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Math Forum » Discussions » sci.math.* » sci.math.independent

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 3:07 PM
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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|.

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