[FOM] A new definition of Cardinality.
T.Forster at dpmms.cam.ac.uk
Wed Dec 2 04:51:53 EST 2009
``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
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)))
>x subnumerous to y <-> Exist f (f:x-->y, f is injective)
>x equinumerous to y <-> Exist f (f:x-->y, f is bijective)
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