[FOM] Cardinality beyond Scott's (cont.)
Zuhair Abdul Ghafoor Al-Johar
zaljohar at yahoo.com
Wed Jan 25 02:59:27 EST 2012
Dear FOMers
This topic come as a continuation to earlier posting
of mine by the same title posted at January 2010.
Here I shall make a correction of an error that
occurred in that post, and also re-present it in
a simpler manner.
Let me state the definition of Cardinality:
Cardinality of any set x is the set of all subsets of the first
supernumerous to x iterative power of the nearest H_kappa set to x,
that are equinumerous to x.
First we define the one place predicate "is an H_kappa set"
Def.) x is an H_kappa set:= Exist d: d is an ordinal & x=H_d
Def.) H_d= {y|y is hereditarily subnumerous to d}
P_i(z) is defined recursively as:
P_0(z)=z
P_i+1(z)=P(P_i(z)) for every ordinal i
P_i(z)=UP_j(z),j<i for every limit ordinal i
Each P_i(z) is termed as an "iterative power of z"
Now an H_kappa set is said to be "in the vicinity of x"
only if there exist an ordinal i such that there exist a set y
such that y=P_i(H_kappa) and y is supernumerous to x.
Now for any H_kappa set in the vicinity of x we define
the two place function "distance" denoted by D as:
Def.) D(x,H_kappa)= min i (P_i(H_kappa) is supernumerous to x)
Now we define the two place predicate "at the least possible
distance from x":
Def.) H_kappa is at the least possible distance from x:=
H_kappa in the vicinity of x and
for all lambda: H_lambda in the vicinity of x ->
D(x,H_kappa) =< D(x,H_lambda)
Now for some x we may have many H_kappa sets each of which is at the
least possible distance from x, now the first one of those (i.e. the
one with the least kappa value) is the "nearest H_kappa set to x".
Now the first supernumerous to x iterative power of H_kappa stands
for the first set y such that y is an iterative power of H_kappa and
y is supernumerous to x.
This completes the explanation of this definition of Cardinality
which is indeed too complex.
The requirement for this definition to work is that for every set
x there must exist at least one H_kappa set in the vicinity of x.
Which is a theorem of ZF.
So this definition works in ZF. However if we axiomatize the above
requirement instead of regularity, then I think we can have a model
of the resulting theory in which there exist an H_kappa set that is
not equinumerous to any well founded set! and if this can be proved,
then this definition would prove to be stronger than Scott's
definition of Cardinality in the sense that it can assign cardinality
to sets that Scott's cardinals cannot while the opposite cannot be
true.
Best regards
Zuhair
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