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

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Charlie-Boo

Posts: 1,582
Registered: 2/27/06
Re: Reducing Incomparability in Cardinal comparisons
Posted: Apr 7, 2013 12:42 AM
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On Mar 13, 3:09 pm, Zuhair <zaljo...@gmail.com> wrote:
> On Mar 12, 10:46 pm, Zuhair <zaljo...@gmail.com> wrote:
>
>
>
>
>

> > 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*)].

>
> > > 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)
>
> Better would be
>
> x Subnumerous* to y iff H(x) <* H(y)
>
> however still this won't concur incomparability completely
>
> However if we define recursively H_n(x) then we can define
> the above relations after those. However still incomparability
> would persist, although the above is still a strong approach
> against incomparability.
>
>
>
>
>

> > 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- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -


Once again, just like I (and others now) said - you throw out stuff
that isn't ready, you make mistakes, you change it, you ponder how to
fix it, you debate possible solutions. It is a total moving target.
It is presented as being a solution but it is just a big problem -
trying to keep it straight as you move things around in a desperate
attempt to clean up the mess.

C-B



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