I have a metaphysical probability question:
Suppose there are an infinite number of parallel Earths. Alice uses a teleporter to teleport to a random Earth. Bob tries to follow Alice, but he has to guess which Earth she teleported to. What are Bob's chances of getting it right? Is there any way for a teleporter machine to randomly select an Earth out of an infinite number of them in a finite amount of time, or is there always going to be, practically speaking, only a finite amount of Earths for Alice to teleport to because of the limitations of the machine? What if I cheat and say the teleporter pokes a hole into the universe and the universe somehow, through a mysterious process, randomly picks an Earth out of an infinitely large ensemble for Alice to teleport to? Are Bob's chances of teleporting to Alice's world zero? — RogueAI
Are there a countable or uncountable infinity of worlds? — fishfry
https://en.m.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem
It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models.
https://victoriagitman.github.io/talks/2015/04/22/an-introduction-to-nonstandard-model-of-arithmetic.html
An easy application of the compactness theorem shows that there are countable nonstandard models of the Peano axioms, or indeed of any collection of true arithmetic statements.
According to Thoralf Skolem's construction, i.e. by injecting infinite cardinalities in the model's structure, which is a countable set of symbols, there is at most a countable number of models of arithmetic. — Tarskian
The strong assumption here is indeed the continuum hypothesis: — Tarskian
https://en.m.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem
It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models.
In her lecture on the subject, Victoria Gitman confirms this:
https://victoriagitman.github.io/talks/2015/04/22/an-introduction-to-nonstandard-model-of-arithmetic.html — Tarskian
An easy application of the compactness theorem shows that there are countable nonstandard models of the Peano axioms, or indeed of any collection of true arithmetic statements. — Tarskian
If the physical multiverse is somewhat structurally similar to the arithmetical multiverse, it should also have a countable number of physical universes. — Tarskian
If we deny the continuum hypothesis, however, then most of the then uncountable universes would be unreachable because there can still only be a countable number of infinite cardinality symbols to do so. — Tarskian
They call this the measure problem. This is not to be confused with the measurement problem of quantum physics, the question of what is a measurement that makes the wave function collapse. — fishfry
The set would have to countable, wouldn't it? You could count the worlds. There's a one-to-one correspondence with each parallel world and the natural numbers. How could the multiverse be uncountably infinite? — RogueAI
https://en.wikipedia.org/wiki/Spectrum_of_a_theory
Saharon Shelah gave an almost complete solution to the spectrum problem.
Roughly speaking this means that either there are the maximum possible number of models in all uncountable cardinalities, or there are only "few" models in all uncountable cardinalities.
In an infinite multiverse, there would be an infinite number of Boltzmann Brain universes, so what are the odds you're in one? 50-50? — RogueAI
Bob tries to follow Alice, but he has to guess which Earth she teleported to. What are Bob's chances of getting it right? — RogueAI
Is there any way for a teleporter machine to randomly select an Earth out of an infinite number of them in a finite amount of time, or is there always going to be, practically speaking, only a finite amount of Earths for Alice to teleport to because of the limitations of the machine? — RogueAI
What if I cheat and say the teleporter pokes a hole into the universe and the universe somehow, through a mysterious process, randomly picks an Earth out of an infinitely large ensemble for Alice to teleport to? Are Bob's chances of teleporting to Alice's world zero? — RogueAI
Mathematical infinity has nothing to do with any study of the universe (except as it may appear in some of the mathematics that describe the physics of the universe). — tim wood
I have a metaphysical probability question:
Suppose there are an infinite number of parallel Earths. Alice uses a teleporter to teleport to a random Earth. Bob tries to follow Alice, but he has to guess which Earth she teleported to. What are Bob's chances of getting it right? Is there any way for a teleporter machine to randomly select an Earth out of an infinite number of them in a finite amount of time, or is there always going to be, practically speaking, only a finite amount of Earths for Alice to teleport to because of the limitations of the machine? — RogueAI
What if I cheat and say the teleporter pokes a hole into the universe and the universe somehow, through a mysterious process, randomly picks an Earth out of an infinitely large ensemble for Alice to teleport to? Are Bob's chances of teleporting to Alice's world zero? — RogueAI
Why not? As to the uni-verse, the predictions are for either a hot death or a cold death. The multiverse I understand as a "place" where universes exist much as galaxies in the universe.If it could predict its end, then the physical universe is effectively a finite structure, which cannot participate in a multiverse. — Tarskian
Is there any way for a teleporter machine to randomly select an Earth out of an infinite number of them in a finite amount of time, or is there always going to be, practically speaking, only a finite amount of Earths for Alice to teleport to because of the limitations of the machine? — RogueAI
Are Bob's chances of teleporting to Alice's world zero?
I get that you're trying to explore statistics, but if one drills down into the OP, as written, it devolves into an exploration of the limits of machines to choose "randomly" among an infinite number of choices. Which is to say: wholly inadequate. Thus the answer is: not zero. — LuckyR
The multiverse I understand as a "place" where universes exist much as galaxies in the universe. — tim wood
As to the uni-verse, the predictions are for either a hot death or a cold death. — tim wood
If alternative universes in the physical multiverse are structurally similar to nonstandard models/universes in the arithmetical multiverse, then alternative universes are not similar to galaxies.
While the distance between galaxies is finite, there is no legitimate notion of distance between universes. The distance in between universes is "infinite" or "inapplicable" (whatever that may mean). It is not possible to reach them by physically traveling to them. With galaxies, you conceivably can.
I just had a train of thought I thought might be vaguely interesting:
1. With an infinite number of options, not all options can have equal probability. — flannel jesus
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