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


Let xinj>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 xinj>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



