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

Thank you, Fishfry. Now we need another quote that it can be continued with absolutely any next number and fitted to any of an infinite number of non-congruent polynomials. That's also true.

I haven’t had time to read all the responses so excuse me if I’m repeating a previous comment.

If we consider the all too human component of this thought experiment, us, we must not fail to consider the fact that our brains are pattern detectors. This somewhat negates the classical notion of randomness in my mind. Ultimately it is nothing more than a lack of information on our part, what we consider randomness essentially is somewhat synonymous with inexplicable. Even when considering this through the lens of probabilistic outcomes, that something with a vanishingly small likelihood of occurring happens, seems more aptly attributed to desirability bias - this shouldn’t have happened! It had a one in all but infinite probability of being the case! - which I think also treats the issue of the original post and addresses what responses I did get through.

That said, another plausible interpretation of randomness (though only subtly different, it is philosophically distinct) is unpredictability. In this sense, the case of 10 5’s appearing and being an unsatisfactory result from the randomness machine would entail you A) had an expectation of what the randomness should look like or B) any result with too much of a “common” pattern would be unsatisfactory and make the machine seem faulty.

Agreed.

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.

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.

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.

Sure.

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.

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).

Excessive pickiness on my part, maybe. Not snark. I'm making a point. I'm disagreeing with your reasoning.

Fair enough.

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.

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.

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.

Maybe. I don't know much about 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.

That's fine. Your speculations are interesting. I'm going to have to read more about Continuum hypothesis. Infinity is interesting.

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?

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? 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?

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.

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. Do you know Bayes Theorem well?

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: $\aleph_0, \aleph_1, \aleph_2, \aleph_3, \dots$.

Now Cantor's famous diagonal argument and also his more general and beautifully simple Cantor's theorem shows that there are $2^{\aleph_0}$ real numbers. And the question is, which Aleph is that??. Perhaps $2^{\aleph_0} = \aleph_5$, or $2^{\aleph_0} = \aleph_{\text{googolplex}}$.

The Continuum hypothesis is the claim that

$2^{\aleph_0} = \aleph_1$

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.

"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?”

For years, the inability to calculate ratios of infinite quantities has prevented the multiverse hypothesis from making testable predictions about the properties of this universe. For the hypothesis to mature into a full-fledged theory of physics, the two-headed-cow question demands an answer."
https://www.quantamagazine.org/the-multiverses-measure-problem-20141103/

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

I take this as a datapoint in favor of my thesis. Guth is a heavy hitter, a physics superstar in both the physics and the popular communities. He's a Big Cheese. And he hasn't spend five minutes -- five seconds -- considering the implications of what he says. Nor is he aware that there's no uniform probability measure on a countable set. This is exactly the kind of thing I'm talking about. You know there are infinitely many multiples of 1000000000 and infinitely many positive integers that aren't multiples of 1000000000, but we can still calculate their respective asymptotic densities (which isn't the same thing as a probability measure). This is not a deep point, it's very trivial. In aa countably infinite multiverse, two-headed cows are like multiples of 1000000000 and one-headed cows are like all the other numbers. What of it? It's shallow masquerading as deep. (I'm not yelling at you, I'm yelling at the world. "Old man yells at cloud.")

https://en.wikipedia.org/wiki/Natural_density

By the way the asymptotic density of the primes (of which there are infinitely many) in the positive integers is zero. Mathematicians have thought about these things. Only the physicists pretend that their ignorance is deep thought.

Superstar physicists who don't know sh*t about the mathematics of infinity and haven't thought about what they're saying. I was shocked when I saw Susskind do it, now I'm no longer surprised. It is in fact a common pattern. And pop-sci reporters write this stuff down and then the general public absorbs these confusions.

Thanks for that link, I will definitely read the article and probably get my blood pressure raised.