjus wrote:
> abo wrote:
> > abo wrote:
> >> Herman Jurjus wrote:
> >>> In a book about foundations of mathematics dating from 1978,
> >>> D. van Dalen wrote:
> >>> "Around 1947 Borel suggested that the set of natural numbers is finite."
> >>> and:
> >>> "Van Dantzig asked in 1956 the question "is 10^10^10 a finite number?""
> >>>
> >>> Does anyone know what van Dalen could have meant, in these two cases?
> >>> Any references?
> >> Sazonov (http://www.csc.liv.ac.uk/~sazonov/papers/lcc.ps) quotes Borel
> >> as saying in 1947, "The very large finite offers the same difficulties
> >> as the infinite." I imagine this is from Borel's book (1946) Les
> >> Paradoxes de l'Infini, but can't be sure, since I can't find a
> >> bibliographic entry for Borel in Sazonov's paper, and don't own Borel's
> >> book.
> >
I have had a look at "Les Paradoxes de L'infini" and, while Borel does
begin with a review of the numbering system with a quasi-ultrafinistic
bent, he quickly retreats into an acceptance of infinity. Two quotes
from the intro pretty much sum it up: "The numbers used in practice by
Man are not bigger than 10 or 12 figures in decimal notation." But:
"The sequence of natural numbers is unlimited...One can always add one
to any natural number already defined, or double the number, or
multiply it by 10..." Although I haven't read all of the book, I don't
think Sazanov's quote is to be found in this book.
> > After doing some Googling there is another possibility, which is an
> > essay called "Definitions in Mathematics," which was published in 1948
> > in a book called "Great Currents of Mathematical Thought: Mathematics:
> > Concepts and Development", edited by F Le Lionnais. Here's the second
> > page of the essay in Google books: tinyurl.com/w87e3">http://tinyurl.com/w87e3
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