The set of all hereditarily x cardinality set.
zuhair
The set of all hereditarily finite sets is provable to exist in
ZFC,but not in Z.
I have a question: for any cardinal x , is the set of all hereditarily
x cardinality provable to exist in ZFC.
To be more precise:
Define: x is hereditarily z cardinality <-> [cardinality of x is
lower or equal to z and every member of the transitive closure of x
has a cardinality that is equal or lower than z].
Transitive closure of x is defined in the conventional manner.
Tc(x)= U{x,Ux,UUx,UUUx,...}.
So the question is, Is the following a theorem of ZFC?
Az( z is a cardinal ->
ExAy ( yex <-> ( |y|<=z & Am(meTc(y)->|m|<=z ) ) ).
Zuhair
zuhair
The question can be simplified to the following:
Define: x is hereditarily sub_ cardinality z <-> [cardinality of x is
lower than z and every member of the transitive closure of x
has a cardinality that is lower than z].
So the question is: Is the following a theorem in ZFC:
Az( z is a cardinal ->
ExAy ( yex <-> ( |y|<z & Am(meTc(y)->|m|<z ) ) ).
So for example:
x is hereditarily sub_aleph_0 cardinality <-> x is hereditarily
finite.
x is hereditarily sub_aleph_1 cardinality <-> x is hereditarily
countable.
so a set is hereditarily sub_aleph_x cardinality means that its
cardinality is less than aleph_x and the cardinality of every member
of its transitive closure is less than aleph_x.
Zuhair
Aatu Koskensilta
> I have a question: for any cardinal x , is the set of all hereditarily
> x cardinality provable to exist in ZFC.
Yes. Recall the inductive definition of the hereditarily finite sets:
o {} is a hereditarily finite set
o If A1, ..., An are hereditarily finite sets, so is {A1, ..., An}
We obtain the sets hereditarily of cardinality < kappa as a simple
generalisation of this inductive definition:
o {} is hereditarily of cardinality < kappa
o If A is a set of sets hereditarily of cardinality < kappa
and |A| < kappa then union of A is hereditarily of cardinalty
< kappa
(The first clause is redundant of course.)
By standard treatment of inductive definitions in ZFC, there is now a
least set closed under these rules, which is the set of sets
hereditarily of cardinality < kappa. Denoting the set of sets
hereditarily < kappa by X<kappa, we have then that the hereditarily
finite sets are the X<omega, the hereditarily countable sets are the
X<omega_1 and so on.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
(I cancelled an earlier version of this article, which contained a silly
error.)
> I have a question: for any cardinal x , is the set of all hereditarily
> x cardinality provable to exist in ZFC.
Yes. Recall the inductive definition of the hereditarily finite sets:
o {} is a hereditarily finite set
o If A1, ..., An are hereditarily finite sets, so is {A1, ..., An}
We obtain the sets hereditarily of cardinality < kappa as a simple
generalisation of this inductive definition:
o {} is hereditarily of cardinality < kappa
o If A is a set of sets hereditarily of cardinality < kappa and |A| <
kappa then A is hereditarily of cardinalty < kappa
zuhair
> (I cancelled an earlier version of this article, which contained a silly
> error.)
>
> > I have a question: for any cardinal x , is the set of all hereditarily
> > x cardinality provable to exist in ZFC.
>
> Yes. Recall the inductive definition of the hereditarily finite sets:
>
> o {} is a hereditarily finite set
> o If A1, ..., An are hereditarily finite sets, so is {A1, ..., An}
>
> We obtain the sets hereditarily of cardinality < kappa as a simple
> generalisation of this inductive definition:
>
> o {} is hereditarily of cardinality < kappa
>
> o If A is a set of sets hereditarily of cardinality < kappa and |A| <
> kappa then A is hereditarily of cardinalty < kappa
>
> (The first clause is redundant of course.)
>
> By standard treatment of inductive definitions in ZFC, there is now a
> least set closed under these rules, which is the set of sets
> hereditarily of cardinality < kappa. Denoting the set of sets
> hereditarily < kappa by X<kappa, we have then that the hereditarily
> finite sets are the X<omega, the hereditarily countable sets are the
> X<omega_1 and so on.
>
> --
>
> "Wovon mann nicht sprechen kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Ok, a question that present itself is what is the cardinality of each
X<kappa?
Now is it true that |X<aleph_x| = aleph_x.
Zuhair