```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.>> ZuhairOne 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 OrExist x*. x*<-bij->x & for all y*. y*<-bij->y -> rank(x*) inrank(y*)].Zuhair
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