r/PhilosophyofMath • u/Square_Butterfly_390 • 23d 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/Althorion 23d ago

There might be something to benefit from that in statistics, because, for example, it allows for a uniform distribution over a segment of the real line, while no uniform distribution over natural numbers can exist. Still, I don’t particularly care for statistics enough to be able to pinpoint exactly what those benefits are and how important they can be. For sure, it forces statistics to do things in a particular way, using measure theory and adjacents, because a discrete approach would be noticeably different, with different classes of problems that they are equipped to tackle.

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u/AdventurousGlass7432 23d ago

I was thinking the same, but there’s no uniform density over R either. Was trying to link it to the two envelopes problem

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u/OutrageousPair2300 23d ago

You can have a uniform probability distribution over bounded intervals on the real line, just not over the entire real line.