My background isn't math, so I can't contribute too much along these lines. The other day, I was reading about proposals to take the infinitely large set of worlds and partition it in some non-arbitrary way so that probabilities can be assigned, but I can't find it now. — RogueAI
Let me know if you find it, I'd be interested. There is no uniform probability distribution on a countable set. That is, there is no way to assign probabilities to, say, the positive integers, in such a way that each one has an equal chance of being picked. There is no conceivable way to do this, and the proof is straightforward.
I concede the point. There might be some fundamental aspect of things that makes a universe of nothing but Boltzmann Brains physically impossible. But that doesn't seem to be the case currently. There doesn't seem to be anything preventing, say, "casino worlds" in Hitchhiker's Guide to the Galaxy (if you haven't read the book, it's a world where random erosion patterns just happened to have created glittering casinos everywhere). — RogueAI
I'm afraid I'm the only one who read that book and thought it was silly. So your point is lost on me, although it would resonate deeply with pretty much everyone else.
This is an assumption, but I think it a fair one. If there are infinite universes, why wouldn't they be countable? But maybe they're not. — RogueAI
Well, there's only one countably infinite cardinality; but there are so many uncountable cardinalities that they're too big to be corralled into a set. So if the probabilities are uniformly distributed -- like your nonsense worlds -- then the odds are unimaginably small that the number of worlds is countable.
But this isn't good reasoning. Mine, meant to be facetious to prove a point; or yours, meant to be serious.
Maybe. I don't know much about the Continuum hypothesis. — RogueAI
Here's all you need to know in order for me to explain my point. Cantor defined infinite cardinalities in such a way that they proceed one after another:
.
Now Cantor's famous diagonal argument and also his more general and beautifully simple
Cantor's theorem shows that there are
real numbers. And the question is,
which Aleph is that??. Perhaps
, or
.
The
Continuum hypothesis is the claim that
That is, that the cardinality of the real numbers is the very next cardinality after that of the natural numbers; or equivalently, that there is no infinite cardinality strictly between that of the naturals and the reals.
The question of whether this is true vexed Cantor and vexed everyone till Cohen proved as recently as 1963 that the question is
independent of the standard axioms of set theory. In other words there are models of set theory in which it's true, as Gödel showed in 1940; and models in which it's false, as discovered by Cohen. In fact Cohen earned the only Fields medal every granted for mathematical logic for his pioneering work in showing how we can cook up arbitrarily weird models of set theory in order to investigate such independence questions.
Now. My point is this. Every time a physicist casually says, "The number of universes might be infinite," or. "The size of the universe might be infinite"; or some breathless pop science writer who knows less than you or I do about the topic makes the same type of claim; they should
immediately exclaim: "This is very exciting! It means that mathematical problems like the Continuum hypothesis, which were formerly relevant ONLY to the realm of pure, abstract, non-physical mathematics, are now potentially amenable to study by physicists!"
But they never say that. Nobody has EVER said that. And in my own opinion, the reason that they don't, is that
they do not take their own suggestion seriously enough to have spent five minutes considering the profound mathematical and physical implications of what they're saying.
That is my point. And I admit that I've probably stated it so many times on this forum, going all the way back many years to the predecessor of this forum, that by now I often state it quickly without providing sufficient context for people seeing it for the first time. For which I take responsibility.
That's fine. Your speculations are interesting. I'm going to have to read more about Continuum hypothesis. — RogueAI
Thank you. I should say that I am often snarky, but was not being snarky (at least intentionally) with you. I'm actually trying to be less snarky these days, and your remark reminded me that I was unsuccessful in this instance.
Infinity is interesting. — RogueAI
Yes it is! And its profound implications are never considered, even momentarily, by all the people, from ignorant pop-sci writers to famous world-class physicists, who casually claim that some aspect of the world might be infinite. Because if infinity is instantiated in the world, then all the set-theoretic questions of infinity immediately become matters of physics; just as the bizarre mathematics of non-Euclidean geometry suddenly became relevant to physics when Einstein developed general relativity.
No, I'm not assuming they're equally likely or distributed uniformly. That's not required to generate the dilemma of have to choose between two infinite sets to figure out which one you're in, but like you said, the true odds may be different. For example, if you're jumping off a tall building, there are two sets to consider: the set of universes where you survive and the set where you don't, and obviously your odds of surviving aren't 50/50, so there's something going on there, and yet, at a fundamental level, reality either is as it appears to be (actual laws of nature, not just fantastic coincidences over and over, we're not Boltzmann brains, etc.) or reality isn't as it appears to be. If there are an infinity of universes of each type, and you don't know what kind of universe you're in, how is it anything other than 50/50? — RogueAI
The jumping off building analogy applies exactly here. Even if you don't know the true odds, it seems (to me) perfectly obvious that it's RARELY the case that the odds of two mutually exclusive events are 50-50. In fact Boltzmann brains are extremely statistically unlikely.
You would have to assert some limiting principle where the multiverse just doesn't produce universes where fantastic coincidence isn't the norm, but what on Earth would that mechanism be? — RogueAI
Well, one point I made was that we DO happen to live in a world of fantastic coincidence leading directly to our existence at this moment. And on the other hand is the building analogy and the Boltzman brain analogy. Boltzman brains are statistically highly unlikely. But then again, which is less likely? A Boltzman brain? Or a fully formed human being? Both are statistically unlikely. In fact it's one of the arguments against Darwinian evolution (among scientifically-minded neo-anti-Darwinists) that there literally hasn't been enough time for pure chance to have produced humans on earth.
After the first exchange, I thought you were making some errors, and I don't have much of a math background, so I asked a probability question about Pi. — RogueAI
What was the question? I honestly don't get it. First you said 123456789 and contrasted that to a random-looking string in order to get me to admit that one string looks random and one doesn't. [Good point actually]. But then the pi example confused me, because then you have two non-random looking strings of digits. So I didn't understand the point being made.
Do you know Bayes Theorem well? — RogueAI
I know Bayes' theorem but not well. I get the idea of priors but I've never been able to get very worked up over the apparent dispute between Baysians and frequentists. I know that people can use Bayes' theorem to show that if you test positive for some awful disease, it may still be much more likely that you have a false positive than that you actually have the disease. That's pretty much all I know.