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

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 Zaljohar@gmail.com Posts: 2,665 Registered: 6/29/07
Reducing Incomparability in Cardinal comparisons
Posted: Mar 11, 2013 4:17 PM

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

Date Subject Author
3/11/13 Zaljohar@gmail.com
3/12/13 Zaljohar@gmail.com
3/12/13 Zaljohar@gmail.com
3/13/13 Zaljohar@gmail.com
3/13/13 Zaljohar@gmail.com
4/7/13 Charlie-Boo