```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:>>>>>>>>>> > 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>> One can also use this relation to define cardinals in ZF.>> |x|={y| for all z in TC({y}). z <* x}>> Of course <* can be defined as:>> x <* y iff [x -inj->y Or> Exist x*. x*<-bij->x & for all y*. y*<-bij->y -> rank(x*) in> rank(y*)].>> ZuhairAll the above I'm sure of, but the following I'm not really sure of:Perhaps we can vanquish incomparability altogetherIf we prove that for all x there exist H(x) defined as the set of allsets hereditarily not strictly supernumerous to x. Where strictsubnumerousity 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 doubtit).Then the Cardinality of a set would be defined as the set of all setsEquinumerous to it of the least possible rank.A Scott like definition, yet not Scott's.Zuhair
```