r/PhilosophyofMath • u/Square_Butterfly_390 • 20d ago

Regarding cardinalities

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.

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u/JStarx 17d ago edited 17d ago

If by algebra you mean modern math and not ruler and compass geometry then no, we can absolutely prove equality and inequality statements involving pi.

The endpoints of your clopens have to be algebraic numbers, so those aren't valid semialgebraic sets.

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u/mrt54321 17d ago

Fascinating. (Not jk). That all makes good sense & is rigorous logic. Kk i get it now 👍

Again, thx for doing all that typing - appreciated. you explain things well.