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 links you supplied do not support your case.
— noAxioms
Sure they do. If A and B are both countably infinite, A=B. — RogueAI
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.Do you dispute this?
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.Is the link I proved wrong?
What point? That a large random number is probably not prime? No, I didn't provide a link for that. Do you dispute it?You also haven't provided any links to back up your point.
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.Do you want to say they're the same size instead of being equal? That's fine with me.
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.Assuming that the universe is infinite, what do you think the probability is that you're a Boltzmann brain? — RogueAI
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.I think your statement above is nonsense, based in the definition of a googolplex. — universeness
Reference? Those three seem mutually contradictory. Any two, fine, but all three? Perhaps this is our disconnect.The geometry of the universe is currently considered flat, and unbounded, not infinite.
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.Assuming that the universe is infinite, what do you think the probability is that you're a Boltzmann brain? — RogueAI
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
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). — Janus
We can only be completely agnostic on the question of if we are a Boltzmann Brain? — Down The Rabbit Hole
In an infinite duration, aren't all possible outcomes equally likely to occur? — Down The Rabbit Hole
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
If the universe is eternal, then it follows that every possible event will occur an infinite number of times. — Wayfarer
If the idea that minds can emerge from mindless stuff is incoherent, this problem goes away. As does simulation theory. — RogueAI
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.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
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.We can only be completely agnostic on the question of if we are a Boltzmann Brain? — Down The Rabbit Hole
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).In an infinite duration, and as all possible existents are of finite duration, then everything would have happened already. — Wayfarer
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.If the idea that minds can emerge from mindless stuff is incoherent, this problem goes away. As does simulation theory. — RogueAI
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.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
This assumes a steady-state hypothesis. It was one of the earliest arguments that our universe is of finite age.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
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:From a mathematical pov, does prime number theorem support or act against the Boltzmann brain proposal? — universeness
I know of no hypothesis where normal minds and BBs have probabilities within a hundred orders of magnitude of each other, let alone equal.This is the basis for my suggestion that Boltzmann brains and human-life are equally likely to occur. — Down The Rabbit Hole
I don't see how we will be able to prove what gives rise to consciousness. — Down The Rabbit Hole
We were talking about subjective probabilities, not actual probabilities — RogueAI
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
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
Any probability has to obey additivity and normalization axioms, otherwise it's not a probability. — SophistiCat
↪jgill
From a mathematical pov, does prime number theorem support or act against the Boltzmann brain proposal? — universeness
In an infinite duration, aren't all possible outcomes equally likely to occur? — Down The Rabbit Hole
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
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
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
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