On Jan 6, 6:12 pm, Butch Malahide <fred.gal...@gmail.com> wrote: > On Jan 6, 2:46 pm, Butch Malahide <fred.gal...@gmail.com> wrote: > > > > > > > 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. > > In this example, according to the definitions proposed by the original > poster, we have both |u| = |w| and |w| > |u|.