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Comment by rvbissell | Hacker News Reader

>He obsessively pursued the ‘continuum hypothesis’, which states roughly that the infinitude of real numbers is next largest to that of natural numbers.

That sounds backwards? There are infinite real numbers between just 0 and 1, alone.

EDIT: my respondents so far seem to be stating the same thing I was. I interpreted the phrase "next largest" from TFA to mean that the infinitude of real numbers was a smaller than that of natural numbers. (So: not the largest, but rather the "next largest"). That, to me, is what sounded backwards.

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This speaks of the mathematical notion of cardinality, which is the number of elements in a set. For infinite sets, it turns out there are multiple sizes. The natural numbers have the smallest infinite size, and it is the same size of set as the integers and perhaps surprisingly, the rationals. But the real numbers have more elements. A quick proof is called Cantor's diagonalization argument, I recommend looking it up!

In fact, I recently learned there is one cardinality for each possible way to order an infinite set. This is slightly confusing because there are many ordinals of each cardinality, but there are just that many cardinals I've decided.

I missed your suggestion about Cantor's diagonalization earlier today, but I thank you for bringing it to my attention. That is quite clever.

Tangential to your point about the poor phrasing is that having "infinite ... numbers between just 0 and 1" is not evidence of a higher infinity.

Specifically, there are infinite rational numbers between 0 and 1, but rationals have the same cardinality as the natural numbers.

That's an interesting statement about rationals. My intuition would be to consider them a subset of the reals, and for natural numbers to be a subset of the rationals. How is my intuition failing me, in this case?

EDIT: found the answer: with Cantor's diagonalization, you can count all the rationals -- effectively mapping each one to a natural number. Since this mapping is demonstrably possible, they have the same cardinality.

two sets can both be infinite and yet one is "larger" than the other (c.f. https://en.wikipedia.org/wiki/Aleph_number).

You can establish two infinite sets are as large as one another by finding a bijection between them. These two sets would have the same "cardinality"

We know the real numbers has larger cardinality than natural numbers, but we don't know if there's anything in between -- can you construct an infinite set that has natural numbers < X < real numbers in terms of cardinality?

OP understands that, the quote says that the "the infinitude of real numbers is next largest to that of natural numbers", which makes it sound like it is next largest as in second largest as in less large as in lower cardinality. So it seems backwards. Maybe the quote means "next largest" as in larger.

Thank you. It seems absurd that 'next largest' could mean 'larger than the largest'. Hence my (OP's) interpretation.

I see. yeah I can see why it's confusing now -- we use "next-to-leading order" to mean a smaller effect than the leading order, but here, "next largest" means the one larger than natural numbers. Language is hard

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