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Topic: Reducing Incomparability in Cardinal comparisons
Replies: 5   Last Post: Apr 7, 2013 12:42 AM

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Posts: 2,665
Registered: 6/29/07
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.


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