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