Not just any set.
Almost any set: only the empty set is excluded. The identities of the elements themselves are irrelevant to the mathematical structure. Any further restrictions are not a part of mathematical definition of probability space, but some particular application you may have in mind.
If elementary event {A} has P(A) = 0, then we can simply not include outcome A into the sample space for simplicity sake.
In some cases this is reasonable, but in others it is impossible. For example, when defining continuous probability distributions you can’t eliminate sample set elements having measure zero or you will be left with the empty set.
There is a potential source of confusion in the “credence” category. Either you mean it as a synonym for probability, and then it follows all the properties of probability, including the fact that it can only measure formally defined events from the event space, which have stable truth value during an iteration of probability experiment.
It is a synonym for probability in the sense that it is a mathematical probability: that is, a measure over a sigma-algebra for which the axioms of a probability space are satisfied. I use a different term here to denote this application of the mathematical concept to a particular real-world purpose. Beside which, the Sleeping Beauty problem explicit uses the word.
I also don’t quite know what you mean by the phrase “stable truth value”. As defined, a universe state either satisfies or does not satisfy a proposition. If you’re referring to propositions that may vary over space or time, then when modelling a given situation you have two choices: either restrict the universe states in your set to locations or time regions over which all selected propositions have a definite truth value, or restrict the propositions to those that have a definite truth value over the selected universe states. Either way works.
Semantic statement “Today is Monday” is not a well-defined event in the Sleeping Beauty problem.
Of course it is. I described the structure under which it is, and you can verify that it does in fact satisfy the axioms of a probability space. As you’re looking for a crux, this is probably it.
Universe states can be distinguished by time information, and in problems like this where time is part of the world-model, they should be. The mathematical structure of a probability space has nothing to do with it, as the mathematical formalism doesn’t care what the elements of a sample space are.
Otherwise you can’t model even a non-coin flip form of the Sleeping Beauty problem in which Beauty is always awoken twice. If the problem asks “what should be Beauty’s credence that it is Monday” then you can’t even model the question without distinguishing universe states by time.
No, I specifically was referring to the Sleeping Beauty experiment. Re-read my comment. Or not. At this point it’s quite clear that we are failing to communicate in a fundamental way. I’m somewhat frustrated that you don’t even comment on those parts where I try to communicate the structure of the question, but only on the parts which seem tangential or merely about terminology. There is no need to reply to this comment, as I probably won’t continue participating in this discussion any further.