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by eigenket 818 days ago
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.
2 comments
cubefox 818 days ago
> The number of physical positions you can be in in space is uncountably infinite

You don't know that. Spacetime may be quantized.

Actually, doesn't the notion of Planck length / Planck time already suggest that shorter distances have no physical meaning?

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eigenket 817 days ago
> 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.

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Vecr 818 days ago
Sure, those infinities. I meant the huge stack of new ones that Cantor started on in his book. What's Aleph 1 actually good for?
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eigenket 818 days ago
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.
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Vecr 818 days ago
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.
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eigenket 818 days ago
I think such an argument would need quite a lot more work, the lack of Solomonoff induction doesn't mean we don't have probability theory.
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Vecr 817 days ago
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.
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