On Jan 6, 11:07 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|.
(1) is correct but (2) if False. Don't forget I said "Characterize" and not "define". The above three statements are AXIOMS to be added to Z or ZF and "| |" here is a PRIMITIVE one place function symbol, also Identity theory is assumed here. Of course all of =,< and > are primitive binary relation symbols also. In reality I'm not so sure if some sets can be bi-surjective but not bijective in this extension of Z or ZF, but if it can then this would supply the basis for (1) being correct. Also accordingly we would have |x| > |y| and |y| > |x|. But with IDENTITY THEORY (2) is forbidden.