The General Backround of Cardinality
zuhair
I was wondering as to what may be the basis for cardinality? more
specifically speaking what is the minimal requirement to have a
defined notion of cardinality for *every* set.
Coret's assumption seems to be thought of as this minimal
requirement, which might or might not be correct.
Coret's assumption is:
Every set is equinumerous to some well founded set.
In symbols:
For all x Exist y ( y is well founded & x equinumerous to y )
However this is not a theorem of ZF.
I am coming to think that the following is really what might be the
minimal requirement for a definition of Cardinality, it is what I call
as:
The General Background of Cardinality:
IF there exist a definable well ordered proper class W , were
W={W_i | Phi(W_i),i is ordinal} , such that V=Union W
then, for every set x
Cardinality(x) is the class of all sets equinumerous to x
that are members of the minimal W_i any set y that is equinumerous to
x appears in.
"V is the class of all sets".
Now it is clear that "Regularity" is only a special case of the above
requirement.
So V must be the union of a definable well ordered proper class.
That's what seems to me to be the minimal requirement to have a
defined notion of Cardinality.
as an example of what W can be other than the Cumulative Hierarchy is:
denote H(x) as the set of all sets hereditarily strictly subnumerous
to x, that are strictly subnumerous to x.
Define(H(x)):-
H(x)={y| all z (z e TC({y}) -> z strictly subnumerous to x}
Now define W as
W={W_i| W_i= H(V_i), i is ordinal}
were V_i is a stage of the Cumulative Hierarchy.
(Note: I don't know weather W is definable
in NBG\MK, and weather W_i is definable in ZF
for every i (it would be interesting to see if ZF can prove that)
, however we are assuming that we are working in
a class theory that can achieve that).
Now, whatever W might be, then it would be
equinumerous to the Cumulative Hierarchy! but
is it the rule that we can find a definable
equinumerous relation between them always?
if that is the case then this might prove
Coret's assumption. if that is not the case
then Coret's assumption might be erroneous
and if we manage to define such W, then
we can define Cardinality even beyond
Coret's assumption.
Zuhair
zuhair
Another example of this way is the following:
recall the definition of H(x)
Define(H(x)):-
H(x)={y| for all z (z e TC({y}) -> z strictly subnumerous to x)}
It is a theorem of ZF that for every Aleph_n, there exist
H(Aleph_n).
Now it would be interesting to know if the following is a theorem
of ZF or not?
For all x Exist n ( x subset of H(Aleph_n) ).
lets call this statement Z.
In case this is a theorem of ZF, then we can define Cardinality in ZF
as:
Cardinality(x) is the class of all sets equinumerous to x
that are members of the minimal H(Aleph_n) any set y that is
equinumerous to
x appears in.
The nice thing about this definition is that we can actually remove
Regularity
and have this definition work ZF-Reg.+Z.
In this way Coret's assumption is not a requirement for that
definition to work.
Zuhair
zuhair
I think this is not a theorem of ZF, it might be of ZFC.
Possibly we can have the following as a theorem of ZF.
For all x Exist i,j ( x subset of H_j(Aleph_i) ).
were H_j(Aleph_i) is the j_th Hereditary set of Aleph_i
for example j=3, i=1
H_3(Aleph_1) = H(H(H(Aleph_1).
anyhow I greatly doubt that.
Rupert
> Hi all,
>
> I was wondering as to what may be the basis for cardinality? more
> specifically speaking what is the minimal requirement to have a
> defined notion of cardinality for *every* set.
>
There was one result which I thought you should be aware of. I don't
know whether anyone has mentioned this to you before. Let ZFU be ZF
with atoms, and consider the theory ZFU-Foundation. If this theory is
consistent, then the notion of cardinality cannot be defined in this
theory; that is there is no formula phi(x,y) which we could interpret
as "y is the cardinality of x" for which we could prove the basic
properties of cardinality which we would want to prove.
I do not know whether this result also applies to ZF-Foundation.
> Coret's assumption seems to be thought of as this minimal
> requirement, which might or might not be correct.
>
> Coret's assumption is:
>
> Every set is equinumerous to some well founded set.
>
> In symbols:
>
> For all x Exist y ( y is well founded & x equinumerous to y )
>
> However this is not a theorem of ZF.
>
It is a theorem of ZF because ZF can prove that every set is well-
founded. Perhaps you wanted to say that it is not a theorem of ZF-
Foundation (assuming that this theory is consistent).
> I am coming to think that the following is really what might be the
> minimal requirement for a definition of Cardinality, it is what I call
> as:
>
> The General Background of Cardinality:
>
> IF there exist a definable well ordered proper class W , were
> W={W_i | Phi(W_i),i is ordinal} , such that V=Union W
> then, for every set x
>
> Cardinality(x) is the class of all sets equinumerous to x
> that are members of the minimal W_i any set y that is equinumerous to
> x appears in.
>
> "V is the class of all sets".
>
> Now it is clear that "Regularity" is only a special case of the above
> requirement.
>
Yes, a theorem of that kind would be sufficient for a satisfactory
definition of cardinality.
> So V must be the union of a definable well ordered proper class.
>
> That's what seems to me to be the minimal requirement to have a
> defined notion of Cardinality.
>
It is certainly a sufficient requirement.
> as an example of what W can be other than the Cumulative Hierarchy is:
>
> denote H(x) as the set of all sets hereditarily strictly subnumerous
> to x, that are strictly subnumerous to x.
>
We still don't know that this can be proved to be a set in ZF, do we?
Am I right there?
> Define(H(x)):-
>
> H(x)={y| all z (z e TC({y}) -> z strictly subnumerous to x}
>
> Now define W as
>
> W={W_i| W_i= H(V_i), i is ordinal}
>
> were V_i is a stage of the Cumulative Hierarchy.
>
> (Note: I don't know weather W is definable
> in NBG\MK, and weather W_i is definable in ZF
There is no problem in ZFC, but AFAIK if we are just in ZF then we do
not know that H(x) is always a set. That is the only problem.
> for every i (it would be interesting to see if ZF can prove that)
> , however we are assuming that we are working in
> a class theory that can achieve that).
>
You mean a class theory with classes of higher type, do you?
We could extend the language of ZF to include classes and classes of
classes, with set-theoretically definable comprehension for each
level, and then your approach will work fine. However, the thing is,
cardinality is already definable in ZF with Scott's trick. In order
for this approach to work for ZF-Foundation, we would need to be able
to prove in ZF-Foundation that every set is equipollent to a set in H
(V_kappa) for some ordinal kappa. I suppose this is conceivable.
> Now, whatever W might be, then it would be
> equinumerous to the Cumulative Hierarchy! but
> is it the rule that we can find a definable
> equinumerous relation between them always?
I don't know if I understand this question. As far as I can tell the
only missing step is proving in ZF-Foundation that every set is
equipollent to set in H(V_kappa) for some ordinal kappa.
zuhair
> On Dec 16, 6:33 pm, zuhair <zaljo...@gmail.com> wrote:
>
> > Hi all,
>
> > I was wondering as to what may be the basis for cardinality? more
> > specifically speaking what is the minimal requirement to have a
> > defined notion of cardinality for *every* set.
>
> There was one result which I thought you should be aware of. I don't
> know whether anyone has mentioned this to you before. Let ZFU be ZF
> with atoms, and consider the theory ZFU-Foundation. If this theory is
> consistent, then the notion of cardinality cannot be defined in this
> theory; that is there is no formula phi(x,y) which we could interpret
> as "y is the cardinality of x" for which we could prove the basic
> properties of cardinality which we would want to prove.
>
> I do not know whether this result also applies to ZF-Foundation.
Yes, Rupert I know that for sure, I will speak about this issue in a
separate
post, I know this issue. That's why I suggested a strong version of
Extensionality to rule out this case. Not only that the axiom that H
(x) exist for every x will also curb these Ur-elements out. Anyhow I
will write in details about this issue.
As to the question weather Quine atoms can do what Ur-elements can do
in Gauntt's models, T.E.Forster says that he is 99% sure that they can
do the job. however I don't know really the answer to this particular
issue, and he doesn't know too.
You understand now why I wanted this strong version of Extensionality
in previous posts to replace the ordinary Extensionality in ZF, in
order to curb Quine atoms as well, however we might have a problem
with the recursive singletons, but I greatly doubt that. since we can
have only countable set of recursive singletons, then I tend to think
they don't pose any problems, unlike Quine atoms and Ur-elements, were
you can have proper classes of them.
By they way if you have Ur-elements, then even if you have Regularity
then the traditional Scott cardinals would be proper classes, however
Scott cardinals can be modified to exclude them.
The existence of Ur-elements weather they are distinct empty objects,
Quine atoms or possibly recursive singletons, is a great threat to any
definition of cardinality beyond Regularity and Choice.
I will write some details about this issue. in another post.
> r
Yes, you are right.
>
> > Define(H(x)):-
>
> > H(x)={y| all z (z e TC({y}) -> z strictly subnumerous to x}
>
> > Now define W as
>
> > W={W_i| W_i= H(V_i), i is ordinal}
>
> > were V_i is a stage of the Cumulative Hierarchy.
>
> > (Note: I don't know weather W is definable
> > in NBG\MK, and weather W_i is definable in ZF
>
> There is no problem in ZFC, but AFAIK if we are just in ZF then we do
> not know that H(x) is always a set. That is the only problem.
Yea, I agree.
>
> > for every i (it would be interesting to see if ZF can prove that)
> > , however we are assuming that we are working in
> > a class theory that can achieve that).
>
> You mean a class theory with classes of higher type, do you?
No, just proper classes alone I didn't mean the higher type proper
classes.
>
> We could extend the language of ZF to include classes and classes of
> classes, with set-theoretically definable comprehension for each
> level, and then your approach will work fine. However, the thing is,
> cardinality is already definable in ZF with Scott's trick. In order
> for this approach to work for ZF-Foundation, we would need to be able
> to prove in ZF-Foundation that every set is equipollent to a set in H
> (V_kappa) for some ordinal kappa. I suppose this is conceivable.
Yes, but ZF-foundation alone doesn't prove that. we need to strengthen
Extensionality to rule out all kinds of Ur-elements, and even that
might not be enough, we might need assumptions like the above.
Bye they way what I call as "the general background of Cardinality" is
a kind of Scott trick essentially, actually Scott trick can be viewed
to be a special case of it.
>
> > Now, whatever W might be, then it would be
> > equinumerous to the Cumulative Hierarchy! but
> > is it the rule that we can find a definable
> > equinumerous relation between them always?
>
> I don't know if I understand this question. As far as I can tell the
> only missing step is proving in ZF-Foundation that every set is
> equipollent to set in H(V_kappa) for some ordinal kappa.
Actually you don't need all of that, the only missing step in ZF-
Foundation
is that every set is equipollent to a well founded set, that's all,
i.e. to
a set of the cumulative Hierarchy, which is Coret's assumption.
Thanks Rupert.
Zuhair
zuhair
>
> > Hi all,
>
> > I was wondering as to what may be the basis for cardinality? more
> > specifically speaking what is the minimal requirement to have a
> > defined notion of cardinality for *every* set.
>
> There was one result which I thought you should be aware of. I don't
> know whether anyone has mentioned this to you before. Let ZFU be ZF
> with atoms, and consider the theory ZFU-Foundation. If this theory is
> consistent, then the notion of cardinality cannot be defined in this
> theory; that is there is no formula phi(x,y) which we could interpret
> as "y is the cardinality of x" for which we could prove the basic
> properties of cardinality which we would want to prove.
>
> I do not know whether this result also applies to ZF-Foundation.
>
> > Coret's assumption seems to be thought of as this minimal
> > requirement, which might or might not be correct.
>
> > Coret's assumption is:
>
> > Every set is equinumerous to some well founded set.
>
> > In symbols:
>
> > For all x Exist y ( y is well founded & x equinumerous to y )
>
> > However this is not a theorem of ZF.
>
> It is a theorem of ZF because ZF can prove that every set is well-
> founded. Perhaps you wanted to say that it is not a theorem of ZF-
> Foundation (assuming that this theory is consistent).
When I meant it is not a theorem of ZF, I meant it is not a theorem of
ZF-Regularity, I thought that the context is clear. the term "ZF" is
used flexibly sometimes to mean
ZF minus Regularity, sometimes it is used even to mean ZFC, and
sometimes it is used to mean ZF (Regularity inclusive).
The official terminology of ZF is inclusive of Regularity of course,
but not of Choice.
However what I largely saw many set theorist use ZF is in the flexible
manner and not in anyway specifically to mean the Regularity inclusive
version.
For example you see Dana Scott writing that it is a model of ZF to
have a proper class of Quine atoms,..etc.. what he actually meant it
is a model of
ZF-minus Regularity. But the context is clear. Also you see T. Forster
writing that
Coret's assumption is not a theorem of ZF, what he meant it is not a
theorem of ZF minus Regularity, but again the context is clear.
Actually I tend to Agree with what once time Moe Blee wrote, that he
like to
write ZFR to mean ZF plus Regularity, ZFRC to mean ZF plus Regularity
and
choice. he was right this would be clearer.
Zuhair
zuhair
I don't know if you are using H(x) in the same way I defined it.
let me recall the definition of H(x) again:
Define(H(x)):-
H(x)={y| for all z (z e TC({y}) -> z strictly subnumerous to x)}
Now
H(V_kappa)={y| for all z (z e TC({V_kappa}) -> z strictly subnumerous
to V_kappa)}
Now lets suppose that we have H(V_kappa) for every V_kappa as a
theorem of
ZF.
were V_kappa is a stage of the Cumulative Hierarchy
Now in ZF the union of all H(V_kappa) for all kappa ordinals
would be actually the cumulative hierarchy itself!
Now we can drop Regularity and add the axiom
For all kappa Exist H(V_kappa)
also add the axiom
For all x Exist kappa ( x subset of H(V_kappa) )
NOTE: this axiom doesn't entail Coret's assumption!
Then we can define Cardinality as:
Cardinality(x) is the class of all sets equinumerous to x
that are subsets of the minimal H(V_kappa) any set y that is
equinumerous to
x is a subset of.
This might work.
However the most important step would be to prove that
For all x Exist kappa ( x subset of H(V_kappa) )
if this is proved, then we can say we have a new definition of
Cardinality beyond
Regularity and Choice.
Zuhair
David Libert
I have several points to catch up on. I am writing this as followup to
Zuhair's most recent article, because I wanted to comment on something
from that, but I will also be taking up points from other recent
articles, maybe even from some other threads.
zuhair (zalj...@gmail.com) writes:
> On Dec 16, 7:07=A0pm, Rupert <rupertmccal...@yahoo.com> wrote:
>> On Dec 16, 6:33=A0pm, zuhair <zaljo...@gmail.com> wrote:
>>
>> > Hi all,
>>
>> > =A0 I was wondering as to what may be the basis for cardinality? more
>> > specifically speaking what is the minimal requirement to have a
>> > defined notion of cardinality for *every* set.
>>
>> There was one result which I thought you should be aware of. I don't
>> know whether anyone has mentioned this to you before. Let ZFU be ZF
>> with atoms, and consider the theory ZFU-Foundation. If this theory is
>> consistent, then the notion of cardinality cannot be defined in this
>> theory; that is there is no formula phi(x,y) which we could interpret
>> as "y is the cardinality of x" for which we could prove the basic
>> properties of cardinality which we would want to prove.
>>
>> I do not know whether this result also applies to ZF-Foundation.
This is interesting. I was claiming my own proof like this in an earlier
thread for ZF - Foundation.
I had not previously heard of any results along such lines. So apparently
there was a previously known result like this for ZFU - Foundation.
Anyway, I think I also have this for ZF - Foundation.
[Deletion]
>> > Now, whatever W might be, then it would be
>> > equinumerous to the Cumulative Hierarchy! but
>> > is it the rule that we can find a definable
>> > equinumerous relation between them always?
>>
>> I don't know if I understand this question. As far as I can tell the
>> only missing step is proving in ZF-Foundation that every set is
>> equipollent to set in H(V_kappa) for some ordinal kappa.
This statement, every set is equipillent to a set in H(V_kappa)
for some ordinal kappa, is a theorem of ZF with Foundation,
but is not a theorem of ZF - Foundation if that theory is
consistent.
We can make a permutation model with atoms arranged as non-well-founded
sets, which are incomparable to all the well-founded sets and so
stay out of all H(V_kappa).
> I don't know if you are using H(x) in the same way I defined it.
>
> let me recall the definition of H(x) again:
>
> Define(H(x)):-
>
> H(x)=3D{y| for all z (z e TC({y}) -> z strictly subnumerous to x)}
>
> Now
>
> H(V_kappa)=3D{y| for all z (z e TC({V_kappa}) -> z strictly subnumerous
> to V_kappa)}
>
> Now lets suppose that we have H(V_kappa) for every V_kappa as a
> theorem of
> ZF.
>
> were V_kappa is a stage of the Cumulative Hierarchy
>
> Now in ZF the union of all H(V_kappa) for all kappa ordinals
> would be actually the cumulative hierarchy itself!
>
> Now we can drop Regularity and add the axiom
>
> For all kappa Exist H(V_kappa)
>
> also add the axiom
>
> For all x Exist kappa ( x subset of H(V_kappa) )
>
> NOTE: this axiom doesn't entail Coret's assumption!
>
> Then we can define Cardinality as:
>
> Cardinality(x) is the class of all sets equinumerous to x
> that are subsets of the minimal H(V_kappa) any set y that is
> equinumerous to
> x is a subset of.
>
> This might work.
>
> However the most important step would be to prove that
>
> For all x Exist kappa ( x subset of H(V_kappa) )
This can't be proved from ZF - Foundation, but as you said above
you could axiomatize it.
> if this is proved, then we can say we have a new definition of
> Cardinality beyond
> Regularity and Choice.
>
> Zuhair
So if you do axiomatize it, and also as you note above
axiomatize that all H(V_kappa) are sets, then this would give
another defintion of cardinality. Since you have added those
axioms, you don;t need to also have Regualrity and Choice, as you
just said.
That is about some of the points of the quoted article above.
I wanted to also note some general background.
We have been going on in various threads about getting H(<#x)
to be a set from ZF.
There was Jech's proof of this for x = omega, and the first
generalization of this so x well-orderable.
I have been going back and forth on general x.
Anyway, to announce I hust flipped again. :) . I have come
up with a new proof attempt to generalize Jech's proof over ZF
(no AC) to all x.
I just thought of this and so far it looks ok. Though I
previously claimed an earlier such generalizing proof and
later retracted it.
Another topic. As I noted above, in an earlier thread I
outline a proof attempt from Con(ZF) constructing a
ZF - regularity model (so no AC) in which there is no
definable cardfinality, ie no definition of a set
(the "cardinality") which was exactly = for equipollent
sets.
I think I have to adjust something from that previous
outline, but once I do I think it is ok.
That original version included sungelton towers. Zuhair
later wrote aboit maybe axiomatizing singeton towers away
to avoid such cases toward a definition.
But that construction just used singleton towers because
they were the easiest to make a simple defintion of the
model.
I can modify the model with a bit of extra complication
to replace the singleton towers by doubleton towers. Or
any other specific von Neumann cardinal.
I can even make a version where there are no infinite
descending towers of membership of well-orderable sets,
but still descending towers of membership of
non-well-orderable sets, where cardinality is
undefinable.
But all my models like this, they always have some
set x with H(x) not a set.
The last point is I can arrange the only such x
with H(x) not a set are all non-well-orderable x.
Above I claimed to generalize Jech's proof to
get H(x) a set for all x. But Jech's proof uses
regularity, and my generalization does also.
These other models, with some H(x) not a set
and cardinality undefinable, don't satisfy regularity
and they don't satisfy AC.
Indeed, if the model satisgies AC, even without
regularity, we can get back cardinals as von Neumann
cardinals.
So my models with cardinality definable, also have
H(x) not a set, for some x.
On the other hand, if H(x) is always a set,
even wothout regularity or AC, we can try Zuhair's
definitions.
Though I just recalled, even when H(x) is always a
set, there was the other problem of Zuhair's
first definitions of amorphous sets having empty
defined cardinality.
Also, I raised a question before, still not settled,
in ZF can there be distinct cardinals with the same
predecessors. Those if they existed would mess up
definitions based on strictly subnumerous everywhere,
but are ok for definitions based on subnumerous
or bu\ijective, as previously discussed.
There were the other approaches, based on minimal
TC, or the one above based on H(V_kappa).
If my models with cardinality undefinable are
correct, these properites muct fail in some models
without regularity and AC.
But as Zuhair notes, another possibility is to
axiomaize these extra properties and get new
definitions that way.
The problem about amorphous sets and empty
defined cardinality can also be axiomatized away,
as Zuhair noted in a previous thread.
But in terms of not adding such special technical
axioms and just having very basic axioms. There are
the ~ regularity ~AC models without definable
cardinality.
And if you have either regularity or AC back you
can get back Scott's trick or von Neumann cardinals
respectively.
Ig my Jecg generalization with H(x) always a set
from regularity and no AC is ok, that looks at
first like it is giving us some H(x) definitions
of Zuhair back. It is overcoming one block to
those.
But it still doesn't solve the other problem
of amorphous sets and empty defined cardinality.
So the main new things from here, if they are
correct, is Jech generalized to all x, and
the undefinable cardinality models can instead
use more complex towers than singleton towers.
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
It cannnot. if H(x) exist for every set and
IF every set is axiomatized to be a subset
of some H(V_kappa) then there is no way
for this approach to fail in defining cardinality, there
cannot be empty cardinals in this approach, you will
always have a minimal kappa were a set equinumerous to
x appears as a subset of H(V_kappa), and
the cardinal is the set of all these equinumerous sets, so
it is always not empty.
Still we can rule out all these towers by axiomatizing that H(x) is a
set for all x.
Actually I am coming to believe that this is an important axiom.
However
the problem of empty cardinals would exist and demand a closer
treatment.
Still I want to see the full prove of generalization of Jecks proof to
all x, in ZF.
IF you prove this point, then the H(V_kappa) approach will be a very
strong
approach, we only need to add to axioms to ZF minus Regularity
and we get a definition of cardinality beyond choice and regularity
and what is more important even beyond Coret's axiom, and this
would be a strong definition, actually even stronger than Scott
cardinals, since
Regularity would be any way a special case of these additional axioms.
Zuhair
zuhair
It would be interesting to me if you simply and in some detail explain
the singleton towers (or doubleton or any towers) and show how can
these affect the definition of cardinality in your models. all your
previous presentations of your models are not that simply presented, I
think you need to simplify them much much more than you did.
I just have the sense they these towers are not a great problem
against cardinality, however I am not sure really. These permutation
models are something that I don't know much of, so I cannot really
make any comment on them.
I just thought that the lemma that for every set y in the transitive
closure of x, there must be a finite sequence x0,x1,x2,.....,xn such
that x0 e x and
xi+1 e xi for x=0,1,2,...,n-1 and y=xn, I thought this lemma
will restrict the permutations that can be done upon these towers.
Observe that the lemma above would entail that there can exist only
countable number of any singleton, doublton, n-ton towers., were n is
finite, however
if you can have a tower for every ordinal d, then of course we'll have
a proper class of them, but I don't know about the permutation models
related to them.
So I don't know how can these be permuted to such a level that make us
have
sets of undefinable cardinality?
I would expect that with the Ur-elements yes, I would expect it with
Quine atoms also , which are in my opinion another version of Ur-
elements. In these cases
we can have proper classes of Ur-elements(without violating
Regularity),
also we can have proper classes of Quine atoms(violating Regularity),
but I don't know how can we have proper classes of n finite towers.
Anyhow, I had the vague sense that all these towers you are speaking
about are also kinds of Ur-elements also, and we should get rid of
them.
Anyhow I don't know the details.
Zuhair
David Libert
zuhair (zalj...@gmail.com) writes:
> On Dec 17, 12:39=A0am, ah...@FreeNet.Carleton.CA (David Libert) wrote:
[Deletion]
>> =A0 Another topic. =A0As I noted above, in an earlier thread I
>> outline a proof attempt =A0from Con(ZF) =A0constructing a
>> ZF - regularity model (so no AC) =A0in which there is no
>> definable cardfinality, =A0ie no definition of a set
>> (the "cardinality") =A0which was exactly =3D for equipollent
>> sets.
>>
>> =A0 I think I have to adjust something from that previous
>> outline, but once I do I think it is ok.
>>
>> =A0 That original version included sungelton towers. =A0Zuhair
>> later wrote aboit maybe axiomatizing singeton towers away
>> to avoid such cases toward a definition.
>>
>> =A0 But that construction just used singleton towers because
>> they were the easiest to make a simple defintion of the
>> model.
>>
>> =A0 I can modify the model with a bit of extra complication
>> to replace the singleton towers by doubleton towers. =A0Or
>> any other specific von Neumann cardinal.
>>
>> =A0 I can even make a version where there are no infinite
>> descending towers of membership of well-orderable sets,
>> but still descending towers of membership of
>> non-well-orderable sets, where cardinality is
>> undefinable.
>>
>> =A0 But all my models like this, they always have some
>> set x =A0with =A0H(x) not a set.
>>
>> =A0 The last point is I can arrange the only such x
>> with H(x) =A0not a set =A0are all non-well-orderable x.
>>
>> =A0 Above I claimed to generalize Jech's proof to
>> get =A0H(x) a set for all x. =A0But Jech's proof uses
>> regularity, and my generalization does also.
>>
>> =A0 These other models, =A0with =A0some H(x) =A0not a set
>> and cardinality undefinable, =A0don't satisfy regularity
>> and they don't satisfy AC.
>>
>> =A0 Indeed, if the model satisgies AC, even without
>> regularity, =A0we can get back cardinals as von Neumann
>> cardinals.
>>
>> =A0 So my models with cardinality definable, also have
>> H(x) =A0not a set, =A0for some =A0x.
[Deletion]
>> =A0 So the main new things from here, if they are
>> correct, =A0is Jech generalized to all x, and
>> the undefinable cardinality models can instead
>> use more complex towers than singleton towers.
>>
>> --
>> David Libert =A0 =A0 =A0 =A0 =A0ah...@FreeNet.Carleton.CA
>
> It would be interesting to me if you simply and in some detail explain
> the singleton towers (or doubleton or any towers) and show how can
> these affect the definition of cardinality in your models. all your
> previous presentations of your models are not that simply presented, I
> think you need to simplify them much much more than you did.
[Deletion]
> Zuhair
I will write out my claimed proof, constructing from a model of ZFC
a model of ZF - regularity in which cardinality is undefinable.
As noted above, these constructed models will satisfy ~regularity,
~AC and there exists sets x such that H_(x) is not a set (is
instead a proper class).
This proof will use methods to construct models of ~regularity, namely
to make a model with atoms which come to represent sets having defined
membership patterns. I used similar methods to constuct a
ZFC - regularity model with a proper class of singleton towers in
order to make H_(<2) a proper class, in
[1] David Libert "A new definition of Cardinality"
sci.logic, sci.math Nov 23
http://groups.google.com/group/sci.math/msg/721cb8170033cf84
I will also be using symmetry methods to construct ~AC models,
as discussed in
[2] David Libert "Cohen symmetric choiceless ZF models"
sci.logic July 6, 2000
http://groups.google.com/group/sci.logic/msg/b4271c2585d2f1e5
I had previously in
[3] David Libert "Why Define Cardinality?"
sci.logic, sci.math Dec 10, 2009
http://groups.google.com/group/sci.logic/msg/0803f45348c83967
posted an outline of a construction of such a model. This is the
outline mentioned in the quote at the start of this article:
>> =A0 Another topic. =A0As I noted above, in an earlier thread I
>> outline a proof attempt =A0from Con(ZF) =A0constructing a
>> ZF - regularity model (so no AC) =A0in which there is no
>> definable cardfinality, =A0ie no definition of a set
>> (the "cardinality") =A0which was exactly =3D for equipollent
>> sets.
>>
>> =A0 I think I have to adjust something from that previous
>> outline, but once I do I think it is ok.
So below I will be giving a more detailed account and also
adjusting the definition as noted above.
We begin working in an assumed ZFC model. (This can be obtained
from a ZF model by Godel's L).
In there I seek to define a proper class of sets, and to define
a binary relation on these which will be the non-standard epsilon
relation on the ZF model we seek. These definitions go on in the
outer ZFC model.
The basic version will be based on singleton towers. I will also
discuss later versions to have alternate towers: doubleton towers,
or non-well-orederable set towers as mentioned above.
Each particular version will just define one ZF model inside
the one starting ZFC model, so we restart in a new ZFC model
to make anoter version. So as we work, we only need to think about
one ZF model at a time.
I will start with the singleton towers version.
I start by defining the class in the ZFC model, which will become
the universe of the constructed ZF model. This part is similar to
[1].
Some elements of the class will play the role originally of
atoms. I begin with these.
These will each be of form <a, b1, ... , bn> some finite tuple
of length >= 2, with first member a ~= 0.
I am building the basic structure of the desired model into these
atoms, so there are various cases put in to help later steps of the
proof. Some of these may have unclear purpose at this beginnning,
but will seen to be useful later as they come up in the proof.
So to start, if n in a finite ordinal > 0 and alpha is
any ordinal, <n, alpha> will be defined to be an atom.
By way of motivation, the constructed ZF will have some
proper classes over it, these proper classes will be indexed
in the outer ZFC model by n varying over finite ordinals
> 0.
Each atom <n, alpha> as just defined will correspond to a
set in the constructed ZF models, these sets will be the
members of those various proper classes. The n'th proper
class will have members corresponding to all
<n, alpha> for that fixed n as alpha varies over all
ordinals.
Next part of defining atoms, for n > 0 and finite
ordinal, alpha arbitrary ordinal, and
m, l finite ordinals
<n, alpha, m, l> will be defined to be an atom.
The intended meaning of these <n, alpha, m> is that
the set coresponding to <n , alpha> will have members
corresponging exactly to <n, alpha, m, l> for that fixed
n, alpha as m varies over finite ordinals and
l varies over finite ordinals > 0.
(I defined atoms <n, alpha, m, 0> but for a technical
reason that will be clear later I don't make these a member
of the set corresponding to <n, alpha> .)
So abusing notation, using an atom to represent the
later corresponding set of the coming constructed model,
so far we have proper classes
{<n, alpha> | alpha ordinal} where each of those
<n, alpha> is a set {<n, alpha, m, l> | m, l finite ordinals
l > 0 }.
These <n, alpha> sets will each be a union of singleton
towers. <n, alpha, m, l> will be {<n, alpha, m, l+1>}.
<m, alpha, m, l> will be {<n, alpha, m, l+1>}
even for l = 0. I still have that, I just leave
<n, alpha, m, 0> out of <n, alpha, m> .
So this will be an infinite descending singleton tower.
That's how we lose regularity.
These comments about mebership are only for motivation.
I will be defining mebership precisely below.
Some more atoms. For n finite ordinal > 0 and alpha1,
alpha2 distinct ordinals, and m finite ordinale,
and l finite ordinal > 0 ,
<n, {alpha1 + 1, alpha2 + 1}, m, l > is an atom.
Note there is no overlap of thise case with the
previous <n , alpha, m, l> case, because a set of form
{alpha1 + 1, alpha2 + 1} is never a von Neumann ordinal:
every non-empty von Neumann ordinal contains {} as member,
and {} = ordinal 0 is not a memeber of
{alpha1 + 1, alpha2 + 1}.
<n, {alpha1 + 1, alpha2 + 2}, m > is intended to represent
eventually {<n, alpha1, m, l> , <n, alpha2, m, l>} : a pair
set.
Next clause : For n finite ordinal > 0 and alpha1,
alpha2 distinct ordinals, and m finite ordinal,
l finite ordinal > 0,
<n, <alpha1 + 1 , alpha2 + 1>, m, l> is an atom.
<alpha1 + 1, alpha2 + 1> is not a von Neumann ordinal
and not a pair set of ordinals so again no overlap with
previous cases.
Finally, some more atoms. To be able to define these was all
the reason we had the last two groups.
For n finite ordinal >0, and alpha1, alpha2 distinct
ordinals <n, <alpha1, alpha2>> is an atom.
Again no overlap with previous cases, since
<alpha1, akpha2> is never a von Neumann ordinal.
<n. <alpha1, alpha2>> is intended to be the function
: <n, alpha1> -> <n, alpha2>, sending member
<n, alpha1, m, l> to <n, alpha2, m, l> for m, l finite
and l > 1.
These are all the atoms.
To recap the above more intuitively.
We have many sets <n, alpha, m, l>, Fixing n, alpha, m
and varying l these are descending singleton towers.
So we have many singleton towers.
We group these together. We form many sets of these,
each is <n, alpha> for fixed n, alpha and consists
of members <n, alpha, m, l> for l > 1.
So basically those many <n, alpha> sets partition all
the <n, alpha, m, l> sets with l > 0.
And specifically they partition them so each singleton
tower hasd all its members thrown into the same <n, alpha>.
These <n, alpha> sets grouping the <n, alpha, m, l>
themselves group into a proper class for each fixed n
as alpha varies.
So a bunch of proper classes, each proper class being
a union of singleton towers.
More though. In proper class n, all the member sets
<n, alpha1> <n, alpha2> are pairwise bijective,
by the <n, <alpha1, alpha2>> atoms.
In order to later easily define the new model, I wanted
the part of it I define by atoms to be a tansitive subclass.
I wanted to make atoms for those bijections, so I also
made atoms to build up the Kuratowski ordered pairs
inside those bijections.
So I define E, a binary relation on this definable
class of atoms, which will interpret epsilon in the
ZF model we will be constructing.
I define E to corresond to the intended meaning I
said aboive of all those atoms.
Among <n, alpha, m, l> 's I define E to make the
singleton towers: <n, alpha, m, l> will have
E member exactly <n, alpha, m, l+1>
<n, alpha> will have members exactly <n, alpha, m, l)
with l > 0.
(All these and below with the conventions: n finite
ordinal > 0, alpha ordinal, m, l ordionals and l > 0).
The the atoms for singletons, doubletons and ordered
pairs. The singleton towers give singletons. That's
why I defined <n, alpha, m, 0> but left it out
of <n, alpha>.
So I define the E relations in the obvious way for
these singletons, doubletons, ordered pairs.
And then those bijections as the sets of ordered
pairs from the atoms.
So that is E on all those atoms.
Next I by transfinite recursion define the sets.
They will be of form <0, t> for t a set of
atoms and previous sets. That's why I had all atoms
with n firast coordinate > 0, so keep distinct
from these sets <0, t>.
I start with the class of all atoms as above.
Ar successor stages I add to the class all sets
<0, t> where t is a subset of the previous class.
Except not really all. I only add in those <0, t>
where t does not have the same members as the E members
of a previous atom or set.
I take unions at limit ordinals.
I define s E <0, t> if s usual member of t.
So this definable class with that E interpreting
epsilon is a preliminary version of our final model.
We turn to permutation methods to lose AC and
cut back that constucted model so far to the final
one. As in [2].
Toward this I will introduce notation for the most
important atoms.
A_n,alpha,m.l will be <n, alpha, m, l> .
A_n,alpha will be <n, alpha> .
f_n,alpha1,alpha2 will be the bijection corresponding
to <n, <alpha1, alpha2> >.
So f_n,alpha1,alpha2 : A_n,alpha1 >->> A_n,alpha2
by sending A_n,alpha1,m.l to A_n,alpha2,m,l.
Also, A_n is not an atom, but will be the proper
class of all A_n,alpha.
I can describe permutations of the inner model so far
by specifying the action on the A_n,alpha,m,l 's.
Everything else is a set over these (in iteration),
and so has its evaluation determined by this base
since E is to be respected.
I am going to specify a group acting on this model.
The group will be generated by subgroups I will specify
in turn by their actions.
First subgroup. Given any permutation of positive
integers from the underlying ZFC, do this permutation
on the n's and leace alpha, m, l fixed as a permutation
of the A_n,alpha,m,l.
Next groups: for each fixed finite n0 > 0, and each
permutation of ordinals represented as a set in ZFC,
so eventrually constant: on A_n0,alpha,m,l
move it by permuting alpha and no change to m,l.
For n ~= n0, the group fixes all A_n,alpha,m,l .
So for each n0 > 0 we have such a subgroup with
that action. We include all all these subgroups
as n0 varies.
Next groups. For each fixed n0 and alpha0,
and each base ZFC model permutation of omega,
act on A_n0,alpha0,m,l by permuting m by
the permutation, and leaving l fixed.
This subgroup leaves all A_n,alpha,m.l
fixed for n ~= n0 or alpha ~= alpha0.
As n0, alpha0 vary we get a definable
class of subgroups.
Our final group and its action is the definable
class sized group, with class sized action (definable)
generated by all these.
So we have a class sized group acting on the inner
model so far. This is lifting the action to the
entire model, including other atoms and sets.
Next the support notion as from [2].
For a any element of the model (an atom or a
set), and for S a set of basic atoms each individually
of form A_n,alpha,m,l or of form f_n,alpha1,alpha2
we define S is a support set for a if every permutation
from that group action which fixes every element of S
fixes a.
We say that an element of the inner model above a has
finite support if that exists a finite set S of
basic atoms of form A_n,alpha,m.l or of form
individually f_n,alpha1,alpha2 which is a support
set for a.
Ie: a support set is allowed to mix forms
A_n,alpha,m,l and f_n,alpha1,alpha2.
Our final model will be all the members of the
previous inner model which individually have finite
support. The E relation interpreting epsilon is the
restriction of the E relatoion of the last inner
model to this submodel.
As [2], this is a model of ZF - regularity.
To argue now that cardinality is undefinable in this
model.
So suppose for contradiction that C is a definable
class sized function over this model taking sets to
sets and sending exactly equipllent pairs to the same
value.
Then C would have suport the union of the supports of
all parameters used to define C. Pick a specific
support set for C to use for the rest.
Note that for each n0 finite ordinal > 0, and each
alpha1, alpha2 ordinals
A_n0,alpha1 and A_n0,alpha2 are equipollent in the
permutation model, since f_n,alpha1,alpha2 for
alpha1 ~= alpha2 is in the model since it has support
{f_n0,alpha1,alpha2}.
I will soon be needing the claim:
for n0, n1 distinct finite ordinals > 0
and alpha1, alpha2 ordinals
A_n0,alpha1 and A_n1,alpha2 are not equipollent in
the permuation model.
To see this claim, suppose not for contradiction,
so let f be a bijection in the permutation model
f : A_n0,alpha1 >->> A_n1,alpha2 .
So f has finite support. Pick a specific such
support for f to use below.
Find m1, m2 distinct finite ordinals
so A_n1,alpha2,m1,l and A_n1,alpha2,m2,l
are out of that support for f for all finite ordinals
l.
Consider the permuation which flips
A_n1,alpha2,m1,1 <-> A_n1,Alpha2,m2,1 and moves
no other A_n,alpha,m,l.
This permutation fixes the f preimage of
A_n1,alpha1.m1,1 and also fixes f. Contrary to
moving A_n1,alpha1,m1,1.
Hence the claim.
We turn toward deriving a contradiction from a
cardinailty function C being definable in the model.
As above, we pick a specific finitie supp(C) for the C
definition, as above.
Let n0 > 0 be a finite ordinal so all
A_n0,alpha,m,l and f_n0,alpha1,alpha2 are out of
supp(C). This for all the alpha,alpha1,alpha2,m,l
under discussion: alpha's oprdinals and m,l
finite ordinals.
Consider an arbitrary ordinal alpha. C(A_n0,alpha)
by our assumptions for contradiction on C is a set in
the permuatation model. The permutation model satisfies
ZF - regularity, which proves transitive closures exist.
So let TC(C(A_n0,alpha)) be that transitive closure
in the sense of the permutation model.
I claim the for all ordinals alpha, and for
supp(C(A_n0,alpha)) a finite support set for
C(A_n0,alpha),
that for n with all
A_n,alpha1,m,l and all f_n,alpha1,alpha2 outside
supp(C),
all A_n,alpha1,m,l in TC(A_n0,alpha)
are are in supp(C(A_0,alpha)
or have some f_n,alpha1,alpha2 or
f_n,alpha2,alpha1
in supp(C(A_n0,alpha)).
Ie: to restate this more intuitively: what can
C(A_n0,alpha) be built from, in the sense of things in
its TC and hence building it from below.
Just considering the A_n outside of supp(C),
we might be able to use supp(C(A_n0,alpha)) members
like this to build up C(A_n0,alpha), and we make
no claims about things not of form A_n',alpha',n',l'
but for the A_n,alpha1,m.l we can only go inside
supp(C_(A_n0,alpha) in the sense above.
To prove this claim, suppose for contradiction
A_n,alpha1,m,l is in TC(C(A_n0,alpha)) and
n is completely uninvolved in supp(C) as above
and and A_n,alpha1 uninvolved in
supp(C(A_n0,alpha1)).
supp(C(A_n0,alpha) is finite, so we can find
a proper class of alpha2 's with A_n,alpha2 similarly
uninvolved with supp(C(A_n0,alpha2)).
So for these consider the group element flipping
A_n,alpha1 and A_n,alpha2 and no other basic moves.
This permutation fixes C(A_n0,alpha) since
A_n,alpha1, A_n,alpha2 were uninvolved in its support.
From A_n,alpha1 being member of TC(C(A_n0,alpha1)
and applying that permuation on this formula (permutations
vrespect E membership) abnd the RS being fixed
we conclude A_n,alpha2 is member TC(A_n0.alpha2).
As we do this in turn over a proper class of different
alpha2, we get a proper class of members in
TC(C(A_n0,alpha)), contrary to TC(C(A_n0,alpha))
being a set.
This contradiction shows the claim.
So to repeat lossely: C(A_n0,alpha) can only be built
from below in sense of TC and it terms of using
A_n,alpha,m,l by such A_n,alpha,m,l involved in
supp(C) or supp(C(A_n0,alpha).
Recall we picked n0 so A_n0 is completely
uninvolved in supp(C).
Now let alpha1 be an arbitrary ordinal.
I claim that TC(C(A_n0,alpha1)) must include some members
of form A_n,alpha,m.l for some n with A_n not involved
in supp(C).
To see this, suppose not for contradiction.
Find some n1 ~- n0 and A_n not involved in supp(C).
Consider the permutation that flips n0 and n and no
other changes.
This preserves C since n0, and n1 were both completely
univolved in supp(C).
This permutes A_n0,alpha1 to A_n1,alpha1.
Consider what this permutation does to C(A_n0,alpha).
Our permutation model was built by sets iteratively over
aroms of form A_n',alpha1,m',l'. This set buolding was
well-founded, all the non-well-foundedness came from those
atoms. So if the the permutation moves C(A_n0,alpha)
you can traqce down the well-founded tree of membership
in TC(A_n0,alpha) and get a path of TC members moved
by the permutation.
(Ok, I am trying to cut corners here. That language sounds
like AC based. Ok do that in the underylnig ZFC model, Or else
recast that as proper languae find an epsilon minimal element
in the TC above suich atoms moved by the perm.)
So we find a A_n',alpha',m',l' form member in TC(C(A_n0,alpha))
moved by the permutation if the permutation moves
C(A_n0,alpha).
But the only A_n',alpha',m'.l' atoms this permutation moved
are for n' = n0 or n1. (The perm just flipped n0 <-> n1
and no other basic changes).
So we must have such example with n' = n0 or n1.
But we picked n0 and n to each have A_n uninvolved
with supp(C).
And our present assumtopn for contradiction is TC(C(A_n0,alpha1))
has no members for form A_n,alpha,m,l with A_n not involved in
supp(C).
So from that none of these atoms could move.
These were moving atoms the transitive closure from the top
moving, so this woud show thw permuation flipping n0, n1
can move C(A_n0,alpha1) after all.
But this permutation fixes C as above, moves A_n0,alpha1
to A_n1,alpha1 and by the last fixes C(A_n0,alpha1).
Hence C(A_n0,alpha1) = C(A_n1,alpha1) .
But n1 was picked as > 0 and ~= n0.
So by the opening claim A_n0,alpha1 and A_n1,alpha1 are
not equipollent.
But C was supposed to be a cardinality definiton, and not
sendf non-equipollent sets to the same value.
This is a contradiction.
We got that constradiction by assuming for contradiction
TC(C(A_n0,alpha1)) had no members of form
A_n,alpha,m,l for some A_n completely uninvolved in
supp(C).
Hence we conclude from this contradicton from that
assumtion that TC(C(A_n0,alpha1)) does include
some members of form A_n,alpha,m,l for
A_n uninvolved in supp(C).
Let A_n'alpha',m',l' be such a member of
TC(C(A_n0,alpha1)) with A_n' uninvolved
in supp(C).
Let alpha2 be an ordinal ~= alpha'.
Consider TC(C(A_n0,alpha2)).
We can make a support set for A_n0,alpha2 with any
member of it for example (A_n0,alpha2,0,1} as suport
set. Use that one for definiteness.
Union in our basic support set from all throught for
C, and obtain a support set for C(A_n0,alpha2))
based on evaluating that as the definition of C
applied to A_n0,alpha2.
And TC(C(A_n0,alpha2)) is definable
from C(A_n0,alpha2) by the definable TC operation
so we take this last support set as also a support
set for TC(C(A_n0,akpha2)).
Our A_n',alpha',m',l' had n' completely
uninvoled in supp(C).
Also alpha2 was picked ~= alpha'
so A_n',alpha',m',n' ~= A_n0,alpha2,0,1
and so not in {A_n0,alpha2,0,1}.
So A_n',alpha',m',l' is not a member of the
support we just made for TC(C(A_n0,alpha2).
A_n',alpha1,m'l' was picked to be a member
of TC(C(A_n0,alpha1)).
The bijection
f_n0,apha1,alpha2 : A_n0,alpha1 >->> A_n0,alpha2
has support {f_n0,alpha1,alpha2} and so is in
our permutation model.
So the permutation model thinks A_n0,alpha1
and A_n0,alpha2 are equipollent.
We assumed for contradiction that C is a cardinality
defintion, in particular equipolent sets get sent to
the same value by C.
Hence C(A_n0,alpha1) = C(A_n0,alpha2).
Hence TC(C(A_n0,alpha1)) = TC(C(A_n0,alpha2)).
So since A_n',alpha',m',l' is a member of
TC(C(A_n0,alpha1)) it is also a member of
TC(C(A_n0,alpha2)).
We picked n' to be completely uninvolved in supp(C)
and above we got A_n',alpha',m',l' to be out
of our new support set for TC(C(A_n0,alpha2)).
Above I had the claim:
> I claim the for all ordinals alpha, and for
>supp(C(A_n0,alpha)) a finite support set for
>C(A_n0,alpha),
>that for n with all
> A_n,alpha1,m,l and all f_n,alpha1,alpha2 outside
> supp(C),
>all A_n,alpha1,m,l in TC(A_n0,alpha)
> are are in supp(C(A_0,alpha)
> or have some f_n,alpha1,alpha2 or
> f_n,alpha2,alpha1
> in supp(C(A_n0,alpha)).
I will apply this claim to TC(C(A_n0,alpha2)).
I have its member A_n',alpha',m',l'.
We picked this A_n',alpha'.m'.l' originally
with A_n uninvolved in supp(C) :
> Hence we conclude from this contradicton from that
>assumtion that TC(C(A_n0,alpha1)) does include
>some members of form A_n,alpha,m,l for
>A_n uninvolved in supp(C).
>
> Let A_n'alpha',m',l' be such a member of
>TC(C(A_n0,alpha1)) with A_n' uninvolved
>in supp(C).
I am going to recopy that claim changing the names of
variables to be the ones we want to apply it to now.
> I claim the for all ordinals alpha2, and for
>supp(C(A_n0,alpha2)) a finite support set for
>C(A_n0,alpha2),
>that for n' with all
> A_n,alphaA,m,l and all f_n,alphaB,alphaC outside
> supp(C),
>all A_n',alpha',m',l' in TC(A_n0,alpha2)
> are are in supp(C(A_0,alpha2)
> or have some f_n',alpha',alphaA or
> f_n',alphaA,alpha'
> in supp(C(A_n0,alpha2)).
We already got A_n',alpha',m',l' to be not
a member of our support set for C(A_0,alpha2)
and A_n' uninvolved in supp(C) as above.
So we must have by these facts and the above claim
some alphaA has f_n',alpha',alphaA or
f_n',alphaA,alpha' in supp(C(A_n0,alpha2).
n' was picked to have A_n' completely uninvolved
in supp(C), so f_n',alpha',alphaA and
f_n',alphaA,alpha' re not in supp(C).
But our support set for C(A_n0,alpha2) was
supp(C) union {A_n0,alpha2,0,1}
see the quote next:
> We can make a support set for A_n0,alpha2 with any
>member of it for example (A_n0,alpha2,0,1} as suport
>set. Use that one for definiteness.
>
> Union in our basic support set from all throught for
>C, and obtain a support set for C(A_n0,alpha2))
>based on evaluating that as the definition of C
>applied to A_n0,alpha2.
So f_n',alpha',alphaA and f_n',alphaA,alpha'
are also not in supp(C(A_n0,alpha2)).
(Ie: union supp(C) with n' uninvolved union
singleon of an A_ atom and not an f_ .)
So this A_n',alpha',m',l' really does
contradict the conclusion of the claim:
>all A_n',alpha',m',l' in TC(A_n0,alpha2)
> are are in supp(C(A_0,alpha2)
> or have some f_n',alpha',alphaA or
> f_n',alphaA,alpha'
> in supp(C(A_n0,alpha2)).
So contrafiction.
From assuming that C had a definition making it
a proper cardinality style definition in the permutation
model.
So: QED cardinality is undefinable in the permuatation
model.
This model had singleton towers : A_n,alpha,m,l as
l varies.
The actual permutations on the model were on n, alpha, m.
The only purpose of the singleton towers was to make these
permutatiions possible.
If i want to permute a set, I have to permute the entire
transitive closure along with it to respect membership.
But with refualrity and well founded transistive closures
and extensionality I can't find the same shape of distinct
transitive closures to send to each other in permutations.
Infinite descending membership towers allow me to put
in distinct copies of isomorphic shapes of singleton towers
(hence allow non-trival perms) without violating
extensionality (by the non-well-foundedness).
This issue was also discussed in [1] & [2].
So the towers on the l's are just there to let the
n, alpha, m 's do their job permuting, and still keep
extensioinality.
So we can get as mentioned above to alternatives
to singleton towers.
I could also do something similar with doubleton
towers. Instead of making a transitive closuree like
w*, I could make it like a binary splitting tree.
So in A_n,alpha,m,l replace fininte ordinal
l by a finite sequence of 0's and 1's, to
make a binary tree and a doubleton tower.
Now I can't use the singletons from the tower
towards the Kuratosski ordered pairs and hence
the f_n,alpha1,aplha2 bijections, so make
new singleton atoms.
The only non-well-founded part is the atoms.
Later we add sets above then in well-founded
fashion.
So if we put in doubleton towers this way,
and don't explicitly add and singleton towers,
the rest of the construction doesn't add any either.
So we obtain a model like this with no singleton
towers (only doubleton ones).
Or similarly use any ordinal instead of 2,
finite sequences of ordinals from an index set.
So you could make versions with any particular
von Neumann cardinal style towers and no others.
I also mentioned earlier having no towers
of any well founded index set at each level.
To do this, make a tower branching like omega
at each level. But put in more permutations
into the group. Allow that index set to be
permuted. If you do the same permutation
action on each level of the tower all the
levels stay isomprphic in the resulting
permutation model.
But they also permute. So this way you
could make towers of amorphous sets at every
level in the tower.
And all well-founded index sets would not
have infinite descending membership towers
with each level isonorphic to that index set.
Or if you wanted to do something similar
for non-well-orderable sets but also
non-amorphous, put in at each tree level
the appropriate permutatiion action to
do whatever you wanted.
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
Well it seems that you proved that we cannot have a *defined*
cardinality over
ZF minus Regularity.
This comes in agreement with Gaunntt's proof , T.E.Forster was talking
about.
If this is the case then we must actually stipulate Cardinality as a
primitive one place function, and axiomatize it as I showed in
previous posts. There is no other resort. right.
I am speaking of ZF and NBG\MK theories of course, and not of NF and
related systems.
I'm actually even becoming to wonder weather the "recursive cardinals"
that I've presented at:
http://groups.google.com.jm/group/sci.logic/browse_thread/thread/d8d3012b6c7ebb87?hl=en
weather they work or not?
Although their I stipulated an anti-foundation axiom, so I am claiming
their that
this recursive cardinality works in ZF-Reg.+Anti-Foundation as
specified in that post, and I think they do work, and I think if I am
correct their, I think we may prove that this is the most we can get
with a "defined cardinality" to do, i.e. every set having definable
cardinality would be defined by those recursive cardinals.
Anyhow that is another subject.
Zuhair
David Libert
[Deletion]
> I will write out my claimed proof, constructing from a model of ZFC
> a model of ZF - regularity in which cardinality is undefinable.
>
> As noted above, these constructed models will satisfy ~regularity,
> ~AC and there exists sets x such that H_(x) is not a set (is
> instead a proper class).
The parent article, quoted from above is
[1] David Libert "The General Backround of Cardinality"
sci.logic Dec 22, 2009
http://groups.google.com/group/sci.logic/msg/16dc3c6329a74e35
When I first thought of this proof, I could not avoid making some
H_(x) a proper class. I was not trying to do that, it was a side
effect of the other things I had to put into the model to make
cardinality undefinable.
I was thinking on this, and thinking even without regularity and
without AC, we can still try Zuhair's early definitions of
cardinality.
One difficulty for those early definitions was the possibility
of H_(x) being a proper class for some x.
If instead H_(x) is always a set, that removes a difficulty
for those early cardinality definitions.
So I wondered if that was why my models with cardinality
undefinable were also getting some proper class H_(x)
by accident.
But then I remembered there was another problem for the
old definitions, apart from some H_(x) being proper classes,
namely some defined cardinalities being empty, as from
[2] David Libert "The magic of Hereditarily Hereditary Cardinals"
sci.logic, sci.math Nov 29, 2009
http://groups.google.com/group/sci.math/msg/1b40b261aeff6e96
So this left a gap. My models in [1] with cardinality undefinable
all had some H_(x) a proper class.
So these models leave open the possibility of some definition of
cardinality if all H_(x) are sets.
The old definition attempts look good if all H_(x) are sets,
but they also still have the problem of empty cardinalities.
So with the proof from [1] at that time if H_(x) are
all sets we couldn't make models with cardinality undefinable
and we couldn't define cardinality either.
So this was a left over unsettled case.
I was stuck on this for a while. But I think I finally got it.
I did find a way to modify the proof from [1] to make
models with cardinality undefinable and all H_(x) a set.
I thought of the [1] proof some time ago. But it took me a
while to actually write it out into article [1].
So when I wrote [1], I actually already knew the extension
of [1]'s proof to make all H_(x) sets.
In [1] though, I quoted earlier articles of mine, from before
finding that extension, and those earlier articles said all
my contructed models had some H_(x) proper classes, which
had been true then.
So [1] seemed complicated enough to write out the old proofs
with some H_(x) proper classes. So I just left that whole
side topic out of [1].
The main point of this current article is to write about
that extension of [1] getting all H_(x) to be sets.
I will discuss that now.
So first, why the [1] methods at first seem to have trouble
making all H_(x) sets.
(Well, literally the [1] proof makes some H_(x) a proper
class, so it is more than just "trouble". By trouble I mean
the first difficulties to modify the [1] proof to change that.)
So [1] makes a bunch of sets like A_n,alpha. Some of these
will be the ones messing up in turn each attempted cardinality
definition.
The details of the [1] proof show that to make a cardinality
for A_n,alpha you need to access some such A_n,alpha.
But it is hard to unformly do this simultaneously over
all cases A_n,alpha as n varies, by AC failures, induced
by permutatiions of the alpha's.
So we can't pick an alpha among the many alpha values
to make the common cardinality of all the A_n,alpha'
that are equipollent to A_n,alpha.
But there could be another try. Instead of picking one
alpha, just use them all. Instead make a definition of
cardinality that says collect all ways to do with all the
alpha.
But that's why I arranged the model to put a proper
class of alpha's in, so you can't collect all those cases
into a set.
If you made an alternative version of [1], where instead
of using all ordinals alpha you only used set many different
alpha, the resulting model from [1] would have cardinality
definable, along the lines as I just said.
This is a bit like Scott's trick. If we do Scott's
trick under ~AC, we don't have the AC to pick
a unique single cardinal representative, but we bundle
together set many of them, avoiding make the choice
by throwing them all in.
With regularity Scott's trick lets you bundle like
this in set size.
[1] messed this up for defining cardinality by
forcing such a bundle to be a proper class.
So we see how [1] needed a proper class of A-n,alpha 's
for fixed n as alpha varies.
On the other hand the ~AC part of the contruction to mess
up picking just one alpha wanted to permute among the alphas.
Permuting means having transitive closures carried to each
other. So we need a proper class of distinct but isomorphic
transitive closures, as [1] did.
And thats how H_(A_n,alpha) ended up being a proper class.
It included all the other A_n,alpha' 's .
So it seems various parts of the [1] construction force us
into some H_(x) proper classes.
With that background I will turn to the solution.
First off, I can't do it with singleton towers as the first part
of [1]. The later parts of [1] were about bigger towers than
singletons, and I must use something like that. But also with
another change from those late parts of [1].
Late in [1] I wrote about changing the singleton tower
version from A_n,alpha,m,l with l finite ordinal
originally to replace such l by finite seqences.
So instead of a linear singleton tower make a branching
tree of descending members.
For my new extended version I will do that: specifically
take the new l's to be finite sequences of finite ordinals,
so on first view an omega branching tree.
But the omega branching of the tree is what the outer ZFC
model sees, with access to the internal details behind
the A_n,alpha,m,l the indexing and so on.
The new definition will put permutations on the omega
copies as the tree branches. So from inside the permutation
model these branching trees will look like an amorphous
set at each branching.
I discusseed a point like this late in [1].
But now we get to the difference from what [1] did, and my
new extension to make all H_(x) sets.
[1] said to make the action of this permutation group
do the same permutation on every level of the tree branching.
As [1] moted, this makes all those copies of how the tree
branches isomorphic in the permutation model.
So here is how I will change that for the extension.
Put a permutaion on all levels of branching down the
tree, but make them all independent of each other.
No longer require the same permutatiin of the omega
copy of branching all through the tree.
This has the effect inside the permutation model
of making all branchings through the tree be
non-isomophic to each other.
Which is exaclty what I want to let H_(x) always
be sets.
But a further adjustment to make this really work.
I also in [1] deliberately made f_n,alpha1,alpha2
bijectiions of A_n,alpha1 to A_n,apha2.
And [1] made A_n,alpha be transive sets, I threw
into A_n,alpha all A_n,alpha,m,l l>0,
including l > 1 so deep down into the singleton
towers.
So the f_n,alpha1,alpha2 in supoport sets still
make isomorhisms all down the transitive closures
of the A_n,alpha1 A_n,alpha2 since they make those
sets isomorphic and they are transitive.
So instead change that only put into A_n,alpha
members A_n,alphja,m,<> the empty sequence
at the top of the tree.
So the bijections are only top level, and below
the new independent permuations throughout the tree
can vary independently.
So where [1] worked with A_n,alpha,m,l members
instread do A_n,alpha,m,<> ie last coord
doesn't vary.
So we have broken the connection of
TC(A_n,alpha1) and TC(A_n,alpha2) even though
at top level we have bijections.
Previously [1] got isomorphisms between those
TC 's in two ways: from the A_n,alpha
deliberate bijections, and from using the same
permutation on the omega branching all down the
tree.
So in the revised version, these TC's are
incomparable to each other. If there were maps
between them go outside of the union of the supports
and permute inputs while not permuting outputs.
Arguments like that show any map between must
have finite range.
So the question is, in this revised definition
of the model, can we still run a version of the
arguments from [1] to show cardinality is
undefinable.
Here is the key realization that allows this
to go through.
We want to permute various A_n,alpha1
to A_n,alpha2. As mentioned previously,
permutationms respect E memnership, so
this means we must be able to permute
TC(A_n,alpha1) to TC(A_n,alpha2).
Those TC are as evaluated inside the
permutation models.
But by the modification to the definition
of the model, such bijections of the
TC are no longer in the permutation model.
When I was stuck before, I would have
never tried a model definition like this,
because I knew I needed such bijections.
Here is the key point. I don't need those
bijections inside the permutation model.
Instread I have and use the permutations
in the outer enveloping AC model. That's
where we define the permutation model and
we reason about it. We can still do
permutation arguments, transferring results
around inside the model, by reasoning from
outside the model for those steps.
Like when you do Galois theory over a
field, the permutations are not members
of the field.
This is so obvious for Galois theory.
Its confusing for models of set theory,
since the permutations in question are the
sort of object that could have been inside
the little model.
But they actually don't need to be.
Some of the steps in the [1] proof go on
inside the permutatiion model, but the
permutation steps step outside to the outer
ZFC.
And the outer ZFC sees all those branching
trees are copies of omega branchuing everywhere.
So it has the required permutations.
The old [1] arguments never used that A_n,alpha
were transtive. So just permute and process as
in [1] on the top level A_n,alpha elements that
remain.
So those parts carry over. And [1] never needed
the permuations we were applying were inside the
little permutation model. So that's ok too.
So with all that, I think [1] repeats on the
new model, and we get cardinality is undefinable.
Now regarding H_(x) always being sets.
The heriditarily subnumerous. The branching
deep in the tree is independent of the top
level. And different parallel versions as alpha
varies have independent parts that are incomparable
to each other.
So we still have a proper class of A_n,alpha
as alpha varies, but they no longer make it into
each others H_(x) sets.
So any set build hereditarily from set many
atoms can't have other side atoms into its
hereditary subnumerous sets.
And building up the set membership levels,
these grow in well-founded fashion inhereted
form the out ZFC model where we have all
H_(x) are sets and have bounded rank.
So we can't add a proper class that way either.
Well this is sketchy but I think something like
this works.
That is as much as I will write for now on
this claimed constructiion of cardinality undefinable
and all H_(x) sets.
I will discuss more about the significance of this.
My [2] article raised another possible question
for attemtped cardinality definitions only using
strict subnumerous. Namely what if in ZF we had
distinct cardinals wit same predecessors.
We don't know if this is possible, but until
we have ruled it out is a problem case.
But for definitions involving at least one
clause of = or <= subnumerous (subnunerous
as opposed to strictly subnumerous) there
were only 2 gaps: if some H_(x) are
proper clases and if some defined cardinalities
are empty.
The proof that those cardinality definitions
work properly is ok if thoose two gaps are axiomatized
away.
So above if it's all correct I gave a ZF - regularity
model with all H_(x) sets, with cardinality
undefinable.
So the only gap not covered in that model is
defined cardinality being empty.
So if everything I just wrote is correct,
my model above must make those definitions
somwtimes define empty cardinality.
And indeed they do, from direct checking. I made
the deep TC levels incomparable to the top level
so not subnumerous to it.
[2] previously raised a question about empty
cardinality for those definitions.
But [2] jsut showed it was possible for these
definitions to define empty cardinality.
It left open the possibility that some variant
on those defintions would fix that problem and
get back nonempty cardinality and everything else
and make a good cardinality defintion.
The new construction from this article is
stronger. Because it shows no definition works,
even if you are allowed to assume H_(x) is
always a set.
--
David Libert ah...@FreeNet.Carleton.CA