Re: Towards a new definition of Cardinality:
zuhair
of cardinality, that was presented earlier to this Usenet
under topic: A new definition of Cardinality, see this link:
http://groups.google.com.jm/group/sci.logic/browse_thread/thread/3e92b5c2185a1450?hl=en
Cardinality was defined as the set of all equinumerous hereditary
sets.
So
cardinality(x) is the set of all hereditary sets equinumerous to x.
a hereditary set is defined as a set having every member of its
transitive closure strictly subnumerous to it.
The THEOREM that
"for every set x there exist cardinality(x)"
is proved in ZFC, but it is not settled yet weather it requires
choice or regularity or neither.
This matter depends on the question weather the existence of
the set of all hereditarily<n sets (a hereditarily<n set is a set that
is strictly subnumerous to n having every member of its transitive
closure strictly subnumerous to n), requires choice or regularity or
neither.
My personal guess is that we don't need choice to prove the above
theorem, however I am not sure of the demand for regularity, my
initial thought was that it doesn't, but I am not sure. This matter is
still not settled yet.
However in this thread I would like to discuss another definition that
seems to be interesting to me at least, which is the following:
"A cardinal is a class of all hereditarily hereditary sets strictly
subnumerous to some set".
To explain this definition: as I said a hereditary set is a set having
every member of its transitive closure strictly subnumerous to it
in symbols:
Define(hereditary):
x is hereditary <->
for all y ( y e Tc(x) -> y strictly subnumerous to x )
Were Tc(x) stands for the 'transitive closure of x' defined
in the standard manner.
Tc(x)=U{x,Ux,UUx,UUUx,......}
were "U" stands for "union" as define in Z set theory.
Now we come to define "hereditarily hereditary"
Define(hereditarily hereditary):
x is hereditarily hereditary <->
(x is hereditary & for all y ( y e Tc(x) -> y is hereditary )).
So a hereditarily hereditary set is a hereditary set were
every member of its transitive closure is also hereditary.
Now we come to define the cardinality of x.
c is said to be the cardinality of x if and only if c is the set of
all hereditarily hereditary sets strictly subnumerous to x.
c=cardinality(x) <->
c={y| y is hereditarily hereditary & y strictly subnumerous to x}
I shall use the symbol "HH" to denote the predicate
"hereditarily hereditary" and the symbol "<" to denote
"strictly subnumerous" , so the above would
be written as:
cardinality(x) = c <-> c={y| HH(y) & y < x }
Define(cardinality(x)):
cardinality(x)=c <-> for all y (y e c <->(HH(y) & y < x)).
This definition gives us a nice result:
Cardinality(0) = 0
Cardinality({0}) = {0}
Cardinality( {0,{0}} ) = {0,{0}}
actually the cardinality of any finite set n would be exactly the
ordinal n that is equinumerous with n.
Actually also the cardinality of any countably infinite set would be
Omega.
In other words if we test this definition on actually any "countable"
set n, then cardinality(n) would identical to the ordinal n that is
equinumerous with n.
The most important observation of this definition on countable sets is
that the cardinality of any countable set is equinumerous to that set,
in symbols:
For all x
(x is countable -> Cardinality(Cardinality(x))=Cardinality(x))
This is an interesting result.
Lets go up further and see what happens.
The question comes what is the cardinality of Aleph_1
(Aleph_1 is the set of all countable ordinals), which is the smallest
of all uncountable sets.
The answer is that Cardinality of Aleph_1 would be the set of all
hereditarily hereditary sets strictly subnumerous to Aleph_1, but what
are these sets?
These sets would be all finite ordinals and also all
Infinite subsets of Omega.
so
Cardinality(Aleph_1) = {y| y is a finite ordinal or y subset_of w}
were "w" stands for "Omega", i.e the set of all finite ordinals.
Now one would understand what is the interesting point in this
definition of cardinality, we see clearly that Cardinality of Aleph_1
is equinumerous to Power(omega), because it is very clear that the set
of *all* infinite subsets of omega is equinumerous to
power omega (we don't need choice nor continuum hypothesis to prove
that), so the set of all infinite subsets of omega and finite ordinals
would be of course a subset of power omega that is equinumerous to
power omega.
We saw that regarding countable sets, there cardinalities are
equinumerous to them, so if this rule extends beyond countable sets,
then actually this would prove that
Aleph_1 is equinumerous to power omega, which is very interesting.
However I personally don't have a proof of the following theorem:
For all x (Cardinality(Cardinality(x))=Cardinality(x))
But just in case this can be proved from ZFC, then this would be
enough to prove that power omega is equinumerous to Aleph_1.
However I greatly doubt that, since I think that the continuum
hypothesis is proved to be independent from ZFC (Godel&Cohen).
Actually I think that the statement that
For all x (Cardinality(Cardinality(x))=Cardinality(x))
is equivalent to Continuum hypothesis.
Which is by itself a nice result!
The nice thing about these cardinals, is that it seems that they are
provable to exist for every set in ZF minus regularity, so they don't
need neither choice, nor regularity ( I guess ).
A nice corollary of the above is the definition of finite and
countable, we can define them in the following manner:
Define(finite):
x is finite <-> Exist! y (HH(y) & y equinumerous to x)
Define(countable):
x is countable <-> Exist! y (HH(y) & y is transitive &
y equinumerous to x)
and this is the first time I see the term "countable" defined
without reference to Omega! although definitely this can be
done easily otherwise.
So for those who like the idea of Randall Holmes of having
a small set theory which he calls "pocket set theory", in which all
sets are countable, and in which all proper classes are of the same
size, then one can have a theory with axioms schemes of
1.Extensionality (as in Z), 2.Class comprehension(as in MK), 3.Pairing
(as in MK), and 4.The axiom that: x is a set iff x is countable. were
countable is defined as above, and 5.The axiom that: all proper
classes are equinumerous.
Summary:
"A cardinal is a class of all hereditarily hereditary sets strictly
subnumerous to some set".
c is said to be the cardinality of x if and only if c is the set of
all hereditarily hereditary sets strictly subnumerous to x.
c=cardinality(x) <->
c={y| y is hereditarily hereditary & y strictly subnumerous to x}
Define(hereditary):
x is hereditary <->
for all y ( y e Tc(x) -> y strictly subnumerous to x )
Were Tc(x) stands for the 'transitive closure of x' defined
in the standard manner.
Tc(x)=U{x,Ux,UUx,UUUx,......}
were "U" stands for "union" as define in Z set theory.
Define(hereditarily hereditary):
x is hereditarily hereditary <->
(x is hereditary & for all y ( y e Tc(x) -> y is hereditary )).
Zuhair
zuhair
> This topic is a continuation to my approach towards a new definition
> of cardinality, that was presented earlier to this Usenet
> under topic: A new definition of Cardinality, see this link:
>
possibly stronger! which would be even more interesting!
Tegiri Nenashi
> Were Tc(x) stands for the 'transitive closure of x' defined
> in the standard manner.
>
> Tc(x)=U{x,Ux,UUx,UUUx,......}
>
> were "U" stands for "union" as define in Z set theory.
This doesn't look (aesthetically) right to me. What about more
symmetric version
TC(x)=U{x,U{x,U{x,U{x,......}}}
?
zuhair
The complete way to write it is;
TC(x)=U{ x , U{x} , U{U{x}} , U{U{U{x}}} ,......}
Zuhair
zuhair
> This topic is a continuation to my approach towards a new definition
> of cardinality, that was presented earlier to this Usenet
> under topic: A new definition of Cardinality, see this link:
>
Correction:
Cardinality(Aleph_1) = {y| y is a finite ordinal or y is infinite