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Concurring incomparability
Posted:
Mar 14, 2013 10:04 AM
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A function F is to be called a Supra-Cardinality Discriminative
Function "SCDF" iff it satisfies the followings:
|x|<|y| ->F(x)<F(y)
|x|>|y| ->F(x)>F(y)
|x|=|y| ->F(x)=F(y)
Of course in ZFC cardinality is itself also an SCDF and no SCDF can be
more discriminative than cardinality. However in ZF alone we may have
SCDFs that have more discriminative power than cardinality.
Example of that is any function F, to be called as quasi-cardinality,
with the following characterizations.
F(x)=F(y) iff |x|=|y|
F(x)<F(y) iff |x|<|y| Or Rank(|x|)<Rank(|y|)
F(x)>F(y) iff F(y)<F(x)
Those functions do not allow incomparability across ranks! However
within the same rank we can have only set amount of incomparable quazi-
cardinals.
To lessen incomparability which as said only occurs within the same
rank, then we can define a new SCDF, that is F* as follows:
F*(x)=F*(y) iff F(H(x)) = F(H(y))
F*(x)<F*(y) iff F(H(x))<F(H(y))
F*(x)>F*(y) iff F*(y)<F*(x)
where H(x) = {y|for all z. z in TC({y}) ->F(z)<F(x)}
Of course H(x) itself behaves as a cardinal in ZF with or without
choice!
Anyhow this doesn't resolve the issue of incomparability completely. A
more harsher approach is to define H_i(x) recursively in a cumulative
manner and then define a function F** as:
F**(x)=F**(y) iff Exist i. F(H_i(x)) = F(H_i(y))
F**(x)<F**(y) iff Exist i.F(H_i(x))<F(H_i(y))
F**(x)>F**(y) iff F**(y)<F**(x)
However this still do not succeed in vanquishing all incomparability
in F** function in ZF.
Zuhair
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