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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 1:16 PM

On Jan 6, 11:07 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
> (1) |x| > |y| does not imply |y| < |x|;
> (2) |x| = |y| does not imply |y| = |x|.

(1) is correct but (2) if False. Don't forget I said "Characterize"
and not "define". The above three statements are AXIOMS to be added to
Z or ZF and "| |" here is a PRIMITIVE one place function symbol, also
Identity theory is assumed here. Of course all of =,< and > are
primitive binary relation symbols also. In reality I'm not so sure if
some sets can be bi-surjective but not bijective in this extension of
Z or ZF, but if it can then this would supply the basis for (1) being
correct. Also accordingly we would have |x| > |y| and |y| > |x|. But
with IDENTITY THEORY (2) is forbidden.

Zuhair

Zuhair