Re probability 0 events, I’d say a good example of one is the question “What probability do we have to live in a universe with our specific fundamental constants?”
And our current theory relies on 20+ real number constants, but critically the probability of getting the constants we do have are always 0, no matter the number that is picked, yet one of them is picked.
Another example is the set of Turing Machines where we can’t decide their halting or non-halting is a probability 0 set, but that doesn’t allow us to construct a Turing Machine that decides whether another arbitrary Turing Machine halts, for well known reasons.
(This follows from the fact that the set of Turing Machines which have a decidable halting problem has probability 1):
I do find myself genuinely confused about how to assign a probability distribution to this kind of question. It’s one of the main things that draws me to things like Tegmark’s mathematical universe/ultimate ensemble, or the simulation hypothesis. In some sense I consider the simplest answer to be “All possible universes exist, therefore it is guaranteed that there is a me that sees the world I see.”
While I agree with the mathematical universe hypothesis/ultimate ensemble/simulation hypothesis, this wasn’t really my point, and it was just pointing out examples of probability 0⁄1 sets in real life where you cannot extend them into something that never/always happens.
This didn’t depend on any of the 3 hypotheses you generated here, 1 follows solely from the uniform probability distribution for real numbers, and the other is essentially measuring asymptotic density.
Re probability 0 events, I’d say a good example of one is the question “What probability do we have to live in a universe with our specific fundamental constants?”
And our current theory relies on 20+ real number constants, but critically the probability of getting the constants we do have are always 0, no matter the number that is picked, yet one of them is picked.
Another example is the set of Turing Machines where we can’t decide their halting or non-halting is a probability 0 set, but that doesn’t allow us to construct a Turing Machine that decides whether another arbitrary Turing Machine halts, for well known reasons.
(This follows from the fact that the set of Turing Machines which have a decidable halting problem has probability 1):
https://arxiv.org/abs/math/0504351
I do find myself genuinely confused about how to assign a probability distribution to this kind of question. It’s one of the main things that draws me to things like Tegmark’s mathematical universe/ultimate ensemble, or the simulation hypothesis. In some sense I consider the simplest answer to be “All possible universes exist, therefore it is guaranteed that there is a me that sees the world I see.”
While I agree with the mathematical universe hypothesis/ultimate ensemble/simulation hypothesis, this wasn’t really my point, and it was just pointing out examples of probability 0⁄1 sets in real life where you cannot extend them into something that never/always happens.
This didn’t depend on any of the 3 hypotheses you generated here, 1 follows solely from the uniform probability distribution for real numbers, and the other is essentially measuring asymptotic density.