[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".