5 ms·
The distinction between the size of {I} and {I}/{I} (and {I}^n) is not the size, but the orderability. (Sorry for the bullshit syntax.) You can map any ration
by yarg 4y ago
The distinction between the size of {I} and {I}/{I} (and {I}^n) is not the size, but the orderability.
(Sorry for the bullshit syntax.)
You can map any rational onto an integer index; you cannot do it so that that r[i - 1] < r[i] < r[i + 1].
This has no impact on cardinality, but there's something here that doesn't sit quite right with me.
- lupire 4y agoWhat doesn't sit right? Your orderability idea is about "monotonic" mappings. Rational numbers are interesting even if they aren't ordered at all (but still can add and multiply, like other non-ordered objects like the integers mod N).
- yarg 4y agoI don't quite know. It feels as if there's some deep structural aspects to the rationals that get casually tossed to the side in order to shoehorn them into integer indices.
- LudwigNagasena 4y agoI guess the point is that you can't map them in a way that preserves "structure" in some sense.
- bawolff 4y agoNote, you also can't do this type of monotonic mapping with the integers mapped to the natural numbers. So its not like the rationals are a dividing point here.
- yarg 4y agoIsn't it really the same thing? The closest mapping of the reals to a countable set is the set of rational numbers. Or am I an idiot?
- zardo 4y agoThere's room for an infinity of closer mappings. E.g ratios of integers and square roots.
- yarg 4y agoOh damn. That's some sneaky code injection. Basically you can add a countable infinity of new countable infinite dimensions to any countably infinite set and get something that's still countably infinite. That shit still feels wrong to me.