Re: Are the number of variants to chess of Aleph nature or not?

Guy Macon  <http://www.guymacon.com/>
wrote:
> Consider the following variants of chess:
> [for i>3, 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 i>3, j>7, 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.

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