On Sunday, October 28, 2012 1:23:17 PM UTC+2, JRStern wrote: > On Sun, 28 Oct 2012 07:31:24 +0000 (UTC), "Peter Webb" > > <webbfamilyDIEspamDie@optusnet.com.au> wrote: > > > > >> I suppose at worst it could be made an optional axiom, or some > > >> statement of principle could be added as an additional axiom to allow > > >> it to be true. > > >> > > >> J. > > > > > >Its a theorem in ZF. It doesn't need an axiom. > > > > It seems a little involved to be the kind of thing we like to call a > > theorem, and I believe it was Poincare who said it (or was he just > > broad-brushing all of Cantor?) only works by assuming its conclusion. > > What would it take to make Poincare happy with this one particular > > theorem?
*** It isn't involved at all. In fact, it is a rather easily-proved theorem in ZF.
Poincaré said lots of things, many of them pretty intelligent and profound, and I think it would have taken way more knowledge of his part in the foundations and in logic to realize his first (not later!) criticism of Cantor wasn't well based. Yet this can hardly be surprising as there were many very good mathematicians that didn't understand correctly Cantor at the beginning, though this number diminished a lot with time and, by the time of Cantor's death, there already was a rather huge wave among mathematicians not only of agreement but also of admiration for Cantor's work.
Of course, ther are *also* false declarations attached to Poincare and to many other great mathematicians and scientists...