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Re: A size criterion: a question
Posted:
Jan 6, 2013 7:14 PM
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On Jan 6, 6:12 pm, Butch Malahide <fred.gal...@gmail.com> wrote:
> On Jan 6, 2:46 pm, Butch Malahide <fred.gal...@gmail.com> wrote:
>
>
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>
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> > On Jan 6, 2:33 pm, fom <fomJ...@nyms.net> wrote:
>
> > > On 1/6/2013 2:22 PM, fom wrote:
> > > > On 1/6/2013 1:35 PM, Zuhair 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.
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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.
>
> > > > it appears to be equivalent
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> > > > from Jech on ZF,
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> > > Yep, missed a C
>
> > > ZFC
>
> > The definitions you quoted below are the same definitions one would
> > use in ZF, that's not the issue.
>
> > > > =================================
> > > > "|x| = |y| if there exists a one-to-one mapping of X onto Y"
>
> > > > injection + surjection
> > > > ===================================
>
> > However, to say that "there exists a one-to-one mapping of X onto Y"
> > means that there is a mapping from X to Y which is both injective and
> > surjective; this is stronger than saying "there exists an injection
> > from X to Y and there exists a surjection from X to Y."
>
> > For example, consider a model of ZF in which no uncountable set of
> > real numbers is well-orderable. Let u be the set of all real numbers,
> > v the set of all countable ordinals, w the union of u and v. Then
> > there is an injection from u to w, and there is a surjection from u to
> > w, but there is no bijection from u to w, because there is no
> > injection from v to u.
>
> In this example, according to the definitions proposed by the original
> poster, we have both |u| = |w| and |w| > |u|.
[Typo corrected.]
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