[FOM] A new definition of Cardinality.
Zuhair Abdul Ghafoor Al-Johar
zaljohar at yahoo.com
Sun Nov 22 17:16:27 EST 2009
Hi all,
As far as I know, all the definitions of cardinality are limited in a
way or another, lets take them one after the other:
1) Von Neumann's Cardinals:
A cardinal is the least of all equinumerous ordinals.
2) Frege-Russell Cardinals:
A cardinal is an equivalence class of sets under equivalence relation
"bijection".
3) Scott-Potter Cardinals:
A cardinal is a class of all equinumerous sets from a common level.
Now lets come to discuss each one of them:
1) Von Neumann's cardinals has the limitation of being dependent on
choice, without choice one cannot know what is the cardinality of
Power(omega) for example.Accordingly in any theory which do not have
the axiom of choice among its axioms most of its sets would be of
indeterminable cardinality, which is a big draw back.
2) Frege-Russell cardinals contradict Z set theory, since their
existence would imply the existence of the set of all sets, which is
in contradiction with Z.
However in NBG and MK class theories, we can define
Frege-Russell cardinals, but by then they would be proper classes and
not sets, which is a great draw back, since proper classes cannot be
members of other classes, and they are hard to handle.
In NF and related theories, Frege-Russell cardinals are sets, but
these theories generally depend on the concept of stratification
of formulas, which is a complex concept, and even finite
axiomatization of NF and NFU and related theories is a complicated
approach, and at the end it also resort to stratification for most of
its inferences. All that make these cardinals undesirable.
3) Scott-Potter Cardinals: depend on the concept of "level" which
depends on the concept of type (Scott) and the iterative concept
(Potter), both concepts of which are complex and difficult to work
with, besides they are not the basic concepts we
use to compare set sizes.
I would like to suggest the following definition:
4) The cardinality of any set x is: The class of all sets
that are equinumerous to x were every member of their transitive
closure is strictly subnumerous to x.
So for any set x, any y is a member of the cardinality of x,
if and only if, y is equinumerous to x and every member of the
transitive closure of y is strictly subnumerous to x.
In symbols:
Define(cardinality(x)):-
z=cardinality(x) <->
for all y (y e z <->
(y equi-numerous to x &
for all m (m e Tc(y)->m strictly subnumerous to x)))
Were Tc(y) stands for the 'transitive closure of y' defined
in the standard manner.
Tc(y)=U{y,Uy,UUy,UUUy,......}
We can actually better define these cardinals through defining the
concept of "hereditary sets"
Define(hereditary):
x is hereditary <->
for all y (y e Tc(x) -> y strictly subnumerous to x)
So a cardinal can be defined in the following manner:
A Cardinal is an equivalence class of hereditary sets under
equivalence relation "bijection".
Or simply
A Cardinal is a class of all equinumerous hereditary sets.
So cardinality of x would be written shortly as:
Cardinality(x) = {y| y is hereditary & y equinumerous to x}
Now it can be proven in ZF that those cardinals would be 'sets', so
they are not proper classes! which makes them easy to handle.
These cardinals don't require choice.
They don't require complex concepts like "stratification,type,
iteration"
They simply depend on the basic concept used to compare set sizes,
which is the presence or absence of injections between the compared
sets.
To me this definition seems to be simpler, more general, and it works
with or without choice, with or without regularity.
So at the end I shall write the definition of cardinal again:
A Cardinal is an equivalence class of hereditary sets under
equivalence relation "bijection".
Or simply:
A Cardinal is a class of all equinumerous hereditary sets.
x is hereditary <->
for all y (y e Tc(x) -> y strictly subnumerous to x)
Cardinality(x) = {y| y is hereditary & y equinumerous to x}
Zuhair
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