a) I’m a little confused by the discussion of Cantor’s argument. As I understand it, the argument is valid in first-order logic, it’s just that the conclusion may have different semantics in different models. That is, the statement “the set X is uncountable” is cashed out in terms of set theory, and so if you have a non-standard model of set theory, then that statement may have non-standard sematics.
More clearly—“X is uncountable” means “there is no bijection between X and a subset of N”, but “there” stilll means “within the given model”.
Exactly (I’m assuming by subset you mean non-strict subset). Crucially, a non-standard model may not have all the bijections you’d expect it to, which is where EY comes at it from.
More clearly—“X is uncountable” means “there is no bijection between X and a subset of N”, but “there” stilll means “within the given model”.
Exactly (I’m assuming by subset you mean non-strict subset). Crucially, a non-standard model may not have all the bijections you’d expect it to, which is where EY comes at it from.
I was, but that’s not necessary—a countably infinite set can be bijectively mapped onto {2, 3, 4, …} which is a proper subset of N after all! ;-)
Oh yeah—brain fail ;)