Given any finite sequence whatever, it can be continued with absolutely any next number and fitted to a polynomial.
https://en.wikipedia.org/wiki/Lagrange_polynomial — fishfry
Ok that's fair. But if we are speculating, isn't it fair for me to point out some things that need to be considered? If the universe instantiates actual infinity in any way: infinitely many sub-universes, infinitely many distinct times within a finite interval of time like 1/2, 1/4, 1/8, ... infinitely many planets, infinitely many anything ... then we must ask ourselves the question: Does the mathematical theory of infinity apply? If yes, then we must ask if things like the Continuum hypothesis and the axiom of choice have now become amenable to physical experiment; and if not, we must then develop a new physical theory of infinity.
I know you weren't thinking of these things, but (in my opinion) the moment one says that there MIGHT be a physical infinity, these questions immediately come to mind. My mind, in any event.
My point was about the ramifications if there are infinitely many universes with different physical constants. IF that is the case, the set of universes "everyone is a Boltzmann Brain" is infinite and the set "everyone is a real person" is infinite,
— RogueAI
This I disagree with. Am I allowed? As Jules played by Samuel L. Jackson says in Pulp Fiction: "Allow me to retort!" The set of positive integers exists. Are there as many numbers equal to 47 as not? No. Are there as many numbers that can be exponents in Fermat's equation? No, 2 is the only one, proven as recently as 1994. Are there infinitely many numbers that are part of a prime pair? Unknown. It is most definitely not the case that every possibility occurs infinitely many times. In the multiverse you have no idea what the actual rules are. Truth is you have no way of knowing that there are infinitely many universes that contain Boltzmann brains. Perhaps there's some as-yet-unknown physical constraint that only allows finitely many such. So your speculation is not fully thought out in my opinion.
Excessive pickiness on my part, maybe. Not snark. I'm making a point. I'm disagreeing with your reasoning.
and they're both countably infinite sets,
— RogueAI
Ah! And you know this, how? This is one of my questions. Let us suppose, arguendo[/ii], that the number of sub-universes in the universe (or universes in the multiverse) is actually infinite. Is it countably infinite or uncountably infinite? Well, you just made an assumption. So if I got you to state one of your unstated assumptions, my objections have not been in vain. And why should the number be countably infinite? And if it's uncountable, what might its cardinality be? Set theorists have some mighty large cardinals these days. So IMO these are the kinds of questions that come immediately to mind whenever someone speculates on physical instantiations of infinity.
After all, if there are even countably many of anything in the physical world, then we can in principle count its number of subsets; and depending on which cardinal number that happens to be, the Continuum hypothesis is therefore amenable to physical experiment. I take it as proof, or at least meta-proof, that physicists don't take infinite universes seriously; else postdocs would be applying for grants to determine the truth of the Continuum hypothesis.
Why are you allowed to speculate about the consequences of physical infinity, but not me? Can you see that I am actually trying to join in your game, by making my own speculations about the implications of physical infinity.
so how would you decide which set you're in if you don't know? It's a coin toss, in that situation.
— RogueAI
Without knowledge of the actual probability distribution, that's like guessing it's 50-50 to land alive after jumping off a tall building. Perhaps some configurations of the multiverse are far more likely than others. You're assuming all configurations are distributed uniformly. Isn't that an assumption?
If the multiverse isn't infinite, none of that applies, of course, but philosophy is about speculation, so I'm speculating here.
— RogueAI
So why can't I play too?
So what was the point of the lottery that comes up with the digits of pi? That example went right over my head.
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
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
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
Maybe. I don't know much about the Continuum hypothesis. — RogueAI
That's fine. Your speculations are interesting. I'm going to have to read more about Continuum hypothesis. — RogueAI
Infinity is interesting. — RogueAI
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
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
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
Do you know Bayes Theorem well? — RogueAI
Guth, a professor of physics at the Massachusetts Institute of Technology, resorts to freaks of nature to pose this “measure problem.” “In a single universe, cows born with two heads are rarer than cows born with one head,” he said. But in an infinitely branching multiverse, “there are an infinite number of one-headed cows and an infinite number of two-headed cows. What happens to the ratio?” — RogueAI
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