How can some infinities be bigger than others?

Re:How can some infinities be bigger than others? by ginnungagap on Thursday April 20, 2023 @05:17AM (#35637853)

There is no need to exhibit bijections, if you have an injection X -> Y and an injection Y -> X, then X and Y have the same cardinality. This is the statement of the Cantor-Schroeder-Bernstein theorem.

If you assume the axiom of choice it's enough to have surjections both way.

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Re:How can some infinities be bigger than others? by yarg on Thursday April 20, 2023 @05:31AM (#35637917)

I get how you can map every value in R -> R^2, but I have yet to see any proof that there's a complete mapping in reverse.

For the most reasonable attempt at flooding a plane with R, I still end up with something where I can trivially define values {r0, r1} that do not map back to R.

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Re:How can some infinities be bigger than others? by ginnungagap on Thursday April 20, 2023 @06:16AM (#35638121)

What are those trivially defined {r0,r1} for the "interleaving digits" map f:(0,1) -> (0,1)^2 defined by taking a real x in (0,1), writing its decimal expansion as

x=0.a1a2a3a4a5a6...

(always pick the nonrepeating decimal expansion in case x has two) and sending it to the pair (y,z) where

y=0.a1a3a5... z=0.a2a4a6...

(Using (0,1) is no issue since we can compose later with bijections (0,1) -> R)

Edit: I'm not sure why I cannot reply to your reply, but {π,π} is not even in (0,1)^2

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Re:How can some infinities be bigger than others? by yarg on Thursday April 20, 2023 @06:23AM (#35638166)

{π, π} will generally do.

Sorry, didn't even look at the bounds - assumed things went to infinity.

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