Date: Mar 12, 2013 3:46 PM
Author: Zaljohar@gmail.com
Subject: Re: Reducing Incomparability in Cardinal comparisons

On Mar 12, 2:49 pm, Zuhair <zaljo...@gmail.com> wrote:
> On Mar 11, 11:17 pm, Zuhair <zaljo...@gmail.com> wrote:
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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.

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> > Define: |x|=|y| iff  x<-bij->y
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> > Define: |x| < |y| iff  x-inj->y Or Rank(|x|) -inj-> Rank(|y|)
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> > Define: |x| > |y| iff |y| < |x|
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> > Define: |x| incomparable to |y| iff ~|x|=|y| & ~|x|<|y| & ~|x|>|y|
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> > where |x| is defined after Scott's.
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> > 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.

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> > However the relation is not that natural at all.
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> > Zuhair
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> One can also use this relation to define cardinals in ZF.
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> |x|={y| for all z in TC({y}). z <* x}
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> Of course <* can be defined as:
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> x <* y iff [x -inj->y Or
> Exist x*. x*<-bij->x & for all y*. y*<-bij->y -> rank(x*) in
> rank(y*)].
>
> Zuhair


All the above I'm sure of, but the following I'm not really sure of:

Perhaps we can vanquish incomparability altogether

If we prove that for all x there exist H(x) defined as the set of all
sets hereditarily not strictly supernumerous to x. Where strict
subnumerousity is the converse of relation <* defined above.

Then perhpas we can define a new Equinumerousity relation as:

x Equinumerous to y iff H(x) bijective to H(y)

Also a new subnumerousity relation may be defined as:

x Subnumerous* to y iff H(x) injective to H(y)

This might resolve all incomparability issues (I very highly doubt
it).

Then the Cardinality of a set would be defined as the set of all sets
Equinumerous to it of the least possible rank.

A Scott like definition, yet not Scott's.

Zuhair