I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
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Yes! The difference between these two types of infinities (the set of non-negative integers and the set of non-negative real numbers) is countability. Basically, our real numbers contain rational numbers, which are countable, and irrational numbers, which are not. Each irrational number is its own infinity, and you can tell this because you cannot write one exactly as a number (it takes an infinite numbers of decimals to write it, otherwise you've written a ratio :) ). So, strictly speaking, the irrational numbers are the bigger infinity between the two.