[roundsquare]

Just for fun... it might not be impossible:

You can reduce R^2 to {x | 0<=x<=1}[1] and thus, by induction, we can reduce R^n to {x | 0<=x<=1} for any integer n.

If I remember correctly, there are about 10^80 particles. The position of each particle is a point in R^3 so the position of all particles is R^(3*10^80). So, the position of every particle could be stored by the position of a particle on a 1 meter (foot, inch, whatever) long stick.

Of course, you run into problems if space is discrete or, in any event, with the Heisenberg uncertainty principle but you can still store a lot of information with each particle.

Horribly impractical of course, but like I said, just for fun...

[1]http://www.earlham.edu/~peters/writing/infapp.htm

[xyzzyz]

Your argument from cardinality of sets is not relevant -- current state of knowledge of nature of the universe prohibits building a device which carries on with computation using real numbers. See Bekenstein bound.

Actually, if we could perform computations using real numbers (think of it as we're back using analog computers and the universe is continuous again and not discrete/quantum), we would for instance be able to solve NP-complete (also #P-complete) problems in polynomial time.

Anyway, the rest of your argument is what philosophers were arguing about two, three and even four hundreds years ago. See Wikipedia pages for "Determinism" or "Mechanism".

[Daniel_Newby]

"The position of each particle is a point in R^3 so the position of all particles is R^(310^80)."

There's a theory that says the the maximum information content of a region of space is proportional to its surface area*. (Honest!) This result is related to black hole physics: information has to be represented by mass and/or energy, and thus curves the space-time continuum according to general relativity. Try to put too much information in a region of space and it gets cloaked by an event horizon.