# [FOM] A new definition of Cardinality.

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Sat Nov 28 16:38:56 EST 2009

```Just few points I wanted to write.

(1) Potter did not introduce "Scott-Potter" cardinals.

This comment was made by Prof. Dana S. Scott

(2) Regarding the rule of Regularity with these cardinals.
The assumption that for every set x there is a set y such
that Cardinality(x)=y , requires regularity, since without
Regularity for some set x, Cardinality(x) might indeed be
a proper class, this observation is due to
Prof.Dana S. Scott , here is his comments:

-----
Without regularity (= Axiom of Foundation),
one can construct in ZF (or ZFC) a model
of ZF (resp. ZFC) where there is a PROPER CLASS of
sets x = {x}.  Call such crazy sets "nodes" for short.

Consider {x, y}, where both elements are nodes and different.
Then, by your definition, (x, y} would be a member of the
cardinality of 2.But there is a proper class of such pairs.
So your cardinal would not be a set.

Prof. Dana S. Scott
------

(3) It is not clear weather ZF (without choice) can prove the existence
the cardinals I defined.

(4) ZFC indeed proves the theorem that
for every set x there exist a set y
such that y=Cardinality(x).

(Permission to copy the notes of Prof. Dana S. Scott, was already taken    prior to posting this message)

Zuhair

```