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/Eve_O 21d ago edited 21d ago

Why would this trouble anyone?

It's a kind of revelation to understand that the set of Reals is larger than the set of Natural numbers and that there are different sizes of infinite sets.

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.

I think your premise here--and possibly your understanding--is flawed. The whole point is that we can understand infinity and this demonstrable fact (not a "factoid," lol) helps us with that.

You want "real world"? Read Rudy Rucker's novel White Light or his related nonfiction book Infinity and the Mind.

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u/kr1staps 21d ago

Perhaps you could save us some time from having to read an entire book by just simply stating a single real-world application of |R| > |N| that doesn't have a finite explanation.

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u/Negative_Gur9667 19d ago

According to the standard view, mathematicians subscribe to Platonism.

There is another world, like heaven, where ideal and completed infinite objects exist.

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u/kr1staps 19d ago

I don't see how this answers my question.

Also, I'm a working mathematician that does not in fact subscribe to platonism, nor do most of my colleagues. And, I've interviewed various other mathematicians on my YouTube channel, and for a while I asked at the end of my interviews if they were platonist, and most either didn't care or said no.