Actual Infinity in Reality?

[polymath257]

3. Hilbert's hotel applies to sets that are countably infinite.  If time is continuous and infinite, it would seem that the set of all possible moments is uncountably infinite.  In that event, Hilbert's hotel simply wouldn't apply.  As long as we're throwing around burden of proof questions, I think you are obligated to either show that time is not continuous, or that even if it is, that the set of all possible moments is a countable infinity.  Otherwise, we can simply dispense with Hilbert's hotel, as it does not cover all the possibilities for a temporally infinite universe that I have raised.  An objection which only applies to some of the possibilities but not all cannot possibly demonstrate that all cases are impossible.

Just a technical point. If we model time using the real numbers (a continuum), then there will be a countable number of *intervals* of any given size (say, 1 second long).
The standard Hilbert Hotel discussion then works just fine with those intervals.

And, in fact, we have

continuum+finite=continuum
continuum+countable infinity=continuum
continuum+continuum=continuum.

Which, again, just shows that continuum-continuum is not well defined as a cardinal number. We can, however, still do set theoretic differences.