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Topic: A size criterion: a question
Replies: 15   Last Post: Jan 9, 2013 1:22 PM

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Michael Stemper

Posts: 671
Registered: 6/26/08
Re: A size criterion: a question
Posted: Jan 9, 2013 1:22 PM
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In article <6f554bbd-8c30-4e0a-8bea-1fd15d8d427f@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
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