Date: Jan 6, 2013 3:33 PM
Author: fom
Subject: Re: A size criterion: a question
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

>

> =================================

> "|x| = |y| if there exists a one-to-one mapping of X onto Y"

>

> injection + surjection

> ===================================

> "|x| <= |y| if there is a one-to-one mapping of X into Y"

>

> injection, surjection undetermined

> ====================================

> "|x| < |y| <-> ((|x| <= |y|) /\ -(|x| = |y|))"

>

> injection, no surjection

>

>