
Re: A size criterion: a question
Posted:
Jan 9, 2013 1:22 PM


In article <6f554bbd8c304e0a8bea1fd15d8d427f@10g2000yqk.googlegroups.com>, Butch Malahide <fred.galvin@gmail.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.)
 Michael F. Stemper #include <Standard_Disclaimer> This email is to be read by its intended recipient only. Any other party reading is required by the EULA to send me $500.00.

