
Re: A size criterion: a question
Posted:
Jan 7, 2013 2:21 AM


On Jan 6, 2:07 pm, Butch Malahide <fred.gal...@gmail.com> wrote: > On Jan 6, 1:35 pm, Zuhair <zaljo...@gmail.com> 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. > > In ZFC there is no difference at all. In ZF there are some striking > differences: under your definition, > (1) x > y does not imply y < x; > (2) x = y does not imply y = x.
In fact, it's consistent with ZF that there are sets x and y such that both x > y and y = x. Also, there can be sets x and y such that x > y and y > x.

