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Reducing Incomparability in Cardinal comparisons
Posted:
Mar 11, 2013 4:17 PM
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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
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