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John Jones · Guest

The "class of all classes" sends us scurrying for alternatives to none,
one, some, or all. We don>t know how to identify this class that is the
class of all classes. Is it found among all classes, or among many, few
or none of them, or is it a class that is on its own, 'somehow' ? I hope
here to resolve the problems raised by the grammar of "class of all
classes".

DISCUSSION

Let>s get back to basics. If we count "all classes" then there are no
classes left. So there can>t be another class. But then comes this
"class" of all classes. It seems to be 'left-over' after our count of
all classes yet, because it is identifiable as a class, it is also among
all classes. What are we to do?

Here is the question we must pose (we need not pose it in the form of a
paradox, for that will waste time): how can something that contains all
things be identified as a thing? The problem is resolvable, at least as
far as we believe that the grammar of "class of all classes" brings us a
problem worth looking at.

The SOLUTION

The confusion we have when faced with the proposal "the class of
classes" is of the same sort we may get from negation. "Not-P" can refer
to identifiable alternatives to P such as "the absence of P", or simply
refer to other variables. However, "not-P" can also refer to the absence
of any P-related identifiable alternatives. This absence is not merely a
contingent "the absence of P", but a necessary absence. This necessary
absence is forged by eliminating the framework that allows us to refer
to P and its identifiable surrogates at all. In this case "not-P" voids
P and its surrogates (identifiable P-like objects). Let>s get used to
this idea of "framework".

The class of all classes is a framework for the manifestation of
identifiable classes, and is not an identiable class among classes. It
is not an object like other class objects. Granted, we may consider the
class-framework as an object if we want to, and I make no argument
against that, but it is not an identifiable object. Then, if we also
jettison identifability as a property of an object then we might
consider the class of all classes as a class object. It>s the best we
can do with the grammar of "class of all classes".

Rounding up; identifiable objects are found among none, one, some
(many), any and all. The class of all classes is a framework for the
manifestation of none, one, some, many and all, but it is not found
among them. So (and returning to the question posed in the discussion
(above)) in that sense only is the class of all classes a class-object;
and only in the sense of being a framework can the class of all classes
be a "container" of "all" classes. That>s as much as the grammar of
"class of all classes" allows.

MoeBlee · Guest

On Oct 2, 3:26 pm, John Jones <jonescard...@aol.com> wrote:

[quote]Let>s get back to basics.
[/quote]
You can>t, since you>ve never visited them in the first place.

MoeBlee

george · Guest

On Oct 2, 6:26 pm, John Jones <jonescard...@aol.com> wrote:
[quote]The "class of all classes" sends us scurrying for alternatives to none,
one, some, or all.
[/quote]
No, it doesn>t.

[quote]We don>t know how to identify this class
[/quote]
We can identify it more or less ANY WAY WE DAMN WELL PLEASE.
THIS IS LOGIC. It is about what follows FROM AXIOMS. This class
will be identified in DIFFERENT ways by DIFFERENT axiom-systems.
As long as you pick one that is consistent, NOBODY WILL OBJECT.
It>s YOUR axiom-system so YOU can do it howEVER YOU like!

So if YOU don>t know how to identify this class, SPEAK FOR YOURSELF.

[quote]Let>s get back to basics.
[/quote]
As MoeBlee has already said, you personally CANNOT do this because
YOU NEVER BOTHERED to familiarize yourself with the basics in the
FIRST place.
YOUR getting down to basics would be getting down FOR THE FIRST time,
NOT getting BACK.

[quote]If we count "all classes" then there are no
classes left. So there can>t be another class. But then comes this
"class" of all classes. It seems to be 'left-over' after our count of
all classes
[/quote]
NO, it does NOT so seem. If it is a class then it IS IN the class of
ALL classes,
so it must be a member of itself.

[quote]yet, because it is identifiable as a class, it is also among
all classes. What are we to do?
[/quote]
NOTHING, of course. This IS NOT a contradiction!
Nothing NEEDS to be done!

Ross A. Finlayson · Guest

MoeBlee wrote:
[quote]On Oct 2, 3:26 pm, John Jones <jonescard...@aol.com> wrote:

Let>s get back to basics.

You can>t, since you>ve never visited them in the first place.

MoeBlee

[/quote]
?

I don>t understand, Moeblee. What does that mean? What are you trying
to say? Confess, you>ve never visited in the first place, or else
don>t. Admittedly this discussion on this group has been going on for
years.

The notion of a class of classes does not exist in ZFC with classes,
because it would just be another irregular infinite structure where
classes contain only sets. It is just another "the" proper class.

So, the class of all classes would be just like the set of all sets,
because quantification is over the class variable instead of over the
set variable. This is used in logic proover systems, to quantify over
class variables, instead of set variables, because free quantification
is over the universe of sets itself, where, it is not true that there>s
a property that evaluates to true for all of them. Model adjustment in
the hierarchy of classes in the group noun game is sequential.

Yet, in ZFC logic proover systems where free quantification is over the
class variable, transfer principle reductions generally apply (where the
transfer principle applies). In MetaMath for example this is so.

So, if there>s reduction in transfer principle applications, that is
kind of like reduction over ordinals in problem space. It>s reflexive,
this model-theoretic model theory where there is the Skolemization over
the product space.

Yet, to admit that, then the standard model implies only true statements
about the objects of the theory, or else it wouldn>t exist, in the
consequential sense. Then it does so and its universe, itself, is
countable.

Given a well ordering of the reals on the unit interval, I known which
function it is. Given over the product space of the square of unit
reals a well ordering, for example the standard well-ordering, in ZFC,
where the reals are a set, that well ordering is a composition of
functions.

Standardly, that may be so.

Regards,

Ross F.

Peter Webb · Guest

"John Jones" <jonescardiff@aol.com> wrote in message
news:gc3hq4$ml5$1@aioe.org...
[quote]The "class of all classes" sends us scurrying for alternatives to none,
one, some, or all. We don>t know how to identify this class that is the
class of all classes. Is it found among all classes, or among many, few or
none of them, or is it a class that is on its own, 'somehow' ? I hope here
to resolve the problems raised by the grammar of "class of all classes".

DISCUSSION

Let>s get back to basics. If we count "all classes" then there are no
classes left. So there can>t be another class. But then comes this "class"
of all classes. It seems to be 'left-over' after our count of all classes
yet, because it is identifiable as a class, it is also among all classes.
[/quote]
Obviously, if A is the class of all classes, A e A, so its not "left over".

[quote]What are we to do?

[/quote]
Understand that classes are not the same as sets.

[quote]Here is the question we must pose (we need not pose it in the form of a
paradox, for that will waste time): how can something that contains all
things be identified as a thing? The problem is resolvable, at least as
far as we believe that the grammar of "class of all classes" brings us a
problem worth looking at.

The SOLUTION

The confusion we have when faced with the proposal "the class of classes"
is of the same sort we may get from negation. "Not-P" can refer to
identifiable alternatives to P such as "the absence of P", or simply refer
to other variables. However, "not-P" can also refer to the absence of any
P-related identifiable alternatives. This absence is not merely a
contingent "the absence of P", but a necessary absence. This necessary
absence is forged by eliminating the framework that allows us to refer to
P and its identifiable surrogates at all. In this case "not-P" voids P and
its surrogates (identifiable P-like objects). Let>s get used to this idea
of "framework".

The class of all classes is a framework for the manifestation of
identifiable classes, and is not an identiable class among classes. It is
not an object like other class objects. Granted, we may consider the
class-framework as an object if we want to, and I make no argument against
that, but it is not an identifiable object. Then, if we also jettison
identifability as a property of an object then we might consider the class
of all classes as a class object. It>s the best we can do with the grammar
of "class of all classes".

[/quote]
Before you prove anything about "classes", you are going to need either
define a "class" in terms of things we all understand (like well founded
sets) or provide some axioms for classes that we can use.

[quote]Rounding up; identifiable objects are found among none, one, some (many),
any and all. The class of all classes is a framework for the manifestation
of none, one, some, many and all, but it is not found among them. So (and
returning to the question posed in the discussion (above)) in that sense
only is the class of all classes a class-object; and only in the sense of
being a framework can the class of all classes be a "container" of "all"
classes. That>s as much as the grammar of "class of all classes" allows.
[/quote]
All you have done is invent a new word ("container") and say the class of
all classes is a container, therefore the problem doesn>t exist.

After you have defined/axiomitised "classes", please do the same for
"container". Just inventing new words without providing a definition is not
doing mathematics; its just wordplay.

MoeBlee · Guest

On Oct 2, 6:26 pm, "Ross A. Finlayson" <r...@tiki-lounge.com.invalid>
wrote:

[quote]The notion of a class of classes does not exist in ZFC
[/quote]
In ZFC we can define 'is a class' (x is a class <-> (x=0 or Ey yex))
and we can express the property of being a class of which all other
classes are members. Though, of course, we prove that there does not
exist such a class.

MoeBlee

John Jones · Guest

Peter Webb wrote:
[quote]
"John Jones" <jonescardiff@aol.com> wrote in message
news:gc3hq4$ml5$1@aioe.org...
The "class of all classes" sends us scurrying for alternatives to
none, one, some, or all. We don>t know how to identify this class that
is the class of all classes. Is it found among all classes, or among
many, few or none of them, or is it a class that is on its own,
'somehow' ? I hope here to resolve the problems raised by the grammar
of "class of all classes".

DISCUSSION

Let>s get back to basics. If we count "all classes" then there are no
classes left. So there can>t be another class. But then comes this
"class" of all classes. It seems to be 'left-over' after our count of
all classes yet, because it is identifiable as a class, it is also
among all classes.

Obviously, if A is the class of all classes, A e A, so its not "left over".
[/quote]
Then there>s no distinction between the class of all classes and another
class. But a distinction has been made. I need to know what is meant by
'of' in the class 'of' all classes. It seems as if you might be wanting
to break altogether with familiar meanings and to enter syntax.

[quote]What are we to do?

Understand that classes are not the same as sets.
[/quote]
Sets and classes are indistinguishable in my example. I am dealing with
their grounds - notions of totality, plurality, and wholeness or
containment.

[quote]Here is the question we must pose (we need not pose it in the form of
a paradox, for that will waste time): how can something that contains
all things be identified as a thing? The problem is resolvable, at
least as far as we believe that the grammar of "class of all classes"
brings us a problem worth looking at.

The SOLUTION

The confusion we have when faced with the proposal "the class of
classes" is of the same sort we may get from negation. "Not-P" can
refer to identifiable alternatives to P such as "the absence of P", or
simply refer to other variables. However, "not-P" can also refer to
the absence of any P-related identifiable alternatives. This absence
is not merely a contingent "the absence of P", but a necessary
absence. This necessary absence is forged by eliminating the framework
that allows us to refer to P and its identifiable surrogates at all.
In this case "not-P" voids P and its surrogates (identifiable P-like
objects). Let>s get used to this idea of "framework".

The class of all classes is a framework for the manifestation of
identifiable classes, and is not an identiable class among classes. It
is not an object like other class objects. Granted, we may consider
the class-framework as an object if we want to, and I make no argument
against that, but it is not an identifiable object. Then, if we also
jettison identifability as a property of an object then we might
consider the class of all classes as a class object. It>s the best we
can do with the grammar of "class of all classes".

Before you prove anything about "classes", you are going to need either
define a "class" in terms of things we all understand (like well founded
sets) or provide some axioms for classes that we can use.
[/quote]
I don>t think an appeal to axioms is appropriate at this point. It adds
another level of inquiry. I also think that that level of inquiry can be
attended to without axioms.

[quote]Rounding up; identifiable objects are found among none, one, some
(many), any and all. The class of all classes is a framework for the
manifestation of none, one, some, many and all, but it is not found
among them. So (and returning to the question posed in the discussion
(above)) in that sense only is the class of all classes a
class-object; and only in the sense of being a framework can the class
of all classes be a "container" of "all" classes. That>s as much as
the grammar of "class of all classes" allows.

All you have done is invent a new word ("container") and say the class
of all classes is a container, therefore the problem doesn>t exist.
[/quote]
We can consider it as a container or as a framework. I argued the latter.

[quote]After you have defined/axiomitised "classes", please do the same for
"container". Just inventing new words without providing a definition is
not doing mathematics; its just wordplay.
[/quote]
My point again is that a plea to axioms merely translates or diverts
difficulties.

You ask for a meaning to 'container'. How far do we allow the concept?
As far as we define it? What informed our definition? do you see my
point here. 'Container' and all words that we recognise, build axioms
that accord with that recognition. Recognition is epistemically prior to
law. There is no appeal to axioms.

Peter Webb · Guest

"John Jones" <jonescardiff@aol.com> wrote in message
news:gc6lif$jt6$1@aioe.org...
[quote]Peter Webb wrote:

"John Jones" <jonescardiff@aol.com> wrote in message
news:gc3hq4$ml5$1@aioe.org...
The "class of all classes" sends us scurrying for alternatives to none,
one, some, or all. We don>t know how to identify this class that is the
class of all classes. Is it found among all classes, or among many, few
or none of them, or is it a class that is on its own, 'somehow' ? I hope
here to resolve the problems raised by the grammar of "class of all
classes".

DISCUSSION

Let>s get back to basics. If we count "all classes" then there are no
classes left. So there can>t be another class. But then comes this
"class" of all classes. It seems to be 'left-over' after our count of
all classes yet, because it is identifiable as a class, it is also among
all classes.

Obviously, if A is the class of all classes, A e A, so its not "left
over".

Then there>s no distinction between the class of all classes and another
class.
[/quote]
Very poor language here.

The "class" of all classes is different to (say) the class of all sets, or
the class of all classes that don>t include themselves.

If you are arguing that the "class of all classes" is conceptually similar
to any other class, then that rather depends upon what you mean by a
"class". Interpreting "class" to mean the same as "set", what you say is
clearly wrong.

However, without some definition of what a "class" is, nobody can possibly
discuss its characteristics, as it is undefined.

[quote]But a distinction has been made. I need to know what is meant by 'of' in
the class 'of' all classes. It seems as if you might be wanting to break
altogether with familiar meanings and to enter syntax.

What are we to do?

Understand that classes are not the same as sets.

Sets and classes are indistinguishable in my example. I am dealing with
their grounds - notions of totality, plurality, and wholeness or
containment.

[/quote]
Well, if classes are indistinguishable from sets, they are sets (Axiom 1 of
ZF).

In this case, there is no "class of all classes" (deriving from Axiom 2 of
ZF).

[quote]Here is the question we must pose (we need not pose it in the form of a
paradox, for that will waste time): how can something that contains all
things be identified as a thing? The problem is resolvable, at least as
far as we believe that the grammar of "class of all classes" brings us a
problem worth looking at.

The SOLUTION

The confusion we have when faced with the proposal "the class of
classes" is of the same sort we may get from negation. "Not-P" can refer
to identifiable alternatives to P such as "the absence of P", or simply
refer to other variables. However, "not-P" can also refer to the absence
of any P-related identifiable alternatives. This absence is not merely a
contingent "the absence of P", but a necessary absence. This necessary
absence is forged by eliminating the framework that allows us to refer
to P and its identifiable surrogates at all. In this case "not-P" voids
P and its surrogates (identifiable P-like objects). Let>s get used to
this idea of "framework".

The class of all classes is a framework for the manifestation of
identifiable classes, and is not an identiable class among classes. It
is not an object like other class objects. Granted, we may consider the
class-framework as an object if we want to, and I make no argument
against that, but it is not an identifiable object. Then, if we also
jettison identifability as a property of an object then we might
consider the class of all classes as a class object. It>s the best we
can do with the grammar of "class of all classes".

Before you prove anything about "classes", you are going to need either
define a "class" in terms of things we all understand (like well founded
sets) or provide some axioms for classes that we can use.

I don>t think an appeal to axioms is appropriate at this point. It adds
another level of inquiry. I also think that that level of inquiry can be
attended to without axioms.

[/quote]

How are you going to prove anything about an object that is completely
undefined? Or are you saying that "class" means the same as "set", in which
case your question is easily answered - there is no class of all classes
because there is "no set of all sets".

[quote]Rounding up; identifiable objects are found among none, one, some
(many), any and all. The class of all classes is a framework for the
manifestation of none, one, some, many and all, but it is not found
among them. So (and returning to the question posed in the discussion
(above)) in that sense only is the class of all classes a class-object;
and only in the sense of being a framework can the class of all classes
be a "container" of "all" classes. That>s as much as the grammar of
"class of all classes" allows.

All you have done is invent a new word ("container") and say the class of
all classes is a container, therefore the problem doesn>t exist.

We can consider it as a container or as a framework. I argued the latter.
[/quote]
I asked you what a class is, and you say its a "container". I ask you what a
container is, and you say it is a "framework".

How about just listing the properties that define a class (or "collection",
or "framework", or whatever you decide to call this thingie that you want to
discuss).

[quote]
After you have defined/axiomitised "classes", please do the same for
"container". Just inventing new words without providing a definition is
not doing mathematics; its just wordplay.

My point again is that a plea to axioms merely translates or diverts
difficulties.

[/quote]
No, with some accepted facts about classes (aka "axioms") and a rule of
inference, you can actually say things about "classes". Without this, how
can you possibly say things about "classes" when they have no stated
properties at all?

[quote]You ask for a meaning to 'container'. How far do we allow the concept? As
far as we define it? What informed our definition?
[/quote]
Its your term, you define it.

[quote]do you see my point here.
[/quote]
The point I see is that you have no idea of what a class is, have not
presented a single fact about classes (ie any axioms), but want to discuss
them anyway.

Tell the rest of us what a "class" is in your mind, and perhaps somebody can
answer your questions. Until then, how are we supposed to discuss a
completely undefined object?

[quote]'Container' and all words that we recognise, build axioms that accord with
that recognition.
[/quote]
Go for it. Build some axioms. How about ZF minus the Ax 2 (the axiom of
regularity)? Whoops, that leads to an inconsistent theory, in which all well
founded statements can be be proved true, and in particular the statement A
&& ~A can be proved true.

[quote]Recognition is epistemically prior to law. There is no appeal to axioms.
[/quote]
This is waffle.

John Jones · Guest

george wrote:
[quote]On Oct 2, 6:26 pm, John Jones <jonescard...@aol.com> wrote:
The "class of all classes" sends us scurrying for alternatives to none,
one, some, or all.

No, it doesn>t.

We don>t know how to identify this class

We can identify it more or less ANY WAY WE DAMN WELL PLEASE.
[/quote]
There>s no big shakes to identifying.

[quote]THIS IS LOGIC. It is about what follows FROM AXIOMS.
[/quote]
Rubbish. Nothing follows from axioms. The only thing that follows from
axioms is the labour of picking them.

[quote]This class
will be identified in DIFFERENT ways by DIFFERENT axiom-systems.
As long as you pick one that is consistent, NOBODY WILL OBJECT.
It>s YOUR axiom-system so YOU can do it howEVER YOU like!

So if YOU don>t know how to identify this class, SPEAK FOR YOURSELF.
[/quote]
You>re saying that we can use the letters c-l-a-s-s to stand in for A,
B, etc.

I think you are saying that the axioms are epistemically prior to their
meaning.

[quote]Let>s get back to basics.

As MoeBlee has already said, you personally CANNOT do this because
YOU NEVER BOTHERED to familiarize yourself with the basics in the
FIRST place.
YOUR getting down to basics would be getting down FOR THE FIRST time,
NOT getting BACK.
[/quote]
I don>t know. I may never get back. I>m on a mission, Stardate 4627.

[quote]If we count "all classes" then there are no
classes left. So there can>t be another class. But then comes this
"class" of all classes. It seems to be 'left-over' after our count of
all classes

NO, it does NOT so seem. If it is a class then it IS IN the class of
ALL classes,
[/quote]
I said that.

[quote]so it must be a member of itself.
[/quote]
Gibberish. There>s no concept of 'a member of itself'. Where does this
term arise? in an axiom? I wonder what it could mean? This term, at
best, is a fond-remembered fiction that failed to work.

[quote]yet, because it is identifiable as a class, it is also among
all classes. What are we to do?

NOTHING, of course. This IS NOT a contradiction!
Nothing NEEDS to be done![/quote]

John Jones · Guest

Ross A. Finlayson wrote:
[quote]MoeBlee wrote:
On Oct 2, 3:26 pm, John Jones <jonescard...@aol.com> wrote:

Let>s get back to basics.

You can>t, since you>ve never visited them in the first place.

MoeBlee

?

I don>t understand, Moeblee. What does that mean? What are you trying
to say? Confess, you>ve never visited in the first place, or else
don>t. Admittedly this discussion on this group has been going on for
years.

The notion of a class of classes does not exist in ZFC with classes,
because it would just be another irregular infinite structure where
classes contain only sets. It is just another "the" proper class.
[/quote]
I am not sure what you are saying here. Are you merely 'translating' the
class of classes into infinite structure and sets? Is there some
structure to these things?
Another question, why would the 'class of classes' notion need to exist
in ZFC? I wonder if ZFC offers only translations of what went before it.

[quote]So, the class of all classes would be just like the set of all sets,
because quantification is over the class variable instead of over the
set variable.
[/quote]
Have you written that correctly? Why 'instead of'?

Also, I have to pull you up on the idea of quantification. What exactly
is it doing?

I have to say that I think these ideas are translations of the obscure.
It>s a bit of a con.

[quote]This is used in logic proover systems, to quantify over
class variables, instead of set variables, because free quantification
is over the universe of sets itself, where, it is not true that there>s
a property that evaluates to true for all of them.
[/quote]
I>m sorry. You>ve invoked "all" as testimony to the "universe". That>s
why you think you can say that "it is not true that there>s a property
that evaluates to true for all of them". You are a fraud, Sir. Don>t
fall into a ditch or you>ll be thrown out of town as a vagrant.

[quote]Model adjustment in
the hierarchy of classes in the group noun game is sequential.

Yet, in ZFC logic proover systems where free quantification is over the
class variable, transfer principle reductions generally apply (where the
transfer principle applies). In MetaMath for example this is so.

So, if there>s reduction in transfer principle applications, that is
kind of like reduction over ordinals in problem space. It>s reflexive,
this model-theoretic model theory where there is the Skolemization over
the product space.

Yet, to admit that, then the standard model implies only true statements
about the objects of the theory, or else it wouldn>t exist, in the
consequential sense. Then it does so and its universe, itself, is
countable.

Given a well ordering of the reals on the unit interval, I known which
function it is. Given over the product space of the square of unit
reals a well ordering, for example the standard well-ordering, in ZFC,
where the reals are a set, that well ordering is a composition of
functions.

Standardly, that may be so.

Regards,

Ross F.[/quote]

John Jones · Guest

Peter Webb wrote:
[quote]
"John Jones" <jonescardiff@aol.com> wrote in message
news:gc6lif$jt6$1@aioe.org...
Peter Webb wrote:

"John Jones" <jonescardiff@aol.com> wrote in message
news:gc3hq4$ml5$1@aioe.org...
The "class of all classes" sends us scurrying for alternatives to
none, one, some, or all. We don>t know how to identify this class
that is the class of all classes. Is it found among all classes, or
among many, few or none of them, or is it a class that is on its
own, 'somehow' ? I hope here to resolve the problems raised by the
grammar of "class of all classes".

DISCUSSION

Let>s get back to basics. If we count "all classes" then there are
no classes left. So there can>t be another class. But then comes
this "class" of all classes. It seems to be 'left-over' after our
count of all classes yet, because it is identifiable as a class, it
is also among all classes.

Obviously, if A is the class of all classes, A e A, so its not "left
over".

Then there>s no distinction between the class of all classes and
another class.

Very poor language here.

The "class" of all classes is different to (say) the class of all sets,
or the class of all classes that don>t include themselves.

If you are arguing that the "class of all classes" is conceptually
similar to any other class, then that rather depends upon what you mean
by a "class". Interpreting "class" to mean the same as "set", what you
say is clearly wrong.

However, without some definition of what a "class" is, nobody can
possibly discuss its characteristics, as it is undefined.

But a distinction has been made. I need to know what is meant by 'of'
in the class 'of' all classes. It seems as if you might be wanting to
break altogether with familiar meanings and to enter syntax.

What are we to do?

Understand that classes are not the same as sets.

Sets and classes are indistinguishable in my example. I am dealing
with their grounds - notions of totality, plurality, and wholeness or
containment.

Well, if classes are indistinguishable from sets, they are sets (Axiom 1
of ZF).

In this case, there is no "class of all classes" (deriving from Axiom 2
of ZF).

Here is the question we must pose (we need not pose it in the form
of a paradox, for that will waste time): how can something that
contains all things be identified as a thing? The problem is
resolvable, at least as far as we believe that the grammar of "class
of all classes" brings us a problem worth looking at.

The SOLUTION

The confusion we have when faced with the proposal "the class of
classes" is of the same sort we may get from negation. "Not-P" can
refer to identifiable alternatives to P such as "the absence of P",
or simply refer to other variables. However, "not-P" can also refer
to the absence of any P-related identifiable alternatives. This
absence is not merely a contingent "the absence of P", but a
necessary absence. This necessary absence is forged by eliminating
the framework that allows us to refer to P and its identifiable
surrogates at all. In this case "not-P" voids P and its surrogates
(identifiable P-like objects). Let>s get used to this idea of
"framework".

The class of all classes is a framework for the manifestation of
identifiable classes, and is not an identiable class among classes.
It is not an object like other class objects. Granted, we may
consider the class-framework as an object if we want to, and I make
no argument against that, but it is not an identifiable object.
Then, if we also jettison identifability as a property of an object
then we might consider the class of all classes as a class object.
It>s the best we can do with the grammar of "class of all classes".

Before you prove anything about "classes", you are going to need
either define a "class" in terms of things we all understand (like
well founded sets) or provide some axioms for classes that we can use.

I don>t think an appeal to axioms is appropriate at this point. It
adds another level of inquiry. I also think that that level of inquiry
can be attended to without axioms.

How are you going to prove anything about an object that is completely
undefined? Or are you saying that "class" means the same as "set", in
which case your question is easily answered - there is no class of all
classes because there is "no set of all sets".

Rounding up; identifiable objects are found among none, one, some
(many), any and all. The class of all classes is a framework for the
manifestation of none, one, some, many and all, but it is not found
among them. So (and returning to the question posed in the
discussion (above)) in that sense only is the class of all classes a
class-object; and only in the sense of being a framework can the
class of all classes be a "container" of "all" classes. That>s as
much as the grammar of "class of all classes" allows.

All you have done is invent a new word ("container") and say the
class of all classes is a container, therefore the problem doesn>t
exist.

We can consider it as a container or as a framework. I argued the latter.

I asked you what a class is, and you say its a "container". I ask you
what a container is, and you say it is a "framework".
[/quote]
WE know what a container is, and that knowledge will serve us here. A
container gathers, whether physically or by reference, objects. A
container is not one of the objects it contains.

[quote]How about just listing the properties that define a class (or
"collection", or "framework", or whatever you decide to call this
thingie that you want to discuss).
[/quote]
I said that a framework describes not the contingent possibility of
objects (like a container) but the necessary possibility of objects.

[quote]
After you have defined/axiomitised "classes", please do the same for
"container". Just inventing new words without providing a definition
is not doing mathematics; its just wordplay.

My point again is that a plea to axioms merely translates or diverts
difficulties.

No, with some accepted facts about classes (aka "axioms") and a rule of
inference, you can actually say things about "classes". Without this,
how can you possibly say things about "classes" when they have no stated
properties at all?

[/quote]
You are putting the problem at a further remove by saying that we must
look at accepted 'facts' about classes before we can talk about classes.
The facts themselves would need looking at just as much as the idea of a
class.

I also don>t know what to make of this idea that there are classes on
the one hand and facts about them on the other. Let>s not strip meaning
from concepts by treating their terms as strings of letters demanding
definition.

[quote]You ask for a meaning to 'container'. How far do we allow the concept?
As far as we define it? What informed our definition?

Its your term, you define it.
[/quote]
It>s the understanding we already have of the term container. A natural
language understanding is already in place. A word is not a string of
letters looking for a definition. There>s no such beast.

[quote]do you see my point here.

The point I see is that you have no idea of what a class is, have not
presented a single fact about classes (ie any axioms), but want to
discuss them anyway.
[/quote]
Classification is so common I wonder what your real intention is.
Classes, sets, groups, collections bags, hold-alls, containers are names
of contingent arrangements of, naturally, identifiable objects. My
argument was that, whichever term you wish to use, the idea of an object
that is, or is not, a "member of itself" is sufficient to scupper the
idea of an identifiable object. An object that is "a member of itself"
identifies the object as both arrangement and thing arranged. I said
that this object was not identifiable.

[quote]Tell the rest of us what a "class" is in your mind, and perhaps somebody
can answer your questions. Until then, how are we supposed to discuss a
completely undefined object?
[/quote]
Careful now, you are almost equating a simple object with a string of
letters. Defining an 'undefined object' is already assuming an
identifiable object. And that>s as much as I need.

[quote]
'Container' and all words that we recognise, build axioms that accord
with that recognition.

Go for it.
[/quote]
Semantics is prior to syntax. Laws are built on the things we recognise.
You can>t build a recognition from an axiom.

[quote]Build some axioms. How about ZF minus the Ax 2 (the axiom of
regularity)? Whoops, that leads to an inconsistent theory, in which all
well founded statements can be be proved true, and in particular the
statement A && ~A can be proved true.

Recognition is epistemically prior to law. There is no appeal to axioms.

This is waffle.
[/quote]
No, you are saying or indicating that we can appeal to axioms to get the
meaning of a term. That>s waffle.

Jan Burse · Guest

John Jones schrieb:
[quote]No, you are saying or indicating that we can appeal to axioms to get the
meaning of a term. That>s waffle.
[/quote]
There were once two fisher men. One was called
smarty and the other was called dumbass.

They had dispute over the neighbour village.
Smarty told dumbass that the neigbour villager
will kill everybody that comes into their fishing
grounds. He told him that this is a written axiom
in the neighbour village.

Dumbass said that there are no such things as
axioms. He said the fish will decide, when I
am near the neighbour village. Because fish are
in the fishing ground, its only possible that
fish can decide whether he will be killed
or not.

So he went into the neighbur village fishing
grounds and was never seen again.

Bye


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