[ark4579]

hmm probably

[contravariant]

Probability theory works, with 100% chance even. Which annoyingly isn't the same as always.

[quantum_state]

The concept of probability is based on the concept of measure in math => limitation in its description of things, e.g., the probability for a real number in [0, 1) being an irrational number is 1.

[eli_gottlieb]

Hold on, what measure over the unit interval assigns probability 1 to the set of irrational numbers in the unit interval? Do irrational numbers on their own even form a proper measurable set?

[omnicognate]

Lebesgue measure [0]. I never did write part 2 and many years have gone by with that blog languishing unmaintained on free wordpress but I wrote a thing that goes through the issues around this [1].

[0] https://en.m.wikipedia.org/wiki/Lebesgue_measure

[1] https://omnicognate.wordpress.com/2013/11/04/sigma-algebras-...

[hgsgm]

Of course. [0,1] has measure 1. Rationals have measure 0. Irrationals have measure 1-0=1

This question is why measure theory exists.

[contravariant]

The irrationals are measurable for any measure that can measure points, which is most of them.

[bmacho]

What is funny thought, that 0% chance events don't can't happen, but must happen. Like when you pick a point on a line, or roll a dice infinitely many times.

[throwme_123]

HMMs are probabilistic models, yes.