The eternally sleeping beauty. (Thought experiment regarding uniform distribution on the natural numbers) A cousin of mine recently confronted me with a thought experiment that in essence contained an analogical situation to the following problem:

Assume you are a beauty with the following properties:

-You know there was a first day on which you woke up.

-You know each time you fall asleep, you lose your memories of the previous times you woke.

-You are immortal and live in an temporally infinite universe.

You are confronted with the question: What probability do you ascribe to the even "Today is the n-th time I woke up."?

It seems to me that there is no answer within Kolmogorov's probability theory, since any day seems equally likely and you cannot have an uniform distribution over the natural numbers. Is the question not well defined? I would love to read your thoughts.
Due to the finite lifetime of princesses (sorry little children but princesses are not immortal) there are a finite number of values this could take (even taking into account some Disney story this could be a large value) so there is a finite but small probability you could define to each day.

Now even with an immortal princess there is a finite age to the planet/universe so there are still limitations on this.

If we now allow the princess some planning she may very well leave notes to herself which could allow her to get the answer with high probability (or zero probability if we follow a storyline like Momento).

In short without greater bounds on the question anyone could mess it around however they like so I think I'll go with your answer than the question is not well defined.
My thoughts:


Since the princess didn't really know how many days she woke up,
So: Her guess is a purely random guess.
->numerator for the probability is 1(1 guess)
She is immortal and lives in a temporary infinite universe,
So: The limit of the numerator is $$\infty$$


$$1/\infty=0$$

So probability for P(Correctly answer"Today is the n-th time I woke up.")=0