On Jan 6, 2:33 pm, fom <fomJ...@nyms.net> wrote: > On 1/6/2013 2:22 PM, fom wrote: > > On 1/6/2013 1:35 PM, Zuhair wrote: > >> If we characterize cardinality in the following manner, How much that > >> would differ from the known cardinality: > > >> |x| < |y| iff there exist an injection from x to y and there do not > >> exist a surjection from a subset of x to y. > > >> |x| > |y| iff there exist a surjection from a subset of x to y and > >> there do not exist an injection from x to y. > > >> |x| = |y| iff there exist an injection from x to y and there exist a > >> surjection from a subset of x to y. > > > it appears to be equivalent > > > from Jech on ZF, > > Yep, missed a C > > ZFC
The definitions you quoted below are the same definitions one would use in ZF, that's not the issue.
> > ================================= > > "|x| = |y| if there exists a one-to-one mapping of X onto Y" > > > injection + surjection > > ===================================
However, to say that "there exists a one-to-one mapping of X onto Y" means that there is a mapping from X to Y which is both injective and surjective; this is stronger than saying "there exists an injection from x to y and there exists a surjection from x to y."
For example, consider a model of ZF in which no uncountable set of real numbers is well-orderable. Let u be the set of all real numbers, v the set of all countable ordinals, w the union of u and v. Then there is an injection from u to w, and there is a surjection from u to w, but there is no bijection from u to w, because there is no injection from v to u.