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.
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.
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
> Here's one mapping from R to R^2:
That's the easy part. It's very easy to map R to R^2, the other direction not so much.
From my searches, based upon what other people said, the mapping is a surjection and not a bijection.
If that's the case, then either they don't have the same cardinality or I'm confused as to what cardinality actually is.