The links you supplied do not support your case.
— noAxioms
Sure they do. If A and B are both countably infinite, A=B. — RogueAI

The mathexchange link never says A=B. It says their cardinalities are the same, with which I fully agree. That means that neither can be said to be more numerous than the other since there's a 1-1 mapping between members of the two sets.

The reply in that article does say about A and B that "they're the same", refering to their cardinality, but is not an assertion that both sets contain the same members.

Do you dispute this?

I dispute that the sets contain the same members (that they're actually the same), or that (to take my first counterexample) a large random positive number is as likely to be prime as not prime, despite the fact that all non-prime whole numbers can indeed be mapped 1-1 with prime numbers. You are drawing invalid probabilistic conclusions from sets based only on their identical cardinality.

Is the link I proved wrong?

I have no problem with it. The primes and non-primes are clearly not the same set, else any member of one would be a member of the other.

You also haven't provided any links to back up your point.

What point? That a large random number is probably not prime? No, I didn't provide a link for that. Do you dispute it?

Do you want to say they're the same size instead of being equal? That's fine with me.

There is a 1-1 mapping between the two sets. It therefore cannot be argued that one set is more numerous than the other. That's 'the same size' when speaking of infinities.

Another example is the set of whole numbers and the set of 1, 10, 100, 1000 ....
In that case the set of whole numbers clearly contains members not in the 2nd set, but the 2nd set is entirely contained in the first set. Nevertheless, they have the same cardinality and neither set can be asserted as having a greater size than the other. The mapping between the two sets is trivial in this case, in either direction.

About the quanta magazine article. There are several errors in it typical of a pop article, but the gist seems to be proving if the real numbers had the next cardinality up from 'countable' or if there was a cardinality in between. Cohen comes along about 60 years ago and proves that the question could not be answered within the framework of set theory. M&S came along in 2016 and proved that there wasn't.

I may have read that wrong. Thanks for the link. I was unaware of the work.

Assuming that the universe is infinite, what do you think the probability is that you're a Boltzmann brain? — RogueAI

It isn't a function of the size of the universe. It is a function of the theory that describes the workings (or the origin) of the universe. Given that, you get a ratio of BB's vs real brains. That ratio should be incredibly close to zero or some huge number. The size of the universe has no impact on that ratio. The odds of the ratio being something else (like say 1) is too small to consider. It's a matter of sorting the theories into two heaps: empiricallly justifiable or not.

I think your statement above is nonsense, based in the definition of a googolplex. — universeness

However large, a googolplex is a finite number. If a finite number of things are spread out evenly in an infinite volume, there would be infinite distance between them on average. You find this nonsense? Perhaps you assume a finite size universe, in which case the question reduces to how finite? It becomes a simple division problem between two finite numbers to get the nonzero density of BBs, but given infinite space, any finite number of objects contained in that volume would have zero density.

The geometry of the universe is currently considered flat, and unbounded, not infinite.

Reference? Those three seem mutually contradictory. Any two, fine, but all three? Perhaps this is our disconnect.

Assuming that the universe is infinite, what do you think the probability is that you're a Boltzmann brain? — RogueAI

I thought I had already answered that question with the suggestion that using my own subjective probability, I think that the universe is NOT infinite.

However large, a googolplex is a finite number. If a finite number of things are spread out evenly in an infinite volume, there would be infinite distance between them on average. You find this nonsense? Perhaps you assume a finite size universe, in which case the question reduces to how finite? It becomes a simple division problem between two finite numbers to get the nonzero density of BBs, but given infinite space, any finite number of objects contained in that volume would have zero density. — noAxioms

You have already agreed that all 'numbers' are finite and in my opinion, there is no infinite volume.
YES, I am suggesting the universe is spatially finite. You can call something unbounded to indicate that it SEEMS to go on forever, and a 2d creature living in flatverse might even think its universe is totally linear, especially, if all of its scientific instrumentation supports that proposal. BUT, it will still accept the possibility that its flatverse is in fact circular but it's so vast that from it's 'light cone' it seems to be linearly unbounded.

Countable infinities are equal, so the infinite set of worlds where we're Boltzmann brains is equal to the infinite set of worlds where we're not. It's a 50/50 chance, epistemically speaking. Given an infinitely large multiverse, of course. — RogueAI

Prime Number Theorem

The Boltzmann brain paradox effectively says, in an infinite duration, we are more likely to be a disembodied brain with false memories than existing as persons within the complexities of our universe. — Down The Rabbit Hole

This seems self-refuting: if we were disembodied brains with false memories there would seem to be no rational justification for believing that we could be such, since the hypothesis that we are more likely to be Boltzmann brains relies on accepted mathematical and physical understandings which are reliant on the assumption that our memories are accurate (enough).

Just because we might expect it in an infinite duration, doesn’t mean we have reason to expect it now.

This seems self-refuting: if we were disembodied brains with false memories there would seem to be no rational justification for believing that we could be such, since the hypothesis that we are more likely to be Boltzmann brains relies on accepted mathematical and physical understandings which are reliant on the assumption that our memories are accurate (enough). — Janus

Yes, I think this is the point raised by Sean Carroll. And it is the same kind of paradox that faces epistemological nihilism - if we can't know things, we can't know that we can't know things.

We can only be completely agnostic on the question of if we are a Boltzmann Brain?

We can only be completely agnostic on the question of if we are a Boltzmann Brain? — Down The Rabbit Hole

Either that, or the idea is groundless and/ or incoherent. I don't know what to think about it.

In an infinite duration, aren't all possible outcomes equally likely to occur? — Down The Rabbit Hole

In an infinite duration, and as all possible existents are of finite duration, then everything would have happened already.

Either that, or the idea is groundless and/ or incoherent. I don't know what to think about it. — Janus

If the idea that minds can emerge from mindless stuff is incoherent, this problem goes away. As does simulation theory.

In an infinite duration, and as all possible existents are of finite duration, then everything would have happened already. — Wayfarer

I don’t understand. Has there already been an infinite duration?

Not according to the cosmological model popularly known as the 'big bang'. According to that model the Universe emerged from the singularity approximately 13.8 billion years ago.

Then why would everything have happened already?

In an infinite duration, and as all possible existents are of finite duration, then everything would have happened already. — Wayfarer

Why is that?

Doesn't it stand to reason that, if the Universe was of infinite duration, and all events in the Universe are of finite duration, then all events would already have occurred? Isn't that deductively valid? (It also seems to map against the idea of the heat death of the universe, which is a hypothesis that the universe will evolve to a state of no thermodynamic free energy, and will therefore be unable to sustain processes that increase entropy.)

Doesn't it stand to reason that, if the Universe was of infinite duration, and all events in the Universe are of finite duration, then all events would already have occurred? Isn't that deductively valid? (It also seems to map against the idea of the heat death of the universe, which is a hypothesis that the universe will evolve to a state of no thermodynamic free energy, and will therefore be unable to sustain processes that increase entropy.) — Wayfarer

That would only seem to hold if the Universe was of finite extent, that is contained a finite number of microphysical constituents. If we consider Nietzsche's 'eternal return' to be more than just a thought-experiment to test for life-affirmation, then this is the physical basis of his idea.

The astrophysicists at the time postulated that if the Universe was of infinite duration and extent, then the night sky should be ablaze with light, given that there would be an infinite number of stars and an infinite amount of time for the light to reach us, and the conclusion was that the Universe must be of finite extent, and it was unclear whether it had been of infinite duration.

I looked it up. The argument is known as the "Eternal Recurrence" and was proposed by various ancient philosophers including Pythagoras, Empedocles, and Heraclitus. The argument is based on the assumption that time is infinite and that the number of possible events that can occur in time is also infinite. If the universe is eternal, then it follows that every possible event will occur an infinite number of times. It was picked up by Nietszche. Where I had misunderstood it was to mean that, if all events are of finite duration, and the Universe is infinitely old, then everything that could occur would have already occurred, because no number of finite events could ever occupy an infinite expanse of time. But I'm not going to press the point!

Ah. I didn’t realize the idea is that the universe is infinitely old.

From a mathematical pov, does prime number theorem support or act against the Boltzmann brain proposal?

If the universe is eternal, then it follows that every possible event will occur an infinite number of times. — Wayfarer

This is the basis for my suggestion that Boltzmann brains and human-life are equally likely to occur. Despite the latter's pattern being more complex.

Other posters have cast doubt on this suggestion. It would be appreciated if @jgill put us straight.

If the idea that minds can emerge from mindless stuff is incoherent, this problem goes away. As does simulation theory. — RogueAI

I don't see how we will be able to prove what gives rise to consciousness.

You're not suggesting substance-dualism are you?

This seems self-refuting: if we were disembodied brains with false memories there would seem to be no rational justification for believing that we could be such, since the hypothesis that we are more likely to be Boltzmann brains relies on accepted mathematical and physical understandings which are reliant on the assumption that our memories are accurate (enough). — Janus

That it is circular like that doesn't result in the conclusion that we're not BBs. It only yields the conclusion that our hypothesis is unjustifiable.
So if you are in fact a BB (there is no 'we' about that, it would be a solipsistic existence), then there is zero evidence of your own nature. If you by freak chance happen to be in a state of suspicion about being a BB, that is just coincidence, completely unrelated to the fact that it happens to be true in this case.
If you are in fact an evolved thing (living in a non-Boltzmann galaxy, if there is such a thing), then you have access to empirical evidence, but you have no way of telling the difference. So you come up with a plausible hypothesis (the BB cannot 'come up with' anything) about the nature of your universe and if the hypothesis predicts that you're more likely to be a BB than not, then there cannot be justification for that hypothesis.

I think that's the summary of the argument.

We can only be completely agnostic on the question of if we are a Boltzmann Brain? — Down The Rabbit Hole

We can do more than that. We can restrict our hypotheses to ones that predict normal existence. If the actual 'way that things are' happens not to correspond to such a hypothesis, then the truth of reality is not something that can be reasonably guessed at.

In an infinite duration, and as all possible existents are of finite duration, then everything would have happened already. — Wayfarer

To use the tense 'would have happened' presumes that there is a present time, and that that present time is after all events (is at the end of infinite time, a contradiction).

If the idea that minds can emerge from mindless stuff is incoherent, this problem goes away. As does simulation theory. — RogueAI

I think that if such is your hypothesis, then like the BB scenario, empirical evidence cannot be trusted, and once again, the result is a completely unjustifiable hypothesis.

Not according to the cosmological model popularly known as the 'big bang'. According to that model the Universe emerged from the singularity approximately 13.8 billion years ago. — Wayfarer

That figure presumes that we can trust empirical evidence, which hasn't been established if we don't start with a hypothesis that allows us to make that assumption.

The astrophysicists at the time postulated that if the Universe was of infinite duration and extent, then the night sky should be ablaze with light, — Janus

This assumes a steady-state hypothesis. It was one of the earliest arguments that our universe is of finite age.

From a mathematical pov, does prime number theorem support or act against the Boltzmann brain proposal? — universeness

I don't see how it is relevant at all, since the BB idea isn't dependent on infinities or primes. It does however illustrate that just because two countable infinities (primes and not-primes say) can be given a 1-1 correspondence, it doesn't follow that random numbers have equally probability of being prime or not. So the following for instance is a non-sequitur:

This is the basis for my suggestion that Boltzmann brains and human-life are equally likely to occur. — Down The Rabbit Hole

I know of no hypothesis where normal minds and BBs have probabilities within a hundred orders of magnitude of each other, let alone equal.

I don't see how we will be able to prove what gives rise to consciousness. — Down The Rabbit Hole

I think you can arrive at a proof via a reductio ad absurdum: the idea that consciousness and mind can from matter leads to absurdities like the following:
https://xkcd.com/505/

We were talking about subjective probabilities, not actual probabilities — RogueAI

Any probability has to obey additivity and normalization axioms, otherwise it's not a probability. If you find that your subjective probabilities add up to more than 100%, then you are being inconsistent.

This is the basis for my suggestion that Boltzmann brains and human-life are equally likely to occur. Despite the latter's pattern being more complex. — Down The Rabbit Hole

You need to be careful about what exactly "equally likely to occur" means in this context. The way cosmologists might pose this question is: "Given an observer, is it more likely to be a regular observer (a human or a similarly evolved creature) or a freak observer like a Boltzmann Brain?" This is a tricky epistemological question involving concepts like reference class, self-location and self-selection.

And yes, infinite, or just very big worlds seem to present a general challenge to observations:

Big World theories, popular in contemporary cosmology, engender a peculiar methodological problem: because they say the world is very big and somewhat stochastic, they imply (or make it highly probable) that every possible human observation is made. The difficulty is that it is unclear how we could ever have empirical reasons for preferring one such theory to another, since they all seem to fit equally well with whatever we observe. — Nick Bostrom

Intuitively though it seems that simply adding "more of the same" to the world (more space or more time or more observers) should not make a difference to a generic observation made by a particular observer at a particular place at a particular time, so the challenge to epistemologists is to explain just how this challenge is only a seeming one. (Bostrom purports to meet it with his Self-Sampling Assumption, which he also uses elsewhere to analyze puzzles like Boltzmann Brains.)

Any probability has to obey additivity and normalization axioms, otherwise it's not a probability. — SophistiCat

Yes, but subjective probabilities are different than objective probabilities.

↪jgill

From a mathematical pov, does prime number theorem support or act against the Boltzmann brain proposal? — universeness

It was a counter to the number of primes being 50%.

In an infinite duration, aren't all possible outcomes equally likely to occur? — Down The Rabbit Hole

An infinite duration could include an infinite number of stages, the possibilities of something happening in a particular stage might not exist in other stages. In other words, its all babble. :smile:

Where I had misunderstood it was to mean that, if all events are of finite duration, and the Universe is infinitely old, then everything that could occur would have already occurred, because no number of finite events could ever occupy an infinite expanse of time. — Wayfarer

Even if all events were of finite duration, and the Universe were infinitely old, that all events that could occur would already have occurred would rely on the Universe being of finite extent; that is the salient point.

I don't know exactly what you are thinking is entailed by "no number of finite events could ever occupy and infinite expanse of time" in this context. That sounds like the standard argument against the possibility of an actual infinite duration.

You could have said, which would have been clearer I think, "no finite number of events could ever occupy an infinite expanse of time", and that, if true, would show that either the Universe has not been of infinite duration or that, if it were, then an infinite number of events would have occurred, with recurrence if the extent of the Universe were finite, but not necessarily if not. That events are finite is a given.

So you come up with a plausible hypothesis (the BB cannot 'come up with' anything) about the nature of your universe and if the hypothesis predicts that you're more likely to be a BB than not, then there cannot be justification for that hypothesis.

I think that's the summary of the argument. — noAxioms

Yes, that's pretty much what I was thinking. Sure, we might be Boltzmann in any case, but if we were then we would have no rational justification for thinking that we are or even might be.

You need to be careful about what exactly "equally likely to occur" means in this context. The way cosmologists might pose this question is: "Given an observer, is it more likely to be a regular observer (a human or a similarly evolved creature) or a freak observer like a Boltzmann Brain?" This is a tricky epistemological question involving concepts like reference class, self-location and self-selection. — SophistiCat

Intuitively though it seems that simply adding "more of the same" to the world (more space or more time or more observers) should not make a difference to a generic observation made by a particular observer at a particular place at a particular time, so the challenge to epistemologists is to explain just how this challenge is only a seeming one. (Bostrom purports to meet it with his Self-Sampling Assumption, which he also uses elsewhere to analyze puzzles like Boltzmann Brains.) — SophistiCat

Yes, I think our meaning of "equally likely to occur" is pivotal. A more agreeable meaning may be from The Principle of Indifference: "A rule for assigning epistemic probabilities. It assumes that if you have multiple plausible scenarios, you should assume each is equally likely till you have evidence otherwise".

Hey, I call dibs on the first Boltzmann USS Enterprise when it pops up!!!

You'll have to fight @universeness for it.