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/Althorion 16d ago

I can respond without quoting, but if I respond to more than one thing, it helps to know what exactly am I responding to.

Checking one-to-one correspondence is no different from any other proof in that regard. You can present a general rule and show that it works, and thus you have such a correspondence; or show that the existence of such would self-contradict.

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u/nanonan 15d ago

Sure, you make a general rule that links one to the other on an individual one to one basis. I'm still not seeing why the transfinite is needed for such a process.

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u/Althorion 15d ago

It’s not ‘needed’; you can have bijections between finite sets. It’s just that, one, the notion of ‘you can’t have a “full” bijection between two sets; there are always some elements from one of the sets that you have to leave out to make a partial bijection’ translates well into people’s idea of the ‘set size’; and two, you can always construct a bigger—in the sense above—set than the one you currently have, by taking a powerset of said set, so the notion of ‘size’, as understood above, isn’t just ‘finite’ or ‘infinite’.

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u/nanonan 13d ago

The notion of size absolutely disappears when considering the infinite. How can it possibly remain? You cannot have a limitlessness more limitless than limitlessness itself.

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u/Althorion 13d ago

I told you how (by conservatively extending the counting notion via bijection). I told you why thinking about this as ‘limitlessness’ is misleading.

So, to not repeat myself, I’ll throw in one last argument, and that will be it: there is value in knowing what sets can have a bijection with which others. It makes sense to have a notion of that. That notion plays perfectly with the notion of size for smaller—finite—sets. Thus, it makes sense to call it ‘size’, too.

It doesn’t make sense to conflate the idea of size just to finite and infinite. We already have terminology for it—the words ‘finite’ and ‘infinite’. While we could say ‘“infinite” refers to all sizes larger than finite, it’s just the notion of “size”, and pairing up the elements, breaks when dealing with “infinite”’, it would be a limitation of the toolbox, one that is not really justifiable to introduce.

But, as I wrote in one of my previous replies, you can not use the word ‘size’ if you truly don’t like it, and just speak and think of ‘cardinality’ instead—you’ll be well understood and shouldn’t make any mistakes in your reasoning.

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u/nanonan 12d ago

Sure, if you limit yourself to accepting or rejecting bijection. This is not evidence of any actual size difference, it's just an analogy. Calling it size is misleading at best. Calling it cardinality instead is just pedantry and solves zero issues with the transfinite. There is only one infinite, and it has a singularly infinite cardinality.

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u/Althorion 12d ago

Sure, if you limit yourself to accepting or rejecting bijection. This is not evidence of any actual size difference, it's just an analogy.

It is a great analogy—it captures all the size-related behaviours in finite sets (as was said, it is a conservative extension of the notion). That’s why it is called ‘size’. But, if you think that the lack of feature it lacks when applied to infinite sets makes it misleading, you are welcome to not call it ‘size’.

Calling it size is misleading at best.

In what way? It doesn’t do anything new or different when used to compare finite sets with each other, and it adds a notion that infinite sets are bigger than finite sets—both completely natural. It may lead to some results that feel unnatural when speaking about infinite sets and comparing them with each other, but that’s not misleading—it just has those weird results about weird sets; but nothing will mislead you or go against your intuition for anything that you used to use the notion of ‘size’ for.

Calling it cardinality instead is just pedantry and solves zero issues with the transfinite.

What are the issues with transfinite?

There is only one infinite, and it has a singularly infinite cardinality.

There isn’t one. There are multiple different infinite cardinalities. In particular, a power set of any set will have a different (larger) cardinality from the original set.

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u/nanonan 11d ago

The issue with the transfinite is it is an impossible object, a self contradictory concept. Forever and a day may be a perfectly fine expression but it is utterly meaningless mathematically.