On 16 Jun., 00:37, PotatoSauce <kiwisqu...@gmail.com> wrote: > On Friday, June 15, 2012 4:54:37 PM UTC-4, WM wrote: > > > Learn what a bijection is. > > > I know how it is defined. > > Obviously you don't, since you go on to say: > > > But it is nonsense, nevertheless. > > For every x you take there are infinitely many you don't take. > > The definition of surjectivity is, "for every x in the codomain, there is a preimage."
The expression: "Every x" is meaningless for an infinite set. > > That is all that means. Nothing more, nothing less.
I know. Nevertheless, the expression: "Every x" is meaningless for an infinite set.
> > You are trying to bring in some weird philosophical interpretation to simple words, finding a contradiction, then complaining about it.
No I am trying to teach you that Cantor needs limits.
If we cover every rational of the unit interval by an interval I_n of length 10^-n, then we get, for every n, at most n Intervals and at most n intervals in the complement. If there was no limit, there could never appear a completely disconnected space in the limit. Look here what a set theorist has to say about that: "I did not talk about what happens at the k -th stage. I talked about what happens after all the finite stages)" ? Asaf Karagila http://math.stackexchange.com/questions/132022/formalizing-an-idea
What is after all finite stages, if not the limt?
Unless we go to the limit, there is no chance to have a cardinality aleph_0 of a set that contains only finite numbers 2n, i.e. that in every finite case contains a number that is larger than its cardinality.