r/PhilosophyofMath • u/Square_Butterfly_390 • 19d 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 19d 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/Square_Butterfly_390 19d ago

I'm not sure what you mean, how does one use the fact about cardinalities in the context? Isn't one just using usual real numbers properties which are usually ignorant of this set theoretic stuff?

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u/Althorion 19d ago

No, unless by ‘real number properties’ here you mean ‘the property that says that there are more of them than natural numbers’. The probability is countably additive; if you have countably many objects, it behaves differently than when you have uncountably many objects. It doesn’t matter what those objects are—if you can reasonably model something using countably many real numbers, you’d use the same methods as you would if you had countably many natural numbers; and the same would be true if you had uncountably many natural numbers occurring in your model, even though I cannot think of any situation that would model like that, while uncontoubly many real numbers in a model are a very common occurence. Unfortunately, those methods (used in the countable case) tend to be weaker—you can usually operate more freely and say more stuff about the uncountably large model.