On Jun 16, 11:46 am, WM <mueck...@rz.fh-augsburg.de> wrote: > On 16 Jun., 18:07, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> > it's not > > uncommon for people to mistakenly think, in a classical context, that > > countable sets come equipped with a designated bijection witnessing > > their countability. > > The proof of countability of S does not require a bijection of S with > N but only an injection of some superset of S into N.
He didn't say "bijection with N" (I take it by N you mean the set of natural numbers or perhaps the set of positive natural numbers).
But, every countable set does have a bijection with N or has a bijection with some member of w (the set of natural numbers). Aatu's point here is that in classical mathematics the definition of 'x is countable' is 'there exists a bijection from x onto the set of natural numbers or onto a natural number', and that definition does not include a clause that we (me, you, whoever) KNOW of a particular bijection that we can specifically mention or define.