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

 Messages: [ Previous | Next ]
 Zaljohar@gmail.com Posts: 2,665 Registered: 6/29/07
Re: A size criterion: a question
Posted: Jan 7, 2013 1:18 PM

On Jan 7, 10:21 am, Butch Malahide <fred.gal...@gmail.com> wrote:
> On Jan 6, 2:07 pm, Butch Malahide <fred.gal...@gmail.com> wrote:
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> > On Jan 6, 1:35 pm, Zuhair <zaljo...@gmail.com> wrote:
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> > > 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.

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> > > |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.

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> > > |x| = |y| iff there exist an injection from x to y and there exist a
> > > surjection from a subset of x to y.

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> > 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|.

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> In fact, it's consistent with ZF that there are sets x and y such that
> both |x| > |y| and |y| = |x|.

True, but not in this extension of ZF!

Zuhair