In article <firstname.lastname@example.org>, Butch Malahide <email@example.com> writes: >On Jan 6, 2:07=A0pm, Butch Malahide <fred.gal...@gmail.com> wrote: >> On Jan 6, 1:35=A0pm, 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|.
Is the last bit why trichotomy requires Choice?
(I promise never to call Choice "unintuitive" again.)
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