Recursive Cardinals.
zuhair
Define(circular): x is circular <-> x e TC(x)
Define(hereditarily non circular):
x hereditarily non circular <->
for all y ( y e TC(x) -> ~ y is circular)
Define(x-recursive):
S is x-recursive <->
S equinumerous to x &
for all y ( y e TC(S) -> y equinumerous to x )&
S hereditarily non circular.
in other words
S is x-recursive <->
S equinumerous to x &
S hereditarily equinumerous to x &
S hereditarily non circular.
Now we can define Cardinality in ZF minus Regularity as:
-----------------------------------------------------------------------
Define(Cardinality(x)):
Cardinality(x) is the class of all x-recursive sets.
-----------------------------------------------------------------------
So for example the Cardinality of the empty set { }
is the class of all sets equinumerous to { }
that are hereditarily equinumerous to { }
and since we have Extensionality then this would be
the class { { } }
So Card({}) = { {} }
Now the cardinality of any singleton set {y} would be
the class of all recursive singletons, i.e
the class of all hereditarily non circular
singleton hereditarily singleton sets.
The cardinality of any doubleton sets {y,z} were ~y=z
is the class of all recursive doubletons, i.e
the class of all hereditarily non circular
doubleton hereditarily doubleton sets.
Same thing is applied for every set x.
Now we may call these cardinals as :
"Recursive Cardinals".
Now what is the proof in ZF minus Regularity
that these cardinals would be *set*s
and not proper classes?
The proof is the following Lemma in ZF minus Regularity:
For all x , for all y
y e TC(x) if and only if there exist a finite sequence
<x0,x1,x2,...,xn> were x0 e x and
xi+1 e xi for every i=0,1,2,...,n-1 and y=xn.
The most important thing is that because of the Lemma
above the class of all x-recursive sets for a particular x would be
a set because the process is exhaustive!, we cannot go
on having increasing numbers of these x-recursive sets
in such a manner as to have a proper class of these
x-recursive sets for a particular x.
While with the Ur-elements and with Quine atoms, we can
have proper classes of them, in such a manner that
for each x there would exist a proper class of sets of Quine atoms
or of Ur-elements that are equinumerous to x.
The reason is that with Ur-elements and Quine atoms there is
nothing to check on their identity, i.e. their identity is not
checked by their membership
(the identity of the members they contain)
they are self-determined identity objects so one can proliferate
them at will to the size of proper classes.
For example we can have the axiom
For all x Exist u
(for all y (y e u -> y is a Ur-element) & u equinumerous to x)
we can have the same axiom with Quine atoms.
For all x Exist u
(for all y (y e u -> y is a Quine atom) & u equinumerous to x)
BUT, we cannot have the same axiom with recursive sets.
i.e. the following is refuted in ZF minus Regularity:
For all m For all x Exist u
(for all y (y e u -> y is m-recursive) & u equinumerous to x)
That's why we can use these recursive sets to define cardinals.
That's why these recursive cardinals are sets!
Of course all of the above needs the axiom:
Axiom of Anti-Foundation:
For all x Exist y ( y is x-recursive ).
I don't think there is a way in ZF minus Regularity to construct
these sets without the above axiom.
The above is an example of actually defining Cardinality without
Regularity and Choice, and without Coret's assumption also.
Of course these recursive cardinals are shunned in ZF.
However to define cardinality that works both in ZF and
ZF minus Regularity one can actually have a joint definition of Scott
cardinals and the above.
--------------------------------------------------------------------------------------------
Define(Cardinality(x)):
Cardinality(x) is the class of all pure well founded sets Equinumerous
to x of the least possible rank, and of all x-recursive sets.
---------------------------------------------------------------------------------------------
"A set is said to be pure iff no Ur-element exist in its transitive
closure".
In ZF this would be reduced to the Scott cardinals.
In ZF minus Regularity + Anti-Foundation
then outside Coret's assumption
this will be reduced to the recursive cardinals,
While when satisfying Coret's assumption
the cardinal would be the union of a Scott cardinal
and a recursive cardinal.
Zuhair
---------------------------------------------------------------
" And treat the sickness by what caused it in the first place"
An ancient saying!
zuhair
> > Define(Cardinality(x)):
>
> > Cardinality(x) is the class of all pure well founded sets Equinumerous
> > to x of the least possible rank, and of all x-recursive sets.
>
> > "A set is said to be pure iff no Ur-element exist in its transitive
> > closure".
>
> > In ZF this would be reduced to the Scott cardinals.
> > In ZF minus Regularity + Anti-Foundation
> > then outside Coret's assumption
> > this will be reduced to the recursive cardinals,
> > While when satisfying Coret's assumption
> > the cardinal would be the union of a Scott cardinal
> > and a recursive cardinal.
>
> > Zuhair
> > ---------------------------------------------------------------
> > " And treat the sickness by what caused it in the first place"
>
> > An ancient saying!
How to construct these cardinals in ZF minus Regularity+Anti-
Foundation axiom above?
Take any set x.
According to the Anti-Foundation axiom, there must exist at least one
set y such that y is x-recursive.
Now we have for every set y, there exist the Transitive closure of y
"TC(y)" which is a set of course, as a theorem of ZF minus
Regularity.
Now we take the P_x(TC(y)) were P_x(z) generally stands for the set
of
all subsets of z that are equinumerous to x.
Since y is equinumerous to x and y is a subset of TC(y), then we have
x subnumerous to TC(y), then P_x(TC(y)) is non empty.
Now we continuing the P_x powering omega times i.e. we continue
powering like that
P_x(TC(y)), P_x(P_x(TC(y))), P_x(P_x(P_x(TC(y)))),....
Let's define these inductively as:
(P_x)0(TC(y)) = TC(y)
(P_x)1(TC(y)) = P_x(TC(y))
(P_x)2(TC(y)) = P_x(P_x(TC(y)))
generally (P_x)i+1(TC(y)) = P_x((P_x)i(TC(y))) for every finite
ordinal i.
Now (P_x)w(TC(y)) = Union(P_x)i(TC(y)) for i=0,1,2,3,.....
Now (P_x)w (TC(y)) would be the Cardinality of x, because it would be
the set of all x-recursive sets. (see the lemma in the first post).
So we can define Recursive Cardinals for *ALL* sets in ZF minus
Regularity + Anti-Foundation.
QED.
A second issue is that it would a theorem that
For all x , Cardinality(x) not subnumerous to x.
Otherwise we'll arrive at a paradox of Cardinality(x) e Cardinality
(x)
while at the same time Cardinality(x) is non circular. A
contradiction.
Thus it is a theorem of ZF-Regularity+Anti-Foundation that
Card(x) not subnumerous to x, for all x.
(of course I am speaking about Cardinality defined in the recursive
manner as detailed in this post, and the Anti-Foundation axiom refer
to the one written here in this post).
Zuhair
zuhair
David Libert, but I think it is better to be mentioned here.
Here I have presented an example of these x-Recursive Cardinals
were x is singleton, It shows that we cannot have
uncountably many recursive singletons, at least at informal level,
here these recursive singletons shall be called "singleton towers":
First, a singleton tower, or what I call a "recursive singleton" can
be defined in the following manner:
x is a singleton tower iff
x is singleton &
For all y ( y e TC(x) -> y is singleton ) &
For all y ( y e TC(x) -> ~ y e TC(y) )
So singleton towers are: hereditarily non circular singleton
hereditarily singletons.
we can symbolize that as x={{{... ...}}} is a singleton tower.
However for the sake of simplicity lets attach a natural number to
each
bracket in these singleton towers, i.e. let's number the brackets in
these towers,
so let's say that x=0{1{2{... ...}}}
i.e. the bracket number sequence of x is <0,1,2,3,.....>
so the outer bracket has the number 0, the one inside it has the
number 1, and the one inside it has the number 2, etc...
Now we can see that the object y defined by the bracket sequence
<1,2,3,....>
will be inside x which is defined by the bracket sequence
<0,1,2,3,....>
So in general the singleton tower xi+1 with bracket sequence of
<i+1,i+2,i+3,....> will be the member of the singleton tower xi with
bracket sequence of <i,i+1,i+2,i+3,...>
I think this is clear, so we have xi+1 e xi for all singleton towers
defined above.
Now lets take the tower were i=0, i.e. x0 with bracket sequence of
<0,1,2,3,....>
Now what is the transitive closure of x0
This would be
TC(x0)= {x1,x2,x3,..........}
right!
Now obviously TC(x0) is countable!
Now let's perform iterative singleton operations on x0, and let's use
the negative integers for that purpose, so we'll have
x(-1) = {x0} ,i.e. the bracket sequence of x(-1) is <-1,0,1,2,3,....>
x(-2) = { x(-1) } , i.e. the bracket sequence x(-2) is
<-2,-1,0,1,2,3,...>
.
.
.
(x(-i)) = {x(-i+1)} for all i=1,2,3,.....
Now how many x(-i) we have?
of course we have countably many of them! that is clear.
Now the set of all singleton towers would be
ST={...,x(-2),x(-1),x0,x1,x2,......}
Which is countable!
We cannot have more than those!
If we take the singleton power of ST
denoted as P1(ST) (i.e. the set of all singleton subsets of ST)
, we'll only have ST itself
ST=P1(ST)
Now because of the following lemma in ZF minus Regularity:
Lemma: For all x , for all y
y e TC(x) if and only if there exist a finite sequence
<x0,x1,x2,...,xn> were x0 e x and
xi+1 e xi for every i=0,1,2,...,n-1 and y=xn.
Then we cannot go down further, nor we can go up further, i.e. we
cannot have for example a singleton tower with a bracket sequence of
<0,1,2,.....,w,w+1,w+2,...> , this is forbidden by the lemma above,
also we cannot have a singleton tower with a bracket sequence of
<-w,.....,-2,-1,0,1,2,....>, or even <...,-2,-1,0,1,2,....>, all of
these cases are forbidden because of the lemma above.
Also we cannot have a set with a bracket sequence of for example
<0,2,3,4,...>, because this would be equal to the x1 i.e. the
singleton
tower with bracket sequence of <1,2,3,....>, also we cannot
have sets with bracket sequences of for example
<1,3,5,7,...>, or <1,2,3,5,7,11,...>, these have gaps in their
sequences
which is not compatible with the definition of singleton powers.
Actually I don't see how we can ever have an uncountable number of
these singleton towers?!
Actually even if we suppose that we can have these gap sequences
above, and by them one might conclude that we can have
an uncountable number of these singleton towers, but still the issue
remains,
how can we have a proper class of them???
Because of the lemma above it is clear that Singleton powering of ST
will stop giving different sets at some point, i.e. the process is
exhaustive!
So the class of all singleton towers, which is the cardinal number one
would be a set!
I might be mistaken though, but I would like to know how can we have
an uncountable number of these singleton towers? and even more
how can we have a proper class of them? I want a proof from within
the model, and not from outside it.
Zuhair
zuhair
I am coming to think that these recursive cardinals cannot do the job
really.
For example what would be the recursive cardinal of Aleph_1, this
would need
2^Aleph_1 of bracket sequences, But we only have maximally Aleph_1
of such sequences.
Actually I am coming to be that only singleton recursives are
possible, one
cannot larger recursive sets.
The reason I am saying that is because I think we actually cannot have
more than countable number of bracket sequences for any recursive
set,but if we say for example we have a recursive doubleton then it
seems that we'll need an uncountable number of them, which is not
possible, therefore we would have only singleton recursive sets.
IF this is correct, then matters would be much easier really. One
would easily define Cardinality if the axiom of strong Extensionality
(see:http://groups.google.com.jm/group/sci.logic/browse_thread/thread/
4bf8472f33458c84?hl=en)
is adopted.
Zuhair
David Libert
> The following is a modified quote from another subject in reply to
> David Libert, but I think it is better to be mentioned here.
>
> Here I have presented an example of these x-Recursive Cardinals
> were x is singleton, It shows that we cannot have
> uncountably many recursive singletons, at least at informal level,
> here these recursive singletons shall be called "singleton towers":
You are responding here to my claims to construct a model
of ZF - regularity with a proper class of singleton towers.
And my other models, or at least the last 2 about cardinality
being undefinable, heavily rested on being able to make a proper
class of singleton towers or something similar, as I wrote
myself recently.
You are skeptical about the possibility of a proper class of
singleton towers (indeed even uncountably many singleton towers).
So as you point out, if there is indeed a problem as you suspect
about a proper class of singleton towers that would show something
is wrong with my models, as you speculated recentlty in another
thread.
I agree with all that. If there is a problem with a proper
class of singleton towers then there is a problem for my models.
I still think a proper class of singleton towers is possible.
I will begin by trying to write out more clearly why I think
so.
I know you write below into the quoted article a claimed proof
there can only be countably many singleton towers.
Below I will preserve the quote of that and give detailed
answers to it.
But to start I will give more background to all this and
try to state my side more clearly. That will be a context
to answering your proof below.
I think a big point in all this is conceptual: what the proofs
are trying to do, what they are really claiming and what are
acceptable methods to be using in such proofs.
So I will try to express those areas as I understand them for
all this more clearly.
I will leave the quote of your article below to respond to it
in detail after my opening discussion.
But to get started, I will jump ahead and quote your closing
comments of the article, because I think they are important
for this whole issue of what is being discussed, and what
constitutes a proper proof.
So the quote from late in your article:
> I might be mistaken though, but I would like to know how can we have
> an uncountable number of these singleton towers? and even more
> how can we have a proper class of them? I want a proof from within
> the model, and not from outside it.
>
>
> Zuhair
This point, am I to give a proof from within the model, or from
outside, is this whole point I have about clarifying the overall
discussion.
So here we go.
We have been interested in the theory ZF - regularity.
You have been seeking to come up with a definition of
cardinailty in that theory or variants, and I have been trying
to produce models of that theory giving us some information
about what can and can't be done.
One possible sort of thing we can do in discussions like this
is to produce proofs inside that base theory ZF - regularity.
Whenever we produce such a proof, we are thereby showing that
every model of the theory satisfies the sentece we just proved,
So we have shown a property uniformly accross all models of the
theory, and have shown that on this issue of deciding that one
sentence all the models are similar to each other.
Another sort of way to understand how a theory like this works
is to argue model theoretically, and show some models of the
theory satisfy a sentence.
That allows the possibilty that some models satisfy that
sentence and some do not. If so we don't have the uniformity.
The models are not all the same in that regard.
To see how one model works, we have to look deeper into it
than merely knowing it satisifies the theory.
In the other case of outright proofs from the theory,
just knowing our model satisfied the theory was enough.
So the model theoretic version has the potential to be
more intricate and delicate than the base theory version.
And indeed, if we are the case of a sentence working
differently in different models, we are in the very case
of incompleteness, when the base theory cannot do the
job of supplying a proof.
So if we are to still get useful information we need
to go beyond that base theory,
As we can do with model theory. We can see the model
also from the outside, and have extra mathematical power
beyond that base theory.
So it is reasonable to expect to go outside the
base theory to make progress, if indeed we are in the
case of incompleteness. As would be the case if
model building arguments really were working.
So it a resonable exercise to step beyond the theory.
It can happen that that way we get new information about
the theory, which the results themselves show would
have been impossible just from inside the theory.
I will give a very simple example like this to illustrate
my point. Later I will work my way back to our real point
of discussion about singleton towers.
But to make suggesitive analogies I will start with
a really simple example, illustrating some of these
points.
So for the moment, replace the theory ZF - regularity
by just the theory of = only, with no other non-logical
axioms.
This theory can express a sentence saying there is
exactly one object in the universe. But it can't decide it.
Suppose you want to rigourously show that can't be decided
by the theory.
One part of showing that could be to construct a two element
model, thereby showing the theory doesn't prove that sentence.
So to mathematically construct such a model, I could be
working inside a larger theory that has 0 and 1 and proves
they are not equal. Then I could construct my model as
universe {0, 1}.
How do I show the sentence is false in that model? I have
to argue that 0 ~= 1. But any argument about properties
of objects must use the definitions of these objects. So
I must be using the definitions of 0 and 1, and my larger
theory information about them, to conclude that.
So I go outside of the theory of =, to get useful
information about that theory.
0 amd 1 are not definable in the simple theory of
equality.
When you go back down to that language. you can't
define them to distinguish them. Indeed the model
admits an automorphism carrying one to the other.
In that language the closest you can state relevant
to this is just the ~= itself, which is not a
satisfying proof.
And the base theory can't prove that anyway, by the
very incompleteness under discussion.
So the final point from this simple example is that
it can be a useful excercise to get information about
a theory to view it from a more powerful theory, and use
proofs in the more powerful language and theory to
study models of the simpler theory.
Let's step up a level, still simplified from
ZF - regularity, but closer to it.
Take the language and = and epsilon for
membership, but lets consider the weaker theory
of just usual extensionality.
We can make a first model of this theory, from
the badckground of conventional math, ie from
outside this weak theory.
Namely make the universe of the model be the usual
integers Z, and make epsilon correspond so
each n+1 is exactly the singleton of n.
Ie: declare the membership relation E interoreting
epilon has n E m <-> m = n+1 .
This would be a model of one singleton tower,
using that to correspond more closely to your
writing than mine: infinite in both directions.
Note that = in the model is interpreted as true
=. The model satifies that object are = or not
according to true =, ie as from the larger theory,
not according to realations to E.
I will illustrate this point with another structure,
which will turn out not to be a model of
extensionality.
Let a, b, c be 3 distrinct objects, and make
the model:
a c
\ /
\ /
\ /
b
where b E a b E c and nothing else.
Then in this structure, we don't suddenly get
a = c just because they have the same members, namely b.
Instread still a ~= c.
And what happens instead is extensionality fails.
So this structure is not a model of the theory of
extensionality.
Let's return back to models of extensinanality.
We previously had the Z model, with one singleton tower.
Now let's consider a 2nd model, made from 2 disjoint
isomprphic copies of Z, for example Z x 2 .
And make each copy be like the previous Z singleton tower
model, with no cross memberships between the 2 copies.
So this is a model of extensionality with 2 distinct
singleton towers.
Why are the singleton towers distinct? For example
why is <5, 0> ~= <5, 1> .
Because those are ~= in the outer theory where we did
the construction. And we are interpreting the = synbol
in the constructed model by the true =, ie that of
the outer working theory if you want to look at it that
way.
Yoy can't prove that from the simple theory of
extesinionalty. Because it can't even define 5, 0, 1
or < , > .
And there is nothing in that theory that tells you
there must be 2 distinct disjoint singleton towers
as this model has. Because the theorty of extensionality
doesn't prove that: by the first model.
Or indeed by a one element model with no singleton
towers.
Nothing from just the theory of extensionality
seems to force the towers distinct. If you try to
use extensionality to see <5, 0> ~= <5, 1>
you seem to first need to show <4, 0> ~= <4, 1>
to show they have different members which in turn
makes you need <3, 0> ~= <3, 1> and so on to
infinite regress.
But that's not how we define the = interpretarion
when we define a model. We define it by = in
a larger theory, and there we have more power.
So I can make this was a model of 2 disjoint
singleton towers.
But I can make another model with 3 towers.
And if I want to make class sized models with
definable classes, I can define a version
of a proper class of copies of Z.
These make models of extensionality, which
is a smaller theory than ZF - regularity.
In
[1] David Libert "A new definition of Cardinality"
sci.logic, sci.math Nov 23, 2009
http://groups.google.com/group/sci.math/msg/721cb8170033cf84
I made models of ZF - regularity, based in the same core
as above.
In there I had various sets in the final model made from
<alpha, n>. As n varies for alpha fixed it is a singleton
tower.
(A picky detail which may be a moment of confusion but
not ultimately importannt: i took n > 0 only and towers
only infinite downward not Z and two directions infinite
as this time I copied from your discussion.)
Why is the <alpha1, n1> tower disjoint from the
<alpha2, n2> tower for alpha1 ~= alpha2 ?
Because no tuples have <alpha1, n1> = <alpha2, n2>
when aloha1 ~= alpha2.
We don't argue these are distinct by invoking
extensionality. Because that involves us in an infnite
regress since the singleton descend infiintely.
We use ~= from the background theory to get ~=
in the constructed model.
And knowing ~=, we get from that to extensionlity.
Because those ~= pairs have ~= members: the singletons
1 step down. And those being ~= just depend on the background
~=, not more magic with extensionality.
All this in analogy to {0, 1} as a two element model of
the theory of equality. And we don't object to such a model
by saying we are allowed to analyse it in the weak theory of
equaltiy, and then complain that theory can't talk about
0 or 1, and can't prove anything ~=.
I feel my previous articles, [1] and the others, already
defined the models and outlined enough to show they have a
proper class of singelons as you were asking.
I have tried to see what could be pitfalls to
interpretting those previous articles this way,
and be explicit about the issues above.
I am about to turn to your further writing. I will
just note an important issue about that to come, is
as I indicated above, it is possible to have disjoint
singleton towers with no membership relations between
them.
So to return to your article:
Yes all ok. And as you say TC(x0) is countable.
> Now let's perform iterative singleton operations on x0, and let's use
> the negative integers for that purpose, so we'll have
>
> x(-1) = {x0} ,i.e. the bracket sequence of x(-1) is <-1,0,1,2,3,....>
> x(-2) = { x(-1) } , i.e. the bracket sequence x(-2) is
> <-2,-1,0,1,2,3,...>
> .
> .
> .
> (x(-i)) = {x(-i+1)} for all i=1,2,3,.....
>
> Now how many x(-i) we have?
>
> of course we have countably many of them! that is clear.
Yes. And this would form one singleton tower, all parts
iteratively connected, ordered like Z by membership
in transtive closure.
> Now the set of all singleton towers would be
>
> ST={...,x(-2),x(-1),x0,x1,x2,......}
No. This numerical notation moves you around in one
connected singleton tower. It doesn't let you move between
disjoint towers unconnected by epsilon.
As I showed in models, the simple ones above for just
theory of extensionality, or [1] for ZF - regularity.
> Which is countable!
>
> We cannot have more than those!
>
> If we take the singleton power of ST
> denoted as P1(ST) (i.e. the set of all singleton subsets of ST)
> , we'll only have ST itself
>
> ST=P1(ST)
>
> Now because of the following lemma in ZF minus Regularity:
>
> Lemma: For all x , for all y
>
> y e TC(x) if and only if there exist a finite sequence
> <x0,x1,x2,...,xn> were x0 e x and
> xi+1 e xi for every i=0,1,2,...,n-1 and y=xn.
>
> Then we cannot go down further, nor we can go up further, i.e. we
> cannot have for example a singleton tower with a bracket sequence of
> <0,1,2,.....,w,w+1,w+2,...> , this is forbidden by the lemma above,
> also we cannot have a singleton tower with a bracket sequence of
> <-w,.....,-2,-1,0,1,2,....>, or even <...,-2,-1,0,1,2,....>, all of
> these cases are forbidden because of the lemma above.
That lemma is ok for ome connected singleton tower.
But it doensn't say what happens with other disjiont
unconnected ones.
> Also we cannot have a set with a bracket sequence of for example
> <0,2,3,4,...>, because this would be equal to the x1 i.e. the
> singleton
> tower with bracket sequence of <1,2,3,....>, also we cannot
> have sets with bracket sequences of for example
> <1,3,5,7,...>, or <1,2,3,5,7,11,...>, these have gaps in their
> sequences
> which is not compatible with the definition of singleton powers.
The notatin is not well defined accross such independent
singleton towers.
> Actually I don't see how we can ever have an uncountable number of
> these singleton towers?!
You can't derive it just from ZF - regularity.
But you can constuct it as [1] into come models.
> Actually even if we suppose that we can have these gap sequences
> above, and by them one might conclude that we can have
> an uncountable number of these singleton towers, but still the issue
> remains,
> how can we have a proper class of them???
By putting a proper class of indices into the definition from outside
in [1] .
> Because of the lemma above it is clear that Singleton powering of ST
> will stop giving different sets at some point, i.e. the process is
> exhaustive!
> So the class of all singleton towers, which is the cardinal number one
> would be a set!
Each singleton tower only has countably many elements, as you wrote.
But there can be many of them. even a proper class as [1].
And they don't generate from each other by such operations.
> I might be mistaken though, but I would like to know how can we have
> an uncountable number of these singleton towers? and even more
> how can we have a proper class of them? I want a proof from within
> the model, and not from outside it.
>
>
> Zuhair
I have to argue from outside. But that's ok. It still shows me there
is a model. From the outside theory which we believe.
How do you show the theory iof equality has 2 element models, if
you are only allowed to argue from inside the pure theory of
equality?
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
> zuhair (zaljo...@gmail.com) writes:
>
> I have to argue from outside. But that's ok. It still shows me there
> is a model. From the outside theory which we believe.
>
> How do you show the theory iof equality has 2 element models, if
> you are only allowed to argue from inside the pure theory of
> equality?
>
> --
> David Libert ah...@FreeNet.Carleton.CA
First I want to thank you a lot a lot really, for writing such a nice
account, that is simple, clear, and straightforward. I appreciate the
time you've spent with all the subjects that I have raised, and I am
very thankful really.
Actually I was suspecting the possibility of singleton towers with
disjoint transitive closures, and I was suspecting that you were using
that.
I was obviously working with an axiom in my mind that I myself didn't
know about , and that is:
For any two singleton towers x,y ( x e TC(y) or y e TC(x) ).
This will cut down the possibility of singleton towers having disjoint
transitive closures.
But still I do think that even if you have these transitively disjoint
singleton towers, still you cannot have a proper class of them!
The reason is the following, Let me explain the bracket sequence
methodology that I began with because I see it easier.
I noticed that you misunderstood them actually.
x_i is the singleton tower with bracket sequence of <i,i+1,i+2,.....>
so suppose that i=0 then x_0 is the *singleton tower* with
bracket sequence of <0,1,2,3,....>
you can represent that as x_0 = 0{1{2{... ...}}}
so the outer bracket has the number 0, the one inside has number 1,
the one inside has number 2, and so one.
to understand this notation notice that the singleton tower x_1 would
have the bracket sequence of <1,2,3,....>
and you will have x_1 e x_0
so x_1 is a *singleton tower* and it is the sole member of the
singleton tower x_0
so each x_i is a singleton tower
Now each x_i+1 is a singleton tower and it is a member of the
singleton tower x_i.
I think this is pretty clear.
Now the transitive closure of x_0 is NOT a singleton tower as you
thought, the transitive closure of x_0 is actually not singleton at
all, it has countably many singleton towers in it (review the
definition of singleton tower as a *singleton* that is hereditarily
singleton and also that is non circular).
So TC(x_0) is NOT a singleton tower, it is a set of singleton towers
yes, but since it is itself not singleton, that's why it is not a
singleton tower itself.
TC(x_0) = {x_1, x_2, x_3,.....}
Notice that Each x_i for i=1,2,3,.... is a singleton tower.
Of course all sets in TC(x_0) are as you said not disjoint at their
transitive closures, i.e. we have the following
For all x,y e TC(x_0) ( x e TC(y) or y e TC(x) ).
That is definitely true. right.
And even my Omega powering of TC(x_0) would at the end yield what you
said
"singleton towers" that are not disjoint at their transitive closures.
However the matter is actually deeper than that!
Even if I suppose the existence of singleton towers that are disjoint
at their transitive closures level, i.e. having disjoint transitive
closures, even if we assume that, don't forget that the identity of
every singleton tower would depend on
the bracket sequence of it, and you know that from the lemma that I
have mentioned, you are only permitted to have a *countable* bracket
sequence for each singleton tower!
Now how much countable bracket sequences we can have?
the answer is:
we can have Power(omega) of these countable bracket sequences ONLY.
So even if you have singleton towers with disjoint transitive closures
still, you cannot have more than Power(omega) of them, because each
singleton tower must correspond to a countable bracket sequence.
All of that is a consequence of the Lemma that I mentioned which is a
LIMITING factor, it limits you from being able to stipulate that you
can have a proper classes of them at will, like the case with Ur-
elements and Quine atoms, or the CIRCULAR singleton towers (a Quine
atom is actually a circular singleton tower of the first degree).
So ONLY What I call as circular sets or Ur-elements can be
proliferated *at will* to the size of proper classes.
The net result is that: you CANNOT have a model of ZF-Reg. with a
proper class of these towers even if you do it from outside, because
the lemma from inside will
contradict this.
If you work from outside, then this should not contradict matters from
inside, lest you will end up with a sort of non-standard singleton
towers, as we have
non-standard naturals, and so on. But we don't want that! aren't we.
If you want to work from outside, then you should work in such a
manner that the lemma that I mentioned do not contradict with your
models, your work from outside must be one that we can test if it
contradict the lemma that I've mentioned or not? If we cannot show how
your work form outside is tested by the lemma I've mentioned, then
your work cannot be tested to know if it contradict this lemma or not,
then we cannot be sure of your models, aren't we.
Actually I came to realize lately that we might not be able to have
x-recursive sets were x>1. or even if we have these, then this must
come to an end,
and I think we cannot have x-recursive sets were x>= power(omega).
All of these results make me think that we can actually define
Cardinality using Scott trick.
Suppose every x such that x-recursive cardinal exist entail that x <
power(omega).
Then the class of *all* x-recursive cardinals is a set! lets call it
RC
Now RC will be the base set, and we build a hierarchy similar to the
Cumulative hierarchy ( a permutation model ) were RC is the base set.
so V0=RC
for every successor ordinal i , Vi+1= Power(Vi)
for every limit ordinal j , Vj = Union(i) Vi were i e j.
Obviously cardinality can be defined using Scott trick.
So it seems that my latest version of strong Extensionality that
shuns circular sets from existence, would actually work to save
defined cardinality.
Zuhair
zuhair
Actually this notation is confusing because one might think that
1,2,.. are members of x_0, but they are not really, they are only
markers for the brackets themselves, may be a better notation would
be:
x_0 = 0_{ 1_{ 2_{... ...}}}
zuhair
Sorry: the set union of the set of all recursive cardinals, would be
the base set, so it would be RC.
David Hartley
<88909956-acd2-407d...@k19g2000yqc.googlegroups.com>,
zuhair <zalj...@gmail.com> writes
>Now how much countable bracket sequences we can have?
> the answer is: we can have Power(omega) of these countable bracket
>sequences ONLY.
How do you distinguish between the sets with the bracket sequences
<1, 2, 3, ...> and <0, 2, 3, ...>? Are they not the same by
Extensionality?
--
David Hartley
zuhair
> In message
> <88909956-acd2-407d-a673-c42065e5a...@k19g2000yqc.googlegroups.com>,
> zuhair <zaljo...@gmail.com> writes
>
> >Now how much countable bracket sequences we can have?
> > the answer is: we can have Power(omega) of these countable bracket
> >sequences ONLY.
>
> How do you distinguish between the sets with the bracket sequences
> <1, 2, 3, ...> and <0, 2, 3, ...>? Are they not the same by
> Extensionality?
Yea, in reality this is a difficult question, if you read what David
was speaking about different models, then we can actually have them
stand for different sets, however I was of the same opinion of yours
really, but this needs to be better defined.
Thanks
Zuhair
>
> --
> David Hartley
zuhair
outside should not come in conflict with inside, and since David gave
nice simple examples, so let me clarify this point using his own
examples:
Lets take identity theory itself,
Lets add to it the following axiom:
For all x,y ( x=y )
Now we cannot work from outside and bring a model which has
Exist x ,y ( ~x=y ) ,
This would be in contradiction to working from inside.
So all models of this theory must not include distinct objects.
This matter is somewhat similar to the singleton towers, the lemma
that I mentioned would prevent us from having proper class of
singleton towers in *ALL* models of ZF-Reg.
However I might be mistaken of course.
I'll see what David Libert would come up with.
Zuhair
David Libert
> In my reply to David Libert above I mentioned that working from
> outside should not come in conflict with inside, and since David gave
> nice simple examples, so let me clarify this point using his own
> examples:
>
> Lets take identity theory itself,
>
> Lets add to it the following axiom:
>
> For all x,y ( x=y )
>
> Now we cannot work from outside and bring a model which has
>
> Exist x ,y ( ~x=y ) ,
>
> This would be in contradiction to working from inside.
Ok, I think I see what you are doing now.
In this identity theory example, you are saying consider instead
of the pure theory of identity, as I said earlier, it supplimented
by the extra axiom For all x,y (x=y).
In your discussions in previous articles about recursive cardinals
at one point you had mentioned an axiom: for every x there is
an x-recursive set.
And more recently you mentioned the axiom you had been thinking
of:
>I was obviously working with an axiom in my mind that I myself didn't
>know about , and that is:
>
>For any two singleton towers x,y ( x e TC(y) or y e TC(x) ).
>
>This will cut down the possibility of singleton towers having disjoint
>transitive closures.
That quoted from your Dec 27 article in this thread.
So I will write about what happens when we add these axioms and
similar ones to ZF - regularity.
First I quote the rest of your parent article:
> So all models of this theory must not include distinct objects.
>
> This matter is somewhat similar to the singleton towers, the lemma
> that I mentioned would prevent us from having proper class of
> singleton towers in *ALL* models of ZF-Reg.
>
> However I might be mistaken of course.
>
> I'll see what David Libert would come up with.
>
> Zuhair
So the question is what happens with the class of x-recursive
sets for various x in various models of ZF - regularity.
For general ZF - regularity models and various x there are
two issues: is the class of all x-recursive sets empty
and is the class of all x-recursive sets a proper class.
If either of these happen for a particular x it makes problems
for defining cardinality of x using x-recursice sets.
On the other hand, if neither of those happen, if that class
is a non-empty set, then we can make a good definition of
cardinality as the set of all x-recursive sets.
In general models of ZF - regularity either of those cases
is possible.
I had in my first article about cardinality being undefinable
written about getting a proper class of doubleton towers and
no singelton towers.
Along similar lines, the question about size of the class
of x-recursice sets as x varies: can be in any of the 3 cases
empty, non-empty set and proper class, in any simulataeous
pattern as x varies.
But as you say, we can also consider variants of
ZF - refularity by adding extra axioms, maybe axiomatizing those
cases away.
So one of your recent;ly suggested axioms was to axiomatize
away the emoty case: aciomatize that the class is always
non-empty. So that does get rid of one problem for the
definition.
You could also just axiomatize the other side too. I think
you did this in a recent article.
Just axiomatize that that class is for all x a set.
Or state them together as I think you did: axiomatize that
it is a non-empty set.
Doing that, you do get a good definition of cardality
as the set of all x-recursive sets,
So of course, if my models with cardinality are correct
they don't satisfy these extra axioms.
Let's consider instead your other recent axiom as I quoted
above:
>For any two singleton towers x,y ( x e TC(y) or y e TC(x) ).
In the singleton towers case as you explcitly said, this
does imply there are only countably many recursive singletons
as you have been saying.
So if you axiomatize that the class is non-empty and also
this comparabilty axiom above, you get a good definition
of cardinality for #x = 1.
There is the possibilty of generalizing your definition
to all x, ie doubleton towers etc.
I think I have come up with models similar to my previous
ones, using similar methods.
First, there can be a model where for every x the class
is a non-empty set. So this would be a model where
your definition of cardinality does work well for all x.
But I think I can also come up with models of the
analogue of your new comparability axioms above for
other cardinalities, ie x of other sizes, where
there is still a proper class of x-recursive sets.
These are for #x > 1.
So this comaparability axiom doesn't seem enough.
But if you want to axiomatize that all those classes
are non-empty sets directly, then it is ok for
definining cardinality.
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
No I am coming to think that this is impossible, because this will
exhaust the bracket sequences,especially if x is supernumerous to power
(omega).
>
> But I think I can also come up with models of the
> analogue of your new comparability axioms above for
> other cardinalities, ie x of other sizes, where
> there is still a proper class of x-recursive sets.
> These are for #x > 1.
>
> So this comaparability axiom doesn't seem enough.
>
> But if you want to axiomatize that all those classes
> are non-empty sets directly, then it is ok for
> definining cardinality.
>
> --
> David Libert ah...@FreeNet.Carleton.CA
No David I think you didn't catch my basic argument in my latest reply
to you.
Even without this axiom that I've wrote lately, I still think that you
cannot have more than Power(omega) of singleton towers. I shall quote
again:
This is the basic issue:
--Quote--
However the matter is actually deeper than that!
Even if I suppose the existence of singleton towers that are disjoint
at their transitive closures level, i.e. having disjoint transitive
closures, even if we assume that, don't forget that the identity of
every singleton tower would depend on
the bracket sequence of it, and you know that from the lemma that I
have mentioned, you are only permitted to have a *countable* bracket
sequence for each singleton tower!
Now how much countable bracket sequences we can have?
the answer is:
we can have Power(omega) of these countable bracket sequences ONLY.
So EVEN if you have singleton towers with disjoint transitive
closures
still, you cannot have more than Power(omega) of them, because each
singleton tower must correspond to a countable bracket sequence.
All of that is a consequence of the *Lemma* that I mentioned which is
--Quote finished---
So the basic issue is actually not related to these additional axioms
of mine you were talking about, it is related to the very basic
concept of having a proper class of singleton towers, which I still
don't see how can it be done EVEN from outside EVEN in absence of
these additional axioms.
I spoke about the resulting inevitable conflict between "outside" and
"inside".
The *lemma* and not the axioms would be the "LIMITING" factor on *all*
models from outside that define a proper class of singleton towers, so
it will contradict all of them.
So still we are differing about the same particular matter, I am
saying we cannot have a proper class of singleton towers in all models
of ZF-Reg., (unless these are non standard singleton towers, like how
we have non standard naturals), and you are saying that we do have
this proper class.
Zuhair
David Libert
> On Dec 29, 5:45=A0am, ah...@FreeNet.Carleton.CA (David Libert) wrote:
>> zuhair (zaljo...@gmail.com) writes:
>> > In my reply to David Libert above I mentioned that working from
>> > outside should not come in conflict with inside, and since David gave
>> > nice simple examples, so let me clarify this point using his own
>> > examples:
Yes, agreed, working from inside or outside should not give confliciting
answers.
>> > Lets take identity theory itself,
>>
>> > Lets add to it the following axiom:
>>
>> > For all x,y ( x=3Dy )
>>
>> > Now we cannot work from outside and bring a model which has
>>
>> > Exist x ,y ( ~x=3Dy ) ,
>>
>> > This would be in contradiction to working from inside.
My original discussion was for pure identity theory. We can also
change the discussion by considering instead a different theory.
And of course, once we are discussing the new theory, as you
say even working from outside we cannot make such a model as above.
None of this constradicts my earlier discussion, which was about
a different theory, namely the pure theory of =.
>> =A0 Ok, I think I see what you are doing now.
>>
>> =A0 In this identity theory example, you are saying consider instead
>> of the pure theory of identity, as I said earlier, it supplimented
>> by =A0the extra axiom =A0 For all x,y =A0(x=3Dy).
>>
>> =A0 In your discussions in previous articles =A0about recursive cardinals
>> at one point you had mentioned an axiom: =A0 for every =A0x =A0there is
>> an =A0x-recursive set.
>>
>> =A0 And more recently you mentioned the axiom you had been thinking
>> of:
>>
>> >I was obviously working with an axiom in my mind that I myself didn't
>> >know about , and that is:
>>
>> >For any two singleton towers x,y ( x e TC(y) or y e TC(x) ).
>>
>> >This will cut down the possibility of singleton towers having disjoint
>> >transitive closures.
>>
>> =A0 That quoted from your =A0Dec 27 article in this thread.
>>
>> =A0 So I will write about what happens when we add these axioms and
>> similar ones to =A0ZF - regularity.
>>
>> =A0 First I quote the rest of your parent article:
>>
>> > So all models of this theory must not include distinct objects.
>>
>> > This matter is somewhat similar to the singleton towers, the lemma
>> > that I mentioned would prevent us from having proper class of
>> > singleton towers in *ALL* models of ZF-Reg.
>>
>> > However I might be mistaken of course.
>>
>> > I'll see what David Libert would come up with.
>>
>> > Zuhair
Here though you are back to just ZF - regularity.
There are 2 cases: ZF - regularity, and ZF - regularity with the
added axiom
>> >I was obviously working with an axiom in my mind that I myself didn't
>> >know about , and that is:
>>
>> >For any two singleton towers x,y ( x e TC(y) or y e TC(x) ).
We get different answers for these 2 cases, so we nust keep the
2 duscussions distinct.
>> =A0 So the question is =A0what happens with the class of x-recursive
>> sets for various =A0x =A0in various models of =A0ZF - regularity.
>>
>> =A0 For general =A0ZF - regularity models =A0and various =A0x =A0there ar=
> e
>> two issues: =A0 is the class of all =A0x-recursive sets =A0empty
>> and =A0is the class of =A0all =A0x-recursive sets a proper class.
>>
>> =A0 If either of these happen for a particular x =A0it makes problems
>> for defining cardinality of =A0x =A0using x-recursice =A0sets.
>>
>> =A0 On the other hand, =A0if neither of those happen, if that class
>> is a non-empty set, =A0then we can make a good definition of
>> cardinality =A0as the set of all x-recursive sets.
>>
>> =A0 In general models of =A0ZF - regularity =A0either of those cases
>> is possible.
>>
>> =A0 I had in my first article about cardinality being undefinable
>> written about getting a proper class of doubleton towers and
>> no singelton towers.
>>
>> =A0Along similar lines, =A0the question about size of =A0the class
>> of =A0x-recursice sets =A0as x varies: =A0can be in any of the =A03 cases
>> empty, =A0non-empty set =A0and proper class, =A0in any simulataeous
>> pattern as x =A0varies.
I still agree with what I wrote above, except I should clarify that last
point. The easy version above would be any pattern of answers for
x a vo Neumann carddinal. So any definable class function on von
Neumann cardinals from the base ZFC model behind the construction
can be made the answers in a constructed ZF - regularity model.
It is hard to say what we generally mean by patterns in general
ZF - regularity models, because as you change the model you change
what all the x's are, so change what you are consdering the full
pattern of answers over.
>> =A0 But as you say, we can also consider variants of
>> ZF - refularity =A0by adding extra axioms, maybe axiomatizing those
>> cases away.
I am turning above to the question of variant theories by adding
axioms, simialr to what you seeemd to suggest above adding an
axiom to the opure theory of equality.
This is a new topic and adds to the discussion about
ZF - regularity, but doesn't conflict with it since they
are about different theories.
>> =A0 So one of your recent;ly suggested axioms was to axiomatize
>> away the emoty case: =A0aciomatize that the class is always
>> non-empty. =A0So that does get rid of one problem for the
>> definition.
>>
>> =A0 You could also just axiomatize the other side too. =A0I think
>> you did this in a recent article.
>>
>> =A0 Just axiomatize that that class is for all x =A0a set.
>>
>> =A0 Or state them together as I think you did: =A0axiomatize that
>> it is a non-empty set.
>>
>> =A0 Doing that, you do get a good definition of cardality
>> as the set of all x-recursive sets,
>>
>> =A0 So of course, if my models with cardinality are correct
>> they don't satisfy these extra axioms.
Anove I meant to write my models with cardinality undefinable.
>> =A0 Let's consider instead your other recent axiom as I quoted
>> above:
>>
>> >For any two singleton towers x,y ( x e TC(y) or y e TC(x) ).
>>
>> =A0 In the singleton towers case as you explcitly said, this
>> does imply there are only countably many recursive singletons
>> as you have been saying.
>>
>> =A0 So if you axiomatize that the class is non-empty and also
>> this comparabilty axiom above, =A0you get a good definition
>> of cardinality for =A0#x =3D 1.
>>
>> =A0 There is the possibilty of generalizing your definition
>> to all x, =A0ie doubleton towers etc.
>>
>> =A0 I think I have come up with models similar to my previous
>> ones, using similar methods.
>>
>> =A0 First, there can be a model where for every x =A0the class
>> is a non-empty set. =A0So this would be a model where
>> your definition of cardinality does work well for all x.
I still think that is correct. I have not yet written out the
details of that model. It is a construction, like my earlier
ones: a permutation model.
> No I am coming to think that this is impossible, because this will
> exhaust the bracket sequences,especially if x is supernumerous to power
> (omega).
I will be writing below about those bracket sequences. I don't think
they are applicable to models such as mine. So I still think that
model is ok, and the bracket sequence don't make a problem for it.
>> =A0 But I think I can also come up with models of the
>> analogue of your new comparability axioms above for
>> other cardinalities, =A0 ie =A0x of other sizes, =A0where
>> there is still a proper class of x-recursive sets.
>> These are for =A0#x > 1.
A different permutation model. This one I have also not yet
described in detail.
>> =A0 So this comaparability axiom doesn't seem enough.
>>
>> =A0 But if you want to axiomatize that all those classes
>> are non-empty sets directly, then it is ok for
>> definining cardinality.
>>
>> --
>> David Libert =A0 =A0 =A0 =A0 =A0ah...@FreeNet.Carleton.CA
I still agree with this last point I wrote above.
> No David I think you didn't catch my basic argument in my latest reply
> to you.
> Even without this axiom that I've wrote lately, I still think that you
> cannot have more than Power(omega) of singleton towers. I shall quote
> again:
I will be replying to that below.
> This is the basic issue:
>
> --Quote--
>
> However the matter is actually deeper than that!
>
> Even if I suppose the existence of singleton towers that are disjoint
> at their transitive closures level, i.e. having disjoint transitive
> closures, even if we assume that, don't forget that the identity of
> every singleton tower would depend on
> the bracket sequence of it, and you know that from the lemma that I
> have mentioned, you are only permitted to have a *countable* bracket
> sequence for each singleton tower!
I don't think these bracket sequence can be used to argue limits on
the number of singleton towers in general ZF - regularity models.
And I still think that my models giving a proper class of singleton
towers are ok.
I will discuss more about these bracket sequences, and why I think
they are incomplete for limiting the number of singleton towers.
To begin, I will review definitions and make a new definition.
From the base article by you of this thread:
>Define(circular): x is circular <-> x e TC(x)
>
>Define(hereditarily non circular):
>
>x hereditarily non circular <->
>for all y ( y e TC(x) -> ~ y is circular)
>
>Define(x-recursive):
>
>S is x-recursive <->
>S equinumerous to x &
>for all y ( y e TC(S) -> y equinumerous to x )&
>S hereditarily non circular.
>
>in other words
>
>S is x-recursive <->
>S equinumerous to x &
>S hereditarily equinumerous to x &
>S hereditarily non circular.
From your Dec 25 article of this thread:
>First, a singleton tower, or what I call a "recursive singleton" can
>be defined in the following manner:
>
>x is a singleton tower iff
>x is singleton &
>For all y ( y e TC(x) -> y is singleton ) &
>For all y ( y e TC(x) -> ~ y e TC(y) )
>
>So singleton towers are: hereditarily non circular singleton
>hereditarily singletons.
I was the one who first used the term singleton towers, and I did not
mean it this way.
Anyway, above, you are maknig the term singleton tower synomous with
recursive singleton.
Later on, in my Dec 26 article in this thread, I continued using
singleton towerr by my original meaning, instead of your stipulation above
that I just quoted.
In your Dec 27 reply in this thread to my Dec 26 article, you noted how
I was using "singleton tower" differently than your quote above and
your recent writing. I will quote that:
>Now the transitive closure of x_0 is NOT a singleton tower as you
>thought, the transitive closure of x_0 is actually not singleton at
>all, it has countably many singleton towers in it (review the
>definition of singleton tower as a *singleton* that is hereditarily
>singleton and also that is non circular).
>
>So TC(x_0) is NOT a singleton tower, it is a set of singleton towers
>yes, but since it is itself not singleton, that's why it is not a
>singleton tower itself.
I want to continue these discussions having clear communicaions.
So from now on I will change and use the definition of
"singleton tower" you have above, and not use my old definition
for it.
I still find it useful to talk about the idea of my old
defintion, so I will define new terms that neither of us have used
before with these definitions. From now on, I will take these to
be the definitions of these new terms.
So above you defined, and I am accepting and using that definition
henceforth, singleton towers to be the same as recursive singletons,
by the definition you gave above.
In that same Dec 27 article of yours above, replying to my Dec 26,
you wrote:
>I was obviously working with an axiom in my mind that I myself didn't
>know about , and that is:
>
>For any two singleton towers x,y ( x e TC(y) or y e TC(x) ).
So one possibility for discussion would be to consider the theory
ZF - regularity + that last axiom.
But you have also said you want to be considering the case of
just ZF - regularity.
So I will be continuing now mainly with that case
ZD - regularity, but I just wanted to recall that axiom to contrast
it with the general case here.
So in ZF - regularity we are considering recursuve singletons,
as you defined that above.
Consider the class of such recursive singletons.
(I know you claim this class cannot be a proper class. I am not
pre-judging to dismiss your claim, so when I call it a class I
am leaving open the possibility it is a set and not a proper
class.)
We can define the binary relation on the class of recursive
singletons: x e TC(y).
This is a partial order on this class.
I will define a chain of recursive singletons to be a set
of recursive singletons which is linearly ordered by this partial
order.
Given any recursive singleton x, we can form the set TC({x}).
This is a chain as I just defined, with the partial order above
on it ordered like the negative intergers.
Given such a recursive singleton x, we can also iteratively
form the iterated singletons above it : {x}, {{x}}, {{{xxx}}},
etc.
In ZF - regularity, we can form the set of all such finite
iterations of singleton over such an x.
That is also a chain and is ordered like omega.
We can union TC({x}) and that set of iterated singletons
above x.
The resulting set is also a chain, and is ordered like Z,
the full usual integers (including 0, positive and negative).
Define a maximal chain of recursive singletons to be a
chain of recursive singletons which is maximal among
all chains of recursive singletons with respect to inclusion.
Then given any recursive singleton x, the Z order type chain
of its treansitive closures and iterated singletons above is
a maximal chain of recursive singletons.
So every recursive singleton is a member of a maximal chain
of recursive singletons.
Any two distinct maximal chains of recuirsive singletons are
disjoint.
So the class of all recursive singletons is partitioned into
these maximal chains, where each chain is ordered like Z.
In your Dec 25 article of this thread you defined the bracket
sequences.
I will be discussing that definition and relating it to
my discussion above about maximal chains of recursive
singletons.
In your Dec 25 article, you began by discussing a bracket
seqence notation for one recursive singleton x
(you called it singleton tower, which by your definition which
i am acceopting is the same as recursvie singleton):
>x is a singleton tower iff
>x is singleton &
>For all y ( y e TC(x) -> y is singleton ) &
>For all y ( y e TC(x) -> ~ y e TC(y) )
>
>So singleton towers are: hereditarily non circular singleton
>hereditarily singletons.
>
>we can symbolize that as x={{{... ...}}} is a singleton tower.
>
>However for the sake of simplicity lets attach a natural number to
>each
>bracket in these singleton towers, i.e. let's number the brackets in
>these towers,
>
>so let's say that x=0{1{2{... ...}}}
>
>i.e. the bracket number sequence of x is <0,1,2,3,.....>
>so the outer bracket has the number 0, the one inside it has the
>number 1, and the one inside it has the number 2, etc...
You next extended this notation to members of TC(x) :
>Now we can see that the object y defined by the bracket sequence
><1,2,3,....>
>will be inside x which is defined by the bracket sequence
><0,1,2,3,....>
>
>So in general the singleton tower xi+1 with bracket sequence of
><i+1,i+2,i+3,....> will be the member of the singleton tower xi with
>bracket sequence of <i,i+1,i+2,i+3,...>
>
>I think this is clear, so we have xi+1 e xi for all singleton towers
>defined above.
A few lines later in the article, you extended these bracket
to iterated singletons above x :
>Now let's perform iterative singleton operations on x0, and let's use
>the negative integers for that purpose, so we'll have
>
>x(-1) = {x0} ,i.e. the bracket sequence of x(-1) is <-1,0,1,2,3,....>
>x(-2) = { x(-1) } , i.e. the bracket sequence x(-2) is
><-2,-1,0,1,2,3,...>
>.
>.
>.
> (x(-i)) = {x(-i+1)} for all i=1,2,3,.....
This was as much as you defined in that Dec 25 article. As far as I
have found that Dec 25 article was the only one defining the bracket
sequences. The other articles discussed this, but did not redefine it
or repeat the definition.
So yopu pick a specific recustrive singleton x (or call it singleton
tower as you did). You define bracket sequences for x and members
of TC(x). You also define bracket sequences for iterated singletons
above x.
So you have defined bracket sequences exactly for members of the
maximal chain of recursive singletons containing as member that
recursive singleton x .
I said the class of recursive singletons is partitioned into
maximal chains of recursive singletons.
So you have defined bracket sequences for exactly the members
of one partition piece.
Above you mentioned the axiom:
>I was obviously working with an axiom in my mind that I myself didn't
>know about , and that is:
>
>For any two singleton towers x,y ( x e TC(y) or y e TC(x) ).
This axioms is equivalent to the statement there is exactly one
maximal chain of recursive singletons.
So in the presence of that axiom, you have defined bracket sequenxces
for all recursive singletons.
In
[1] David Libert "A new definition of Cardinality"
sci.logic, sci.math Nov 23, 2009
http://groups.google.com/group/sci.math/msg/721cb8170033cf84
I constructed a ZFC - regularity model with a proper class of
recursive cardinals.
I still think that construction is ok.
You could modify that construction, to only put in one maximal
chain of recrusvie singletons: just take the alpha as 0 instead of varying overll all
ordinals as [1] did. (In [1] I was using my old definiton of singleton towers
So that would give a ZF - regularity model with exaclty one maximal chain,
and hence a model where your discussion above defined bracket sequences for all
recursive singletons.
But the original [1], or variants of it making alpha vary over less than a
proper class but still over more than one value, would give models of
ZF - regularity with more than one maximal chain of recursive singletons.
And even if you are suspicious of my [1] construction, until you have disproven
such cases your proof should consider the possibility of such cases.
And if my [1] and variant models are correct, you can't actually disprove those
cases anyway.
Your quoted discussion only gave those cases, using TC operations or singletons
above, so you are staying inside one maximal chain.
And you have no argument that there is only one maximal chain.
After you introduced those notations as I quoted above, in the Dec 27 article
you wrote
>Now the set of all singleton towers would be
>
>ST={...,x(-2),x(-1),x0,x1,x2,......}
There was no argument for this. It only follows if every recursive cardinal is
one of those listed, ie in that one maximal chain.
So you were implicitly assuming only one maximal chain.
But the whole basis of my claims of a proper class of recursive singletons
was a proper class of maximal chains, and you have proved no argument against
that. You are just implicitly assuming it doesn't happen.
I gave a construction in [1].
In other articles you did raise questions about that construction. You
asked what the atoms were in the construction, and if they messed up usual
extensonality. You also asked about doing the proofs from inside the
constructed model and not outside.
I answered each of these in 2 articles. I think my [1] account was
complete. My later articles were not changing anything from [1].
Just repeating what [1] had done in cleareer language.
Apart from those points which I answered, you did not make specific
criticisms of [1]. You just said maybe there is something wrong with
my models, because the conclusion goes against your discussion of
bracket sequences.
But now I have pointed out a gap in the discussion about bracket
sequences.
So I have the [1] construction which has no specific detailed
criticisms standing.
Against that is the bracket sequences, which I have criticisized.
I return you your parent article:
> Now how much countable bracket sequences we can have?
> the answer is:
> we can have Power(omega) of these countable bracket sequences ONLY.
This only limits the number of recusive singletons if you
had defined bracket sequences for all recursive sequences.
> So EVEN if you have singleton towers with disjoint transitive
> closures
> still, you cannot have more than Power(omega) of them, because each
> singleton tower must correspond to a countable bracket sequence.
You are assertnig that now, but your background discussion has not
shown it.
> All of that is a consequence of the *Lemma* that I mentioned which is
> a
> LIMITING factor, it limits you from being able to stipulate that you
> can have a proper classes of them at will, like the case with Ur-
> elements and Quine atoms, or the CIRCULAR singleton towers (a Quine
> atom is actually a circular singleton tower of the first degree).
Your lemma from the Dec 25 article:
>Lemma: For all x , for all y
>
>y e TC(x) if and only if there exist a finite sequence
><x0,x1,x2,...,xn> were x0 e x and
>xi+1 e xi for every i=0,1,2,...,n-1 and y=xn.
only puts limits on moving around within one maximal chain.
I have already agreed each maximal chain is countable.
> So ONLY What I call as circular sets or Ur-elements can be
> proliferated *at will* to the size of proper classes.
I showed how to do this in [1].
> The net result is that: you CANNOT have a model of ZF-Reg. with a
> proper class of these towers even if you do it from outside, because
> the lemma from inside will
> contradict this.
My [1] construction got all the axioms of ZF - regularity.
If you are claiming your lemma is provable from ZF - regularity
then it can't contradict my model which got all those axioms.
This from general principles and I have also noted a specific
problem with the discussion from your lemma.
> If you work from outside, then this should not contradict matters
> from
> inside, lest you will end up with a sort of non-standard singleton
> towers, as we have non-standard naturals, and so on.
> But we don't want that! aren't we.
Yes, inside and outside should give non-conflicing answers.
And I have said why there is a problem for inside opposing
outside.
> If you want to work from outside, then you should work in such a
> manner that the lemma that I mentioned do not contradict with your
> models, your work from outside must be one that we can test if it
> contradict the lemma that I've mentioned or not? If we cannot show
> how
> your work form outside is tested by the lemma I've mentioned, then
> your work cannot be tested to know if it contradict this lemma or
> not,
> then we cannot be sure of your models, aren't we.
The lemma is fine. The problem is using it to conclude there are
only countable manmy recursive singleons. Since the lemma is
only applicable to countably many singletons.
So the discussion using it only shows that countable set of
recursuve singletons is indeed countable.
> --Quote finished---
>
> So the basic issue is actually not related to these additional axioms
> of mine you were talking about, it is related to the very basic
> concept of having a proper class of singleton towers, which I still
> don't see how can it be done EVEN from outside EVEN in absence of
> these additional axioms.
See [1] and the other articles. I will give further references
below.
> I spoke about the resulting inevitable conflict between "outside" and
> "inside".
They will not conflict. If they do it shows the theories used are
inconsistent.
> The *lemma* and not the axioms would be the "LIMITING" factor on *all*
> models from outside that define a proper class of singleton towers, so
> it will contradict all of them.
Not if the lemma doesn't apply to all cases.
> So still we are differing about the same particular matter, I am
> saying we cannot have a proper class of singleton towers in all models
> of ZF-Reg., (unless these are non standard singleton towers, like how
> we have non standard naturals), and you are saying that we do have
> this proper class.
>
> Zuhair
I am saying it is consistent to have such a proper class, if ZF is
consitent.
It is also consistent to not have a proper class.
It is consistent to have models of ZF - regularity + ~ regularity
+ there are no recursive singletons.
This was discussed in an article among those I will reference below.
A few more points.
Later in you article as quoted above, you write about the possiblilty of
have P(omega) recursive cardinals. (You called them singleton towers).
Here is a possible approach along suich lines.
Above, you had picked a recursive singleton x, and used it to define
a bracket sequence for x's maximal chain.
Maybe you could extend such a definition to a second maximal chain,
by picking a member rectrusive singleton y there, and making a similar
defintion.
But now the two parts of the definition, on eahc maximal chain,
depeneded on the arbitrary choide of x or y, so its this dependency
that allows diostinct recursice singletons to get the same bracket
sequence.
For example, the most straigtforward copy over y of what you did
to x would be to assign both <0, 1, 2, ... > . But x and y are
still distinct, and each bracket sequence also depernded on base point
in the singleton tower.
You might try to make bracket sequences distinct by skipping numbers
in the bracket labelling inside y.
So you might want to keep going that way, same idea with different
skips on different maximal chains to get different bracket sequences.
So that gets you a P(omega) limit like you said.
But these were not intrinsic defnitions. They depended on different
skip patterns for different maximal chains.
So a plan like that being possble to apply to all maximal chains
only manages to apply to all of them if there were at most
P(omega) many maximal chains.
So again, to make a plan like that work, you have to assume the
very thing you were trying to prove.
This is the sort of thing that happens when you search for a proof
of something false. Proof strategies just expand out in circles.
And if my models are right then these claims are false.
And my models don't have standing criticisms.
Well the same thing I said above for the base case discussion.
I am saying the same applies for this extended version to use
P(omega).
Other issues. That outline I said just now depends on picking
a base point x or y or whatever in each maximal chain.
My [1] models admit automophisms. If you map each recursive
singleton to its own singleton you get a non-identity bijection
of the class of all recursove singletons to itself. This can
be lifted by transginfinite rwecursion up the ranks over those
to the entire model.
So there is no definable way to picj the x y etc from each
maximal chain.
So at very least, the bracket sequeebce assignment will
not be deinfable without parameters.
It depends on picking x y etc, anf then defining over
those parematers.
But our discussiin was about ZF - regulairty, not
ZFC - regularity.
[1] didn't dp this, but you could make a variant of [1],
similar to my other models, where you permute independly
within each maximal chain by shifitng the chain up or
down together, possibly different shifts on different chains.
Obtain a model where there is no choice set to pick
a base point in each maximal chain.
Given bracket sequences you can define the least recursive
singleton with leading positive bracket label. So from
bracket labelling you can recover a choice set of
recuirsuve singletons.
So this would be a model with no bracketing notation
inside as the recent discussion on infinitely many
maximal chains.
So the bracketing sequence won't be definable,
and in ~AC cases it might not even exist in the model as
a set.
And even when it exists, it can fail to extend to all
maximal chains.
The limitations of the notations don't show the
underlying objects are limited, they just show the
labellng of them is limited.
So these are the various problems with braceting
sequences as an attack on many recursice cardinals.
I will close with some references about my models.
I have had some other models besides [1].
These and [1] and background were referenced in
[2] David Libert " Extensionality and Circular objects"
sci.logic, sci.math Dec 23, 2009
http://groups.google.com/group/sci.logic/msg/f3ed79cb6bf9fb5e
I recently in this thread had those two followup articles on points
about these conistructions:
[3] David Libert "Recursive Cardinals"
sci.logic, sci.math Dec 26, 2009
http://groups.google.com/group/sci.logic/msg/2a5d12e702092b36
[3] David Libert "Recursive Cardinals"
sci.logic, sci.math Dec 29, 2009
http://groups.google.com/group/sci.logic/msg/a1b67d758208e941
In [3] I discussed how these constructions use atoms toward defining
sets in the final model.
In [4] I noted how and why these arguments work both from inside
and outside the constructed models.
--
David Libert ah...@FreeNet.Carleton.CA
David Libert
Correcting references:
David Libert (ah...@FreeNet.Carleton.CA) writes:
[Deletion]
> In
>
> [1] David Libert "A new definition of Cardinality"
> sci.logic, sci.math Nov 23, 2009
> http://groups.google.com/group/sci.math/msg/721cb8170033cf84
>
> I constructed a ZFC - regularity model with a proper class of
> recursive cardinals.
>
> I still think that construction is ok.
[Deletion]
> I will close with some references about my models.
>
> I have had some other models besides [1].
>
>
> These and [1] and background were referenced in
>
> [2] David Libert " Extensionality and Circular objects"
> sci.logic, sci.math Dec 23, 2009
> http://groups.google.com/group/sci.logic/msg/f3ed79cb6bf9fb5e
>
>
>
> I recently in this thread had those two followup articles on points
> about these conistructions:
>
>
> [3] David Libert "Recursive Cardinals"
> sci.logic, sci.math Dec 26, 2009
> http://groups.google.com/group/sci.logic/msg/2a5d12e702092b36
>
>
>
> [3] David Libert "Recursive Cardinals"
> sci.logic, sci.math Dec 29, 2009
> http://groups.google.com/group/sci.logic/msg/a1b67d758208e941
>
>
> In [3] I discussed how these constructions use atoms toward defining
> sets in the final model.
>
> In [4] I noted how and why these arguments work both from inside
> and outside the constructed models.
>
>
> --
> David Libert ah...@FreeNet.Carleton.CA
The article about the atoms was actually
[5] David Libert " Extensionality and Circular objects"
sci.logic, sci.math Dec 23, 2009
http://groups.google.com/group/sci.logic/msg/ecfd5cc4f0f3a072
The article about inside and outside the model was the first [3]
above. (I see when I cut and pasted I forgot to change [3] to [4]).
The second [3] above = [4] was about adding other axioms to
ZF - regularity.
--
David Libert ah...@FreeNet.Carleton.CA
David Libert
I will add some points to my account from the parent article
[0] David Libert "Recursive Cardinals"
sci.logic, sci.math Dec 29, 2009
http://groups.google.com/group/sci.logic/msg/8d9476e07a2e0b11
I wrote in [0] :
> So the class of all recursive singletons is partitioned into
> these maximal chains, where each chain is ordered like Z.
I did not mention explicitly in [0], but still true, is there
are no membership relations between distinct maximals chains.
(I had nade a similar comment to this in an older article before
I had used the word "chains" to talk about these).
So if you draw each maximal chain as a vertical line, with that
viertical line representing the Z ordering on the maximal chain
membership, the case of many maximal chains can be visualized
as a forest of vertical lines.
In [0] I wrote about making models with various numbers of
maximal chains:
> In
>
> [1] David Libert "A new definition of Cardinality"
> sci.logic, sci.math Nov 23, 2009
> http://groups.google.com/group/sci.math/msg/721cb8170033cf84
>
> I constructed a ZFC - regularity model with a proper class of
> recursive cardinals.
>
> I still think that construction is ok.
>
> You could modify that construction, to only put in one maximal
> chain of recrusvie singletons: just take the alpha as 0 instead of varying overll all
> ordinals as [1] did. (In [1] I was using my old definiton of singleton towers
>
> So that would give a ZF - regularity model with exaclty one maximal chain,
> and hence a model where your discussion above defined bracket sequences for all
> recursive singletons.
>
> But the original [1], or variants of it making alpha vary over less than a
> proper class but still over more than one value, would give models of
> ZF - regularity with more than one maximal chain of recursive singletons.
I went on to note in [0] that your main opening discussion earlier of
bracket sequences only defined them on one maximal chain.
I went on to suggest how it might be possible to extend that to more
maximal chains, involving P(omega) as you had been writing:
> A few more points.
>
> Later in you article as quoted above, you write about the possiblilty of
> have P(omega) recursive cardinals. (You called them singleton towers).
>
> Here is a possible approach along suich lines.
>
> Above, you had picked a recursive singleton x, and used it to define
> a bracket sequence for x's maximal chain.
>
> Maybe you could extend such a definition to a second maximal chain,
> by picking a member rectrusive singleton y there, and making a similar
> defintion.
>
> But now the two parts of the definition, on eahc maximal chain,
> depeneded on the arbitrary choide of x or y, so its this dependency
> that allows diostinct recursice singletons to get the same bracket
> sequence.
>
> For example, the most straigtforward copy over y of what you did
> to x would be to assign both <0, 1, 2, ... > . But x and y are
> still distinct, and each bracket sequence also depernded on base point
> in the singleton tower.
>
> You might try to make bracket sequences distinct by skipping numbers
> in the bracket labelling inside y.
>
> So you might want to keep going that way, same idea with different
> skips on different maximal chains to get different bracket sequences.
>
> So that gets you a P(omega) limit like you said.
I went on in [0] to discuss some ~AC issues about carrying on the
above plan:
So in term of the forest visualisation above, the laat quoted
permutation model is making independ vertical shifts in each
vertical line.
This makes ~AC troubles to pick a base point from each
vertical line, and jence as noted trouble to have sets in the
constructed model assigning bracket sequences to infinitely
many veritcal lines.
The above is the background to this present article.
I am writing this now to note an additional related point
I had not thought of when writng [0].
Above I was permuting vertically in each verrical line.
Actually not an arbitrary permutation: a vertical shift
respecting E membership.
My new point is we can also permute horizonally, permuting
one maximal chain to another.
The [1] models with more than one maximal chain have such
automorphisms.
And if there are infinitely many maxamal chain we can make
a permutation model based on such permutations.
If we do that with full permutations on omega many
maximal chain we get an amorphouse set of maximal chains.
In general, we could copy other permutation groups and
actions onto the maximal chains, and make the set of maximal
chains be as various sets made in permutation models.
Return to the case of full permuations on omega,
ie an amorphous set of maximal chains, for definteness.
In the resulting contructed ~AC model, there is no
injection of infinitely many maximal chains into P(omega),
by the usual sorts of permutation arguments.
(Ie in the finite support model, any function from
an infinite subset of the amorphous set into P(omega)
must have finite range).
So this makes an additional difficulty for extending the
definition of bracket sequences to infinitely many
maximal chains.
I earlier in [0] discussed vertical permutations.
I just introduced horizontal ones.
We can combione them anyway way we please. We could put
both into one model. Or we could put either one in and
leave out the other.
If both are left out, we construct a ZFC - regularity
model, and the current round of difficulties disappear.
We could make a set (though undefinable as discussed
above) assigning unique P(omega) bracket sequences
to to the members of #P(omega) or fewer many maximal
chains.
But the permutations if present make 2 distinct problems
for the construction. To pick a base point in each
maximal chain, and to assign uniqye P(omega)
differences across maximal chains.
In particular, if we use horizontal permutations and
no vertical ones, we get a model which has a choice
set picking a base point for each maximal chain, yet
we still can't have a set assigning bracket sequences to
infinitely many maximal chains by the trouble to assign
P(omega) elements to chains.
That concludes my comments about difficulties for
bracket sequences.
I will note further, I had said above to make the
horizontal permutations correspond to other ~AC
permutation constructions, and so get many kinds
of models like this.
I will note further this is ever more gereral for
all ~AC type sets.
Given any ZF + ~AC model and any set A in the model
we could redo a variant of the [1] construction, with that
~AC model replacing the ZFC base model from [1].
Redo the construction so alpha defining ther various
maximal chains varies over A members, instead of all
ordinals.
So obtain a ZF - regularity model where there maximal
chains are isomorphic to that A.
So any way that AC can fail involving a set A can be copied
over to maxinal chains.
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
wrote:
Thanks David, now matters are clearer. I shall elaborate on that also.
Zuhair
Transfer Principle
> Here I have presented an example of these x-Recursive Cardinals
> were x is singleton, It shows that we cannot have
> uncountably many recursive singletons, at least at informal level,
> here these recursive singletons shall be called "singleton towers"
As I've said earlier, I wanted to avoid these set theory debate
threads until February. But I've been lurking this thread for a
while now, and this discussion is turning out to be rather
interesting, and so I make this post now (as it may be hard to
find this thread in February).
In this thread, zuhair declares that he is trying to define a
new type of cardinality, "recursive cardinality." We already
know that many so-called "cranks" would also like to redefine
cardinality as well, so perhaps zuhair's "recursive cardinality"
will turn out to be more acceptable to "cranks" than standard
cardinality as defined by Cantor.
Now zuhair begins by defining a "singleton tower":
> First, a singleton tower, or what I call a "recursive singleton" can
> be defined in the following manner:
> x is a singleton tower iff
> x is singleton &
> For all y ( y e TC(x) -> y is singleton ) &
> For all y ( y e TC(x) -> ~ y e TC(y) )
> So singleton towers are: hereditarily non circular singleton
> hereditarily singletons.
Then zuhair attempts to construct a "recursive singleton":
> So in general the singleton tower xi+1 with bracket sequence of
> <i+1,i+2,i+3,....> will be the member of the singleton tower xi with
> bracket sequence of <i,i+1,i+2,i+3,...>
> I think this is clear, so we have xi+1 e xi for all singleton towers
> defined above.
But then, as zuhair noted earlier in this thread, most
standard theories, in particular ZF, prove that no recursive
singleton exists. The well-known proof of the nonexistence of
recursive singletons uses Foundation/Regularity, and so zuhair
proceeds by working in ZF-Regularity rather than ZF.
Yet I've mentioned in previous threads that Regularity alone
doesn't prove the nonexistence of most illfounded sets, but
usually requires additional axioms of ZF to prove. After all,
the axiom of Regularity only directly prescribes that every
nonempty be disjoint with one of its elements (its lone
element in the case of singletons). So we know that a set x
such that:
x = {x}
can't exist using Regularity alone, since the set x fails to
be disjoint with its lone element, namely itself. But to
prove the nonexistence of distinct sets x,y such that:
x = {y}
y = {x}
Regularity alone fails, since both x and y are disjoint with
their respective lone elements, namely each other. Of course,
we know that it's neither x nor y, but the set:
z = {x, y}
that fails to be disjoint with either of its elements. Notice
that this set is the _transitive closure_ of both x and y,
and its existence is guaranteed by the Axiom of Pairing. In
general, it's not the illfounded set itself that fails to be
disjoint with all of its elements, but its transitive closure
(or a subset thereof) that fails, and so we need other axioms
in addition to Regularity that guarantee the existence of the
transitive closure. This is why zuhair frequently refers to
the transitive closure in his posts, going so far as to
introduce the notation TC(x) for the transitive closure of x.
So to prove that no recursive singleton exists, we must look
first at its transitive closure, as zuhair does here:
> Now lets take the tower were i=0, i.e. x0 with bracket sequence of
> <0,1,2,3,....>
> Now what is the transitive closure of x0
> This would be
> TC(x0)= {x1,x2,x3,..........}
> right!
And we see that TC(x0) fails to be disjoint with any of its
elements, since (TC(x0) intersect xi) = xi+1 for any natural
number i. Thus, we can contradict the existence of x0 by
showing that its existence implies that of TC(x0). The usual
proof in ZF thereof begins by using the Axiom of Infinity to
derive the existence of an inductive set, omega. Then we use
the Replacement Schema to replace the natural number i in the
set omega with xi+1. The resulting set is TC(x0).
Therefore the proof that no singleton tower exists requires
not just Regularity, but Infinity and Replacement Schema. So
we now have choice of three theories, namely ZF-Regularity,
ZF-Infinity, and ZF-Replacement Schema (in other words, the
theory Z+Regularity) in which to work. In any of these three
theories, the proof that no singleton tower exists falls
through, and so we may attempt to work with singleton towers.
Of course, zuhair chose to drop Regularity, and this is what
most set theorists would do in this situation. But recall how
at the start of this post, my goal was to present zuhair's
recursive cardinality as acceptable to "_cranks_." And, given
a choice among Regularity, Infinity, and Replacement Schema
to drop, most "cranks" would choose _Infinity_ as the axiom
they'd like to drop. (Note that nearly every set theoretic
"crank" thread is basically an argument over the Axiom of
Infinity in ZF.) Notice that the general proof in ZF that
every set has a transitive closure requires Infinity.
Thus, instead of ZF-Regularity, I will choose to work in
ZF-Infinity for the remainder of this post. This theory
should be more acceptable to "cranks," including even the
ultrafinitists, since even an ultrafinitist should have no
problem with the existence of _singletons_, since all of
these sets xi are merely singletons! (On the other hand,
sometimes I'd prefer to work with sets in which not only is
xi+1 an element of xi, but a _subset_ of xi as well, so that
we can mimic von Neumann ordinals. But we won't do so here.)
I've attempted this approach in several previous threads,
but always seem to run into trouble. The problem is that
I want to look at the theory:
ZF-Infinity+"a singleton tower exists"
but the problem is that there appears to be no way to write
"a singleton tower exists" using only FOL and the language
of ZF-Infinity. For zuhair gives two definitions of his
concept "singleton tower," one using transitive closures and
the other using natural numbers as indices, but neither of
these are permitted in ZF-Infinity. Thus, even though that
there is likely a (nonstandard, of course) model of the
theory ZF-Infinity in which singleton towers exist, there's
no way to add an axiom prescribing there existence. (At
least I haven't found a way, yet.)
In another thread, someone -- possibly either of the other
posters is this thread, Hartley or Libert -- attempted to
give a theory of nonstandard arithmetic but noting that if
Goldbach's Conjecture is undecidable in PA, then GC would be
true in the standard model of PA, and so PA+~GC would give a
theory of nonstandard arithmetic. The language of the theory
would add an additional symbol "w" to the language of PA,
and w would be treated as a nonstandard natural number and a
counterexample of GC. But to me, this is unsatisfactory as
we don't _know_ whether GC is decidable in PA or not. I'd
prefer a statement that is _known_ to be undecidable in PA,
perhaps Goodstein's Theorem, and let "w" be a counterexample
to Goodstein rather than Goldbach.
And so we'd like to do the same in this set theory, adding
the symbol "ST" to the language of ZF-Infinity and stipulate
that ST is a singleton tower. But we can't do so unless we
know how to write "there exists a singleton tower" as a
(first-order) axiom in the language of ZF-Infinity.
Thus concludes this post for now. For February, I've already
stated that I'll return to working on the problem of finding
a rigorous definition of the "crank" ellipsis, and so I'll
work of that before returning to "recursive cardinality" for
the "cranks."
(BTW, someone asked me what a "crank" ellipsis is, so let me
answer that. We know that in standard analysis, some real
numbers, such as 1/3, cannot be written in decimal notation
with only finitely many digits, and so one often resorts to
using an ellipsis, so 1/3 = 0.333... now. But some "cranks"
want to extend the standard ellipsis in order to include
nonstandard reals such as 0.333...3, and often numbers such
as 0.999...9 as distinct from unity, with the difference
being something like 0.000...1. The "crank" ellipsis problem
entails finding a rigorous definition of ellipsis so that
these claims are provable.)
David Libert
> On Dec 25 2009, 6:28=A0am, zuhair <zaljo...@gmail.com> wrote:
[Deletion]
> Now zuhair begins by defining a "singleton tower":
>
>> First, a singleton tower, or what I call a "recursive singleton" can
>> be defined in the following manner:
>> x is a singleton tower =A0iff
>> x is singleton &
>> For all y ( y e TC(x) -> y is singleton ) &
>> For all y ( y e TC(x) -> ~ y e TC(y) )
>> So singleton towers are: hereditarily non circular singleton
>> hereditarily singletons.
>
> Then zuhair attempts to construct a "recursive singleton":
>
>> So in general the singleton tower xi+1 with bracket sequence of
>> <i+1,i+2,i+3,....> will be the member of the singleton tower xi with
>> bracket sequence of <i,i+1,i+2,i+3,...>
>> I think this is clear, so we have xi+1 e xi for all singleton towers
>> defined above.
>
> But then, as zuhair noted earlier in this thread, most
> standard theories, in particular ZF, prove that no recursive
> singleton exists. The well-known proof of the nonexistence of
> recursive singletons uses Foundation/Regularity, and so zuhair
> proceeds by working in ZF-Regularity rather than ZF.
>
> Yet I've mentioned in previous threads that Regularity alone
> doesn't prove the nonexistence of most illfounded sets, but
> usually requires additional axioms of ZF to prove. After all,
> the axiom of Regularity only directly prescribes that every
> nonempty be disjoint with one of its elements (its lone
> element in the case of singletons). So we know that a set x
> such that:
>
> x =3D {x}
>
> can't exist using Regularity alone, since the set x fails to
> be disjoint with its lone element, namely itself. But to
> prove the nonexistence of distinct sets x,y such that:
>
> x =3D {y}
> y =3D {x}
>
> Regularity alone fails, since both x and y are disjoint with
> their respective lone elements, namely each other. Of course,
> we know that it's neither x nor y, but the set:
>
> z =3D {x, y}
>
> that fails to be disjoint with either of its elements. Notice
> that this set is the _transitive closure_ of both x and y,
> and its existence is guaranteed by the Axiom of Pairing. In
> general, it's not the illfounded set itself that fails to be
> disjoint with all of its elements, but its transitive closure
> (or a subset thereof) that fails, and so we need other axioms
> in addition to Regularity that guarantee the existence of the
> transitive closure. This is why zuhair frequently refers to
> the transitive closure in his posts, going so far as to
> introduce the notation TC(x) for the transitive closure of x.
>
> So to prove that no recursive singleton exists, we must look
> first at its transitive closure, as zuhair does here:
>
>> Now lets take the tower were i=3D0, i.e. x0 with bracket sequence of
>> <0,1,2,3,....>
>> Now what is the transitive closure of x0
>> This would be
>> TC(x0)=3D {x1,x2,x3,..........}
>> right!
>
> And we see that TC(x0) fails to be disjoint with any of its
> elements, since (TC(x0) intersect xi) =3D xi+1 for any natural
> number i. Thus, we can contradict the existence of x0 by
> showing that its existence implies that of TC(x0). The usual
> proof in ZF thereof begins by using the Axiom of Infinity to
> derive the existence of an inductive set, omega. Then we use
> the Replacement Schema to replace the natural number i in the
> set omega with xi+1. The resulting set is TC(x0).
>
> Therefore the proof that no singleton tower exists requires
> not just Regularity, but Infinity and Replacement Schema. So
> we now have choice of three theories, namely ZF-Regularity,
> ZF-Infinity, and ZF-Replacement Schema (in other words, the
> theory Z+Regularity) in which to work. In any of these three
> theories, the proof that no singleton tower exists falls
> through, and so we may attempt to work with singleton towers.
Yes, that's right. In ZF, refuting singleton towers depends
on all three of regularity, infinity and replacement.
As you say, by dropping any one we can then get models
with singleton towers (if ZF is consistent).
You are right. It is hard in this theory to even define
what a singleton tower is.
Zuhair have a definition above. But he was working in
ZF - regularity, so he had infinity and replacement.
Based on those two, he had transitive closures exist
and are definable, so he could use them in his definition
of singleton tower.
I think I have a reasonable way to proceed in the theory
ZF - infinity.
In this theory we can't prove transitive closures exist
as a set.
It is not even so easy at first to define transitive
closures as a definable class.
The obvious way to define transitive closure of x,
TC(x) as a class would be the class of all y such that
there is a finite sequence of membership from x to y.
But that depends on having a definition of finite.
The usual definition of finite, with axiom of infinity
available, would be having cardinality a member of omega.
This doesn't work in ZF - infinity.
I think I do have a reasonable definition of TC(x)
as a class in ZF - infinity.
It is trying to formalize the idea above: a finite
chain of membership from x to y.
Any first order formalization will never fully capture
this full definition as seen from outside models.
But I think this one is a reasonable first order
approximation, just as the usual ZF definition
of omega is a reasonable first order approximation
to the notion of the set of finite ordinals, even
though there are nonstandard models of ZF with
interpretion of the omega definition including more
than finite ordinals as viewed from outside the model.
So here is the proposed definition of the class
TC(x) in ZF - infinity.
TC(x) is the class of those y such that there exists
a set A such that
A is linearly ordered by set inclusion &
that linear ordering on A has smallest and largest
elements &
{x} is a member of A &
{x} is the smallest member of A by set inclusion &
y is a member of the largest element of A by set
inclusion &
every member of A other than the largest element has
an immediate successor by set inclusion &
every member of A other than {x} has an immediate
predecessor by set inclusion &
for every members B, C of A, if B is immediate
predecessor of C then there is some z
such that C = B union {z} and z is not member
of C &
for every C, D members of A if C is immediate
predecessor of D and for z1, z2 such that
(C = {x} and z1 = x
or
there exists B in A such that B is immediate
predecessor of C and C = B union {z1}
and z1 not member of B) &
D = C union {z2} and z2 not member C
then z2 member of z1 &
if C is the largest member of A by set inclusion
and B is the immdediate successor of C then
C = B union {y} &
every subset E of A has a smallest element with
respect to set inclusion .
The set A above will code a finite sequence of distinct
members in a chain from x to y.
The coding above assumes A has more than one member.
If x member x we would want to use instead {{x}}
as the sequence, which my coding above won't allow for.
But I am working over ZF - infinity, so x member x
is imposible.
If you want this to work also without regularity and
handle x member x, easiest is disjunct the above
with disjunct : x member x and y = x .
That last conjunct above about E is trying to say in
first order that A is well-founded with respect to
set inclusion.
So a "well-founded" linear order with immediate
successors and predecessors except at endpoints,
this is trying to formalize A being finite.
This definition is strong enough to allow proofs
by induction on all TC(x) members by level of
membership nesting below x.
So I think this is a reasonable first order coding
of TC(X).
If you add back in infinity, and have the usual
definition of TC(x), that theory proves the two
TC(x) definitions are equivalent.
So accepting for the moment this formalization
of TC(x), we can now repeat Zuhair's definition
of singleton tower, using this TC(x) where Zuhair's
definition used TC(x).
So we come to a definition in ZF - infinity
of what a singleton tower is.
And it is possible to construct a model of
ZF - infinity + there is a singleton tower.
This even with regularity in the constructed model.
I did that in
[1] David Libert
"Axiom of infinity and the set of all hereditary finite sets"
sci.logic Oct 3, 2007
http://groups.google.com/group/sci.logic/msg/7593d4adf17732b7
Note, as [1] explictly noted, this is also a model with a set
x having no set TC(x).
[Deletion]
> And so we'd like to do the same in this set theory, adding
> the symbol "ST" to the language of ZF-Infinity and stipulate
> that ST is a singleton tower. But we can't do so unless we
> know how to write "there exists a singleton tower" as a
> (first-order) axiom in the language of ZF-Infinity.
I think above was a reasonable try at that.
[Deletion]
--
David Libert ah...@FreeNet.Carleton.CA
Transfer Principle
> Transfer Principle (lwal...@lausd.net) writes:
> > I've attempted this approach in several previous threads,
> > but always seem to run into trouble. The problem is that
> > I want to look at the theory:
> > ZF-Infinity+"a singleton tower exists"
> > but the problem is that there appears to be no way to write
> > "a singleton tower exists" using only FOL and the language
> > of ZF-Infinity. For zuhair gives two definitions of his
> > concept "singleton tower," one using transitive closures and
> > the other using natural numbers as indices, but neither of
> > these are permitted in ZF-Infinity. Thus, even though that
> > there is likely a (nonstandard, of course) model of the
> > theory ZF-Infinity in which singleton towers exist, there's
> > no way to add an axiom prescribing there existence. (At
> > least I haven't found a way, yet.)
Thanks for the information! Wow, this does seem long, but I
realize that this is the cost of dropping Infinity.
So how will this information in Libert's post above help us
define recursive cardinality for the so-called "cranks"? The
answer is, now that we have a singleton tower, we may go
back to zuhair's post and proceed as he does to define his
recursive cardinality. The hope is that for at least one "crank,"
recursive cardinality will be an acceptable alternative to the
standard (Cantorian) cardinality.
According to zuhair, singleton towers of uncountable "height"
may be impossible. The "cranks" may appreciate this, since
many of them are opposed to the notion of uncountability in
the first place. So we already have at least one point of
agreement between zuhair and the "cranks."
zuhair
Thanks lwal, that was a nice account.
Zuhair
zuhair
I think you meant ...having *finite* as a member of omega.
Actually the definition of "finite" has nothing to do
with weather omega exist or not.
The standard definition which is after one of Tarski's definitions of
'finite' is:
x is finite iff x is equinumerous to some natural number.
The definition of 'natural number' has no reference to omega at all.
x is a natural number iff
x is ordinal &
for all y ((y subset of x & y is ordinal & ~y=0) -> Uy e y ).
as one can see, Omega has nothing to do with this definition at all.
Or even a better definition of "finite" which is equivalent to the
above in Z would be:
x is finite iff
Exist R ( R is well ordering on x & converse(R) is well ordering on
x )
besides many other ways to do that.
Zuhair