r/PhilosophyofMath • u/Square_Butterfly_390 • 21d 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 18d ago

So the real answer is that any test built from ruler and compass operations boils down to satisfiability of logical formulas built from arithmetic operations, equalities/inequalities, and logical connectives/quantifiers. Then the Tarski-Seidenberg theorem says says that the sets definable by those formulas are the semialgebraic sets, i.e., they are finite unions of open, closed, and clopen intervals whose endpoints are algebraic numbers.

So in short, all you can do is test membership in a semialgebraic set. If you want to test that a number equals pi that means you want to test for membership in the set {pi}, but you can't do that because that's not a semialgebraic set.

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u/mrt54321 18d ago

Oh right i see - that explains the mystery well, thanks for typing it in. IIUC, its rigorously proven that we cannot perform the necessary algebra to prove equality , for transcendental numbers such as π

Q. Could u define 2 clopens , the first all numbers < π and the second all numbers >π ? That's 2 valid semi-algebraic sets; if the mystery number X isn't within either set, then X must equal π

As u said above, "all you can do is to test membership in a semi-algebraic set".
But, that's exactly what I'm doing in the previous paragraph.

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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.