Anton Sherwood (
Mon, 17 Nov 1997 18:37:42 -0800 (PST)

John K Clark wrote, of the contention that
the cardinality of the reals is equal to 2 ^ aleph-null:

: That's the problem. To prove it you'd have to find a one to one correspondence
: between a set with 2^alepha-null members and the points on a line. You'd have
: little difficulty with the rational numbers, those that can be expressed as
: ratios of whole numbers, like 5/7, but then you'd also have to deal with
: numbers that are not fractions but are the solution to some polynomial
: equation, like the square root of 2, after that you'd also have to deal with
: numbers that are not fractions or the solution to any polynomial equation
: but can be expressed as the sum of an infinite sequence, such as PI or e.
: After you've done all that your problems would have just begun because then
: you'd have to deal with Chaitin's numbers such as... well, actually only one
: of them has a name but there are an awful lot of them.

Is there any reason to think that some real numbers can *not* be expressed
as infinite strings of bits? And if not - if you can't get arbitrarily
close to the value of such numbers - what, if anything, does it mean to
say they have a value?

Anton Sherwood *\\* +1 415 267 0685 *\\*