If you thought this was too hard or too technical or too weird, I recommend that you take a look at https://www.lesswrong.com/posts/Een2oqjZe6Gtx6hrj/an-elementary-introduction-to-infra-bayesianism , which is intended as a companion piece to mine for those with less in the way of mathematical chops, or who'd simply rather have a less technical overview.
That sounds like it also works. I've seen the proof both ways and I think I was mixing them together in my head.
The number of such sets is specifically uncountable. Each set is of itself countable. Apologies, I'll fix the OP.
Consider the following example for the interval X = (0, 1) (which is homeomorphic to R). Suppose we wanted to assign measures to all of its subsets, and do so in accordance with the ordinary desiderata of sigma-additivity and m(X) = 1.
Now partition the interval into an uncountable family of countable sets X_i such that two numbers live in the same subset iff they differ by a rational number. (Make sure you fully understand this construction before continuing!)
What measure should we assign to any such X_i? We can quickly see that the X_i are all of equal cardinality (that of Aleph-null) and even have natural maps to each other (given by adding irrationals mod 1).
We can't assign them measure 0 - by sigma-additivity that gives us m(X) = 0. We can't assign them positive measure - again by sigma-additivity that gives us m(X) >> 1.
Thus we cannot assign such subsets any measure, so we must have been wrong from the start that 2^X was a reasonable sigma-algebra to pick as the foundation of our measure in X.
This seems approximately correct as the motivation, which IMO is expressible/ cashable-out in several isomorphic ways. (In that, in Demiurgery, in distributions over game-tree-branches, in expected utility maximinning...)
If you have questions/requests for [explanation/clarification], comments, feedback, or job offers, I'd be happy for any of those. I promise I don't generally bite.
Some even worse meta-effects: I have had some fairly bad experiences already in my attempts to get grants or a research position. I wish I could detail them more here, but I am not stupid, and I know that the people who deal with those grant applications or sit on those hiring panels come here and read. Probably this is already too much to have said. If you want, you can reach out to me privately and I'll happily speak on this.
Gonna put a few more recommendations here, given that I seem to have broadly overlapping puzzle taste (especially Stephen's Sausage Roll, which I even got to beta test!):
Clearly in the same vein: Antichamber, English Country Tune
Less clearly in the same vein for various reasons but still very very good: Return of the Obra Dinn (ish), Contradiction (the FMV game, less good), Gorogoa (very weird, not quite complete information, stylistically gorgeous)