Date: Mar 12, 2013 7:49 AM
Author: Zaljohar@gmail.com
Subject: Re: Reducing Incomparability in Cardinal comparisons

On Mar 11, 11:17 pm, Zuhair <zaljo...@gmail.com> wrote:
> Let x-inj->y stands for there exist an injection from x to y and there
> do not exist a bijection between them; while x<-bij-> means there
> exist a bijection between x and y.
>
> Define: |x|=|y| iff  x<-bij->y
>
> Define: |x| < |y| iff  x-inj->y Or Rank(|x|) -inj-> Rank(|y|)
>
> Define: |x| > |y| iff |y| < |x|
>
> Define: |x| incomparable to |y| iff ~|x|=|y| & ~|x|<|y| & ~|x|>|y|
>
> where |x| is defined after Scott's.
>
> Now those are definitions of what I call "complex size comparisons",
> they are MORE discriminatory than the ordinary notions of cardinal
> comparisons. Actually it is provable in ZF that for each set x there
> exist a *set* of all cardinals that are INCOMPARABLE to |x|. This of
> course reduces incomparability between cardinals from being of a
> proper class size in some models of ZF to only set sized classes in
> ALL models of ZF.
>
> However the relation is not that natural at all.
>
> Zuhair


One can also use this relation to define cardinals in ZF.

|x|={y| for all z in TC({y}). z <* x}

Of course <* can be defined as:

x <* y iff [x -inj->y Or
Exist x*. x*<-bij->x & for all y*. y*<-bij->y -> rank(x*) in
rank(y*)].

Zuhair