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Yes, the set of integers is infinite in size, but we can completely describe them with that set.
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Actually, we can't.
See Gödel's incompleteness theorem. We can't describe the integers completely, consistantly and decideably. The integers cannot be described.
Even if you had a simple equation describing the universe, look at gravity. The three body problem can't be solved analytically -- yet, gravity is a rather simple mathematical equation.
It is quite possible that there are games which are inheritly complex. Even if we knew the laws of physics, it wouldn't help us figure out how to play chess. If humankind runs out of things 'in the universe' to play with, we can play with things that are not. Or so I suspect.