[kazga]

"All triangles have 3 sides" might be an uninteresting triviality, but "The sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse in a right-angled triangle" is neither trivial nor uninteresting and yet it has a probability of 1.

[paulsutter]

I love this example the most. It's the exception that proves the rule.

If you need to dig this hard to find something interesting with a probability of 1, that's pretty good evidence that the vast majority of interesting statements are not of the true/false variety.

Although I don't find it interesting, I am open-minded enough to ... embrace.. the .. uh.. diversity of the world, that allows some people, to find that interesting.

[kazga]

Pythagorean Theorem is "digging hard" and "not interesting"? Mind explaining?

The language that contains all Turing Machines that halt on all inputs is not decidable.

Or

e^(iy) = sin(y) + i * cos(y)

Are those uninteresting trivialities to you?

[jholman]

Yes, exactly. And "this code has a mathematical error in it" is often interesting, often non-trivial, and often has probability 1 (and often probability 0).

And these things are exactly the sort of thing that "differ from [trivialities about triangles] only in degree of complexity".

Note that "all triangles have 3 sides" is probably an axiom, but "all triangles have less than 11 sides" is a trivial theorem.