Mason  🏃‍♂️ ✂️

And what the practical users pull keeps changing, the foundations of that mathematics have changed many times over since Euclid, and will continue to change again. Believing in the axiom of choice isn't a matter of rigor: it's a choice, is it useful? Does it feel right?

In a world in which this statement has no objective truth value: P ∨ ¬P I don't know what this means, do we not know the answer objectively for lack of rigor, or because the ground truth eludes us? Or is the answer a choice we make? https://twitter.com/webdevMason/status/1126324353648349184?s=19 …

The answer to what question? What ∨ means generally? Are you implying that it's a choice that we note it that way, or that it's a choice that the concept exists at all? If my society has developed a concept for v and yours hasn't, does ∨ exist on my land and not yours?

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.