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wnoise comments on The Absolute Self-Selection Assumption - Less Wrong
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This is a meme I keep seeing, and it's just not true. You need a lot more assumptions to justify that, such as "randomly generated", or very very strong versions of the cosmological principle.
The real line is infinite, but there's only one copy of the number 7.
The randomness of quantum mechanics is enough to guarantee under very weak conditions that, in most Everett branches, there are infinitely many copies of any pattern which occurs with positive probability.
The paper I linked justifies this assumption for one set of cosmological beliefs.
Also, though I made this claim as fact, you could generously consider it to be the assumption of the least convenient possible world. Are you sufficiently confident that there are only finitely many copies of you that you are OK with anthropics that would collapse if there were infinitely many copies?
So you're going with "randomly generated". Which is fine, but it needs to be spelled out.
You need to be very careful pulling intuitions about randomness from the finite case and applying it to the infinite case. In particular, it is no longer true that just because something happened, it has a positive probability. Any given real number has probability zero of being picked from the uniform distribution on [0,1) yet one certainly will be picked. And we can pick an infinite number of times and never encounter a duplicate.
I'm not attacking this assumption in order to attack your final conclusion, I'm just attacking this assumption.
I have actually never observed a real number picked at random. I have often observed rational numbers picked at pseudo-random, though.
Observing a Geiger counter near a piece of radioactive material was one of the highlights of my undergraduate physics labs. And the time distribution of clicks is random in the same sense that the OP was using.
I think the bigger problem is not randomness vs. pseudorandomness, but rather the question of whether uncountable probability spaces actually exist in physical situations.
I believe they do for the same reasons I take seriously the existence of other Everett branches. In fact the mapping is rather straightforward: I can't observe or directly interact with them in full generality, but the laws governing them and what I can observe are so very much simpler than laws that excise the unobservable ones. Whether I can actually exhibit most real numbers is besides the point.
Is there a demonstration that a physics based on the computables is more complex than a physics based on the reals?
This is a complicated question. In practice, it is difficult in this particular context to measure what we mean by more or less complicated. A Blum-Shub-Smale machine which is essentially the equivalent of a Turing machine but for real numbers can do anything a regular Turing machine can do. This would suggest that physics based on the real is in general capable of doing more. But in terms of describing rules, it seems that physics based on the reals is simpler. For example, trying to talk about points in space is a lot easier when one can have any real coordinate rather than any computable coordinate. If one wants to prove something about some sort of space that only has computable coordinates the easiest thing is generally to embed it in the corresponding real manifold or the like.
As Sniffnoy notes, the bigger problem is about the observation of an actual real number. Any observable signal specifying the instant at which the particle triggered the counter has finite information content, unlike a true real number. This includes the signal sent by your ears to your brain.
I shouldn't have mentioned pseudo-random number generation in the grandparent -- it's a red herring.
Not in a finite amount of time.
What do you mean?
Drawing from a continuous distribution happens fairly often, so your comment confuses me. Or maybe you'd say that those aren't "really infinite" and are confined to a certain number of bits, but quantum mechanics would be an exception to that.
As Cyan pointed out, when you choose a number confined to a certain number of bits, you are actually choosing from among the rationals.
I don't understand your reference to QM. I wasn't objecting to the randomness aspect. I was simply pointing out that to actually receive that randomly chosen real, you will (almost certainly) need to receive an infinite number of bits, and assuming finite channel capacity, that will take an infinite amount of time. So that event you mentioned, the one with an infinitesimal probability (zero probability for all practical purposes) is not going to actually happen (i.e. finish happening).
It was a minor quibble, which I now regret making.
I believe there are probably only countably many distinguishable observer moments, in which case this can't happen by countable additivity.
But you are certainly correct, that a lot goes into this assumption. I should be more clear about this; in particular, I should probably add a bunch of "may"'s.