3 ms·
> The proper definition of ‘countably infinite’ is simply that the set can be placed in bijection with the natural numbers. This has nothing (directly) to do wi
by LegionMammal978 2y ago
> The proper definition of ‘countably infinite’ is simply that the set can be placed in bijection with the natural numbers. This has nothing (directly) to do with ordering.
Does it not imply that there exists an ordering which reaches each value in finite time? That's how I'd straightforwardly read GP's comment. A given countable set under its 'natural' ordering might not have order type ω, but there will always exist some alternative ordering with type ω.
(And indeed, the existence of such an ordering implies that the set is countable.)
- xanderlewis 2y agoYes. You're correct. I realised this some time after writing my comment. So it actually does work as a definition, if you phrase it properly. I guess it's just equivalent to the observation that ω has the property that every initial segment (i.e. every subset of the form {0, 1, ..., n}) is finite. So you could say: a set S is countable if it admits an ordering that makes it 'covered by all finite initial segments'. ...but this boils down to saying the set admits an ordering that makes it isomorphic to some n, or ω... which just means it's countable in the usual sense.