Mason  🏃‍♂️ ✂️

I'm confused, and I don't want to assume you're confused or that you don't know what P∨¬P means, could you clarify for me?

I haven't done anything with logic in 10+ years, so tell me if I'm mistaken, but: for any P, P∨¬P is not a claim until I hold it true/false, assert equivalence, assert an implication, etc. There may be general features of the pattern P∨¬P, I dunno.

Mathematicians are strongly divided on P∨¬P, also known as the excluded middle. Some, and any who accept the axiom of choice (AC) assert it is vacuously true. For any given statement P, "P or Not P" is true. Constructivists (who implicitly reject AC) allow it to be false.

Or rather, constructivists are agnostic to P∨¬P. Either way this is a fundamental belief about classical logic. And it turns out... You get different mathematical universes if you deny the excluded middle or not, if you accept the axiom of choice or not.

If you're familiar with the Banach Tarski Paradox of being able to take a sphere, cut it up into pieces, move the pieces, and get two spheres, that's a consequence of the Axiom of Choice.

Ah, re: P∨¬P, I was confused about whether an assertion was implied. You've brought up set theory a few times, but it's not something I'm familiar with. It'll be helpful to me if we can stick to mathematical concepts that someone without a math background can work with.

This gets to the heart of it. Mathematicians believe in wild, absurd things that not only have no bearing on the rules of the universe, but that it would be patently absurd to believe they did. Banach Tarski, if a real physical sphere in the universe behaved like that, 1 + 1 = 1

Aaron, I'm not familiar with this topic.