Para-Cardinality:
zuhair
I would like to introduce a new concept that I called as
"para-cardinality".
Para-Cardinality is not Cardinality, so they must not be confused.
Para-Cardinality shall be denoted by the symbol | |"
Define( |x|" ):
A=|x|" <->
for all y (y e A <-> Exist u,z( u strictly subnumerous to x &
z=|u|" & y subset of z & y equinumerous to z)).
From Extensionality |x|" would be unique for each set x.
Now I am not sure if the following is a theorem of ZF minus Reg.
For all x Exist y ( y=|x|" )
If it is not, then this must be axiomatized in order to define
Para-Cardinality.
The essence of this is to destroy incomparability of set size in
ZF i.e. outside Choice.
The basic idea is that for every two sets x and y, either
|x|" is a subset of |y|" or |y|" is a subset of |x|",
so para-cardinalities are always comparable.
Now we define the following characteristics of para-cardinalities:
(1) para-cardinality(x) <= para-cardinality(y) iff
|x|" subset of |y|".
(2) para-cardinality (x) >= para-cardinality(y) iff
|y|" subset of |x|".
Here there is no incomparability of para-cardinalities,
that is unlike the case of cardinality outside choice.
Now we have the following relationship with Cardinality:
(1) |x| = |y| -> |x|" = |y|"
(2) |x| < |y| -> |x|" < |y|"
(3) |x| > |y| -> |x|" > |y|"
I am not sure if para-cardinality can help in reaching into
the most possible general definition of cardinality in
ZF-Reg.
Zuhair
zuhair
> Hi all,
>
> I would like to introduce a new concept that I called as
> "para-cardinality".
>
> Para-Cardinality is not Cardinality, so they must not be confused.
>
> Para-Cardinality shall be denoted by the symbol | |"
>
> Define( |x|" ):
>
> A=|x|" <->
> for all y (y e A <-> Exist u,z( u strictly subnumerous to x &
> z=|u|" & y subset of z & y equinumerous to z)).
>
> From Extensionality |x|" would be unique for each set x.
>
> Now I am not sure if the following is a theorem of ZF minus Reg.
>
> For all x Exist y ( y=|x|" )
>
> If it is not, then this must be axiomatized in order to define
> Para-Cardinality.
>
An important issue: I am also not sure if this definition
can work in ZF, i.e. with Regularity.
I think the issue might depend on weather
we can have a well founded para-cardinality
of a well founded set that is both Tarski infinite
and Dedekind finite? definitely it's
para- cardinality would be *not well orderable*,
but is it consistent that it is well
founded also?
Zuhair
zuhair
>
> I would like to introduce a new concept that I called as
> "para-cardinality".
>
> Para-Cardinality is not Cardinality, so they must not be confused.
>
> Para-Cardinality shall be denoted by the symbol | |"
>
> Define( |x|" ):
>
> A=|x|" <->
> for all y (y e A <-> Exist u,z( u strictly subnumerous to x &
> z=|u|" & y subset of z & y equinumerous to z)).
>
> From Extensionality |x|" would be unique for each set x.
>
> Now I am not sure if the following is a theorem of ZF minus Reg.
>
> For all x Exist y ( y=|x|" )
>
> If it is not, then this must be axiomatized in order to define
> Para-Cardinality.
>
> The essence of this is to destroy incomparability of set size in
> ZF i.e. outside Choice.
>
> The basic idea is that for every two sets x and y, either
> |x|" is a subset of |y|" or |y|" is a subset of |x|",
> so para-cardinalities are always comparable.
On second look, I am actually no sure
of this, perhaps if these para-cardinalities
can work in ZF, then perhaps this can be
proved.But I don't know if it can
be proved in ZF minus Reg.
>
> Now we define the following characteristics of para-cardinalities:
>
> (1) para-cardinality(x) <= para-cardinality(y) iff
> |x|" subset of |y|".
>
> (2) para-cardinality (x) >= para-cardinality(y) iff
> |y|" subset of |x|".
>
> Here there is no incomparability of para-cardinalities,
> that is unlike the case of cardinality outside choice.
If there can be a case were ~ |x|" subset of |y|"
and ~ |y|" subset of |x|" , then we can
indeed have incomparability between para-cardinalities.
Zuhair