On Mon, 29 Oct 2012 11:29:46 -0600, David C. Ullrich <email@example.com> wrote:
>On Sun, 28 Oct 2012 04:23:28 -0700, JRStern <JRStern@foobar.invalid> >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. > >I don't believe that. Any evidence at all?
It was in something I've read over the last few weeks, as I happened to wander into this area. I was hoping it would already be familiar to those here.
Well, this isn't the one I recall directly, but it seems to refer to the same thing, in "Set Theory And Its Logic", Quine, Cambridge MA 1969, p 241, Chapter XI, "Russell's Theory of Types":
Poincare tried to account for Russell's paradox as the efect rather of a subtle fallacy than of a collapse of irreducible principles. He attributed it to what he called a vicious circle. The defining characteristic of the paradoxical class y is '(x)(x member y .=. x not member y)', and the paradox comes, as we know, of letting the quantified variable 'x' here take y itself as a value ... He called the suspect procedure *impredicative*. We must not presuppose y in defining y.
I know Quine is off the main path of set theory and formal logic. I started my little expedition here with Boolos' "Logic, Logic, Logic" and Cohen's "Set Theory and the Continuum Hypothesis". I don't claim to understand any great amount of any of these, but they were picked up more as an experiment and wild tangential curiosity to something else, that lead me back to the old diagonal argument that I'd simply been walking around and reciting for 30 years as I was once taught. Back then they didn't mention Poincare, either.