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

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 Butch Malahide Posts: 894 Registered: 6/29/05
Re: A size criterion: a question
Posted: Jan 6, 2013 3:46 PM

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.

>
> >> |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
>
> > from Jech on ZF,
>
> 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.