Weird multi-sized infinities pop up in physics in a few places. The number of physical positions you can be in in space is uncountably infinite, the number of protons in the universe appears to be countably infinite.
There are interesting physical differences between quantum systems whose spectra are discrete (countably infinite eigenvalues) and continuous (uncountably infinite spectrum) and even combinations of both.
> Actually, doesn't the notion of Planck length / Planck time
Nope, we have absolutely no evidence of that. Space-time may be either discrete or continuous. Based on our current understanding we have no evidence either way.
Aleph 1 turns up when you try to (very) formally deal with probability theory, specifically when dealing with probability measures over the reals. This is sort of useful for physics for example when trying to be very careful about operators like position and momentum in quantum mechanics but it isn't really central. Its sort of nice to know you can do this stuff "properly" but physicists don't care much.
It's arguable that there's no such thing as a probability measure over the reals, because Solomonoff induction only works over computable programs, and the reals (in the sense needed) are not computable.
No, I mean even if you had (perfect, non-approximated) Solomonoff induction, you could only generate probabilities for computable "theories" (programs that predict all your past and future input), but I suppose it's possible that the impossibility proofs actually depend in some way on Aleph 1, so you would need it for consistency.
There are interesting physical differences between quantum systems whose spectra are discrete (countably infinite eigenvalues) and continuous (uncountably infinite spectrum) and even combinations of both.