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Re: Reducing Incomparability in Cardinal comparisons
Posted:
Mar 12, 2013 3:46 PM


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



