Guy Macon http://www.guymacon.com/ wrote:
Consider the following variants of chess:
[for i3, Variant i: standard set of men, 8xi board.]

The above set of variants is clearly infinite and maps to the set of
integers. [...]

Now consider these variants of chess:
[for i3, j7, Variant i.j: standard set of men, jxi board.]

The above set of variants is also clearly infinite, larger than the
previous infinite set, and maps to the set of fractions.

These are properly called the positive rational numbers (i.e., the set
of numbers that can be written as i/j for positive integers i and j).
The set of positive rationals is *not* larger than the set of integers:
it has the same cardinality.

Proof. (Writing N for the positive integers, Q' for the positive
rationals and |S| for the cardinality of the set S.) Every positive
integer n can be written n/1, so is a positive rational. Therefore,
|N|=|Q'|. Any positive rational m/n can be coded unambiguously by
the positive integer 2^m x 3^n, so |Q'|=|N|. QED.


Dave.

--
David Richerby Solar-Powered Expensive Book (TM):
www.chiark.greenend.org.uk/~davidr/ it's like a romantic novel but it'll
break the bank and it doesn't work in
the dark!