On Apr 16, 6:48 pm, Guy Macon http://www.guymacon.com/ wrote:
Now consider these variants of chess:

Variant 3: standard set of men, 8x6 board. -- same as in list above
Variant 3.1: standard set of men, 9x6 board.
Variant 3.2: standard set of men, 10x6 board.
Variant 3.3: standard set of men, 11x6 board.
Variant 3.4: standard set of men, 12x6 board.
Variant 3.5: standard set of men, 13x6 board.
Variant 3.6: standard set of men, 14x6 board.
...

The above set of variants is also clearly infinite,
larger than the previous infinite set, and maps
to the set of fractions. Eventually you get to
Variant 3.141592653589793238462643383: standard
set of men, 141415926535897932384626433811x6 board
and on to any other fraction you choose.

The question of *meaningful* differences is more
interesting. I don't see any meaningful difference
between playing on an 8x1000 board and playing on an
8x1002 board. But the loss of meaningfulness is
gradual; where exactly does it reach zero?

Meaningful differences is very important. One could argue that the
look of pieces, or their names, could be considered changes, and that
could be about infinite. But it has no effect on gameplay.

What I do see so far is several things that would lend to chess
variants be unbounded:
1. Time control, being infinite. One could do an infinite range of
time delays for a Bronstein clock. Not practical.
2. Size of board, being infinite. This then means an infinite number
of shapes. Unless the size of the board is infinite, then the number
of boards is finite.
3. Recursion. Here is an example. Say you can Calvinball the rules.
Let's say to implement a new rule, like Gipf introduces a new pieces,
one has to win another game to do this. If the number of possible
games one can play is infinite, then there is an infinite number of
varieties of chess. If then someone else where to go about wanting to
change the rules for the new game to see about the old game, they have
to play yet another game, then it is possible to cause an infinite
recursive set of action in place. This is not practical, but is
arguably meaningful to the game experience as a whole.

Pretty much here, either the parameters of a rule are unbound, or the
number of rules is unbound, or there is an infinite recursive rule
that can take place. If so, then such would be unlimited. The first
two be Heraclitian, and the last one being Calvinball. If anyone can
find any others, please say so.

Now, whether or not there is an infinite number of piece types, or
rules, that is another issue that would need to be considered here.
Anyone have evidence that there is an infinite number of rules that
can come in existence for a game like chess?

- Rich