Re: The modern mathematical concept of infinity is ...

On Feb 19, 8:51 pm, Ralf Bader <ba...@xxxxxxxxxx> wrote:
lwal...@xxxxxxxxx wrote:
One can only imagine how many more mathematicians
WM must find having similar or even identical
ideas as himself, before Bader, Chandler, and the
others finally accept that WM can have a theory
that isn't the "Most Stupid Philosophy" ever.
That is now getting completely ridiculous.
The point is that Isles' or Nelson's work has mathematical content, and
Mueckenheim's has not.

Yet WM implies that he agrees with Isles. Therefore
WM has mathematical content if and only if Isles has
mathematical content.

We already know that WM makes many inferences that
are invalid in ZFC, but are apparently invalid in
the WM-Isles theory. So WM is wrong to say that ZFC
proves the existence of only finitely many naturals,
but right to say that WM-Isles proves the existence
of only finitely many naturals.

What's provable in ZFC may not necessarily be
provable in WM-Isles, and vice versa. Yet WM-Isles
is just as valid a theory as ZFC. Indeed, if and
when Nelson can ever complete his inconsistency
proof of PA, WM-Isles would be a _more_ valid theory
than ZFC, since Nelson's Paradox would prove the
latter, but not the former, to be inconsistent.

Nelson hopes to show that "n is a natural (or
counting) number," for a particular large natural n,
implies some P and not P. If Nelson succeeds, then
any theory implying the existence of the natural
number n, such as PA and ZFC, would be inconsistent,
but WM-Isles would not be proved inconsistent,
provided n is larger than 2^65536 (or 2^^5).
.

Relevant Pages

  • Re: The modern mathematical concept of infinity is ...
    ... Yet WM implies that he agrees with Isles. ... So WM is wrong to say that ZFC ... proves the existence of only finitely many naturals, ... but right to say that WM-Isles proves the existence ...
    (sci.math)
  • Re: Skolems Paradox
    ... >>>your plain ordinary set theoretic truth. ... When I said it wasn't clear to me whether or not the meaning of P ... the axioms of ZFC. ... >subsets of naturals than M', then M is closer to the truth than M'. ...
    (sci.logic)
  • Re: Infinite Induction and the Limits of Curves
    ... for all finite natural n, but for the infinite case as well, given certain ... The rationals are equinumerous with the irrationals. ... I'd be interested in the exact reason why this doesn't work in ZFC. ... S_n = {even naturals up to n} ...
    (sci.math)
  • Re: Infinite Induction and the Limits of Curves
    ... The rationals are equinumerous with the irrationals. ... I'd be interested in the exact reason why this doesn't work in ZFC. ... S_n = {even naturals up to n} ...
    (sci.math)
  • Re: Infinite Induction and the Limits of Curves
    ... for all finite natural n, but for the infinite case as well, given certain ... The rationals are equinumerous with the irrationals. ... I'd be interested in the exact reason why this doesn't work in ZFC. ... S_n = {even naturals up to n} ...
    (sci.math)