Working with the set of real numbers (was: Finding size of Variable)

[Rustom Mody]

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