[FOM] Cardinality in weaker set theories

[T.Forster@dpmms.cam.ac.uk]

On Jan 18 2011, Colin McLarty wrote:

>Thanks to everyone for helpful answers.
>
>I am not sure how to apply the Scott trick to get a set of
>representatives of each cardinality, without replacement.  I think the
>key issue is whether you can prove that when all elements of a set S
>have well defined ordinal rank then there is a supremum to their
>ranks, so that S also has well defined rank.
>


I suspect the answer to this might depend on whether you are thinking
of ordinals as von Neumann ordinals, or perhaps have some other 
implementation in mind.   It might be that, for what you are doing, you 
don't need a single global implementation of ordinal arithmetic, but will
be satisfied with local implementations.  We might have to look very very 
closely at the proof you are trying to formalise.


Scott's trick won't give you *representatives*; what it gives you is an
*implementation of cardinals*.  The Scott's trick cardinal |x| is not the
same size as x.  But you know all that.  If you want representatives you 
have to use choice, rather than replacement.  Let X be a random large set.
Quotient out the power set of X by equipollence and use choice to pick
representatives from the quotient. Thus you can have `local' families of 
representatives.  You *might* want these families to *cohere* in nice 
ways, and then you might have to start worrying about whether or not every 
set has a transitive superset, and that is surely not a theorem of ZC.

Executive summary;

In ZC one cannot define a two-place relation R(x,y) with the property
that for all x there is a unique y s.t. R(x,y) and y is the same size
as x. I think this remains true even if ZC includes foundation, and this 
is because ZC doesn't prove that every set has a canonical transitive 
superset (secretly the fist V_\alpha to which it belongs).  This last
fact means that Scott's trick cardinals (or Scott's trick anything-else 
for that matter) are not available *globally*, though they are often 
available *locally*.


>_______________________________________________
>FOM mailing list
>FOM at cs.nyu.edu
>http://www.cs.nyu.edu/mailman/listinfo/fom
>