[FOM] A new definition of Cardinality.

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

So: 
   ``for all x there is y the same size as x whose transitive closure is of
    minimal size among y the same size as x.''  

My guess is that this is a weak choice principle. I'll look into it and 
report back.



On Dec 1 2009, Zuhair Abdul Ghafoor Al-Johar wrote:

>Dear Mr. Forster:
>
> You seem to be right regarding the concept of not every set is 
> equinumerous to a well founded set. However the way how I see matters, is 
> that every set MUST have a cardinality that is a set despite choice and 
> regularity. We need to find a general definition of cardinality in ZF 
> minus Regularity (and without choice).
>
> However as a first step, Scott's trick is acceptable, for at least it 
> defines cardinality for all well founded sets despite choice.
>
> By the way, I have the following question in my mind, which might be 
> related to Scott's trick?
>
>We know from ZF that for every set x there exist a transitive closure 
>TC(x) that is a set. 
>
>Now does ZF(with Regularity of course)
>prove or refute the following? 
>
>For all x Exist y 
>(y equinumerous to x  and
>not Exist z (z equinumerous to x and TC(z) strictly subnumerous to TC(y))) 
>
>were 
>
>x subnumerous to y  <-> Exist f (f:x-->y, f is injective) 
>x equinumerous to y <-> Exist f (f:x-->y, f is bijective) 
>
>
>Zuhair 
>
>
>
>
>
>      
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