(Epistemic status: I know basic probability theory but am otherwise just applying common sense here)
This seems to mostly be a philosophical question.
I believe the answer is that then you’re hitting the limits of your model and Bayesianism doesn’t necessarily apply.
In practical terms, I’d say it’s most likely that you were mistaken about the probability of the event in fact being 0. (Probability 1 events occuring should be fine).
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.
(Epistemic status: I know basic probability theory but am otherwise just applying common sense here)
This seems to mostly be a philosophical question. I believe the answer is that then you’re hitting the limits of your model and Bayesianism doesn’t necessarily apply. In practical terms, I’d say it’s most likely that you were mistaken about the probability of the event in fact being 0. (Probability 1 events occuring should be fine).
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.