> You'll see that this definition corresponds exactly to the usual notions of size for finite sets, and makes intuitive sense.
This has always been my problem with the idea of some infinities being "bigger" than others. This definition does not correspond to the usual notion of size for finite sets, because finite sets actually have size. So rather than making "intuitive sense", it makes no sense (to me).
Having no surjective function from one set to another makes sense, but defining one set necessarily as "less than" the other does not, as the word "less" cannot, in such a context, mean what its definition implies.
> (if for every y we have an x, there must be at least as many xs as ys)
The lack of a surjective function does not mean that we cannot have an X for every Y, it just means we can't map an X for every Y with a surjective function. It doesn't mean a pairing cannot exist (at least not abstractly, which is how all infinite sets exist anyway). The pairing can simply be random and undefinable. There, now a pairing exists and both sets are the same "size". (It's not a surjective function, but if the point is simply to compare "sizes", then what does it matter?)
This has always been my problem with the idea of some infinities being "bigger" than others. This definition does not correspond to the usual notion of size for finite sets, because finite sets actually have size. So rather than making "intuitive sense", it makes no sense (to me).
Having no surjective function from one set to another makes sense, but defining one set necessarily as "less than" the other does not, as the word "less" cannot, in such a context, mean what its definition implies.
> (if for every y we have an x, there must be at least as many xs as ys)
The lack of a surjective function does not mean that we cannot have an X for every Y, it just means we can't map an X for every Y with a surjective function. It doesn't mean a pairing cannot exist (at least not abstractly, which is how all infinite sets exist anyway). The pairing can simply be random and undefinable. There, now a pairing exists and both sets are the same "size". (It's not a surjective function, but if the point is simply to compare "sizes", then what does it matter?)