On Wednesday, March 5, 2014 9:11:13 AM UTC+5:30, Steven D'Aprano wrote: > On Wed, 05 Mar 2014 02:15:14 +0000, Albert van der Horst wrote:
> > Adding cf's adds all computable numbers in infinite precision. However > > that is not even a drop in the ocean, as the computable numbers have > > measure zero. > On the other hand, it's not really clear that the non-computable numbers > are useful or necessary for anything. They exist as mathematical > abstractions, but they'll never be the result of any calculation or > measurement that anyone might do. There are even more extreme versions of this amounting to roughly this view: "Any infinity supposedly 'larger' than the natural numbers is a nonsensical notion." See eg http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory and Weyl/Polya bet (pg 10 of http://research.microsoft.com/en-us/um/people/gurevich/Opera/123.pdf ) I cannot find the exact quote so from memory Weyl says something to this effect: Cantor's diagonalization PROOF is not in question. Its CONCLUSION very much is. The classical/platonic mathematician (subject to wooly thinking) concludes that the real numbers are a superset of the integers The constructvist mathematician (who supposedly thinks clearly) only concludes the obvious, viz that real numbers cannot be enumerated To go from 'cannot be enumerated' to 'is a proper superset of' requires the assumption of 'completed infinities' and that is not math but theology -- https://mail.python.org/mailman/listinfo/python-list
