Sets are

[Zuhair]

fictitious selective containers.
A selective container is a container that possess
a rule according to which it allows objects to be
inside it and do not allow others from entering it.
A member of a set is to be imagined
as an object allowed by that fictitious container
to enter it even if it cannot enter that container
so for example we can understand that
x is a member of x to mean that the fictitious
container x allows x to enter it despite weather
x can actually enter itself or not. Also
we can understand that x is an element of y
and y is an element of x to mean that
x allows y to enter it and also y allows x
to enter it. In general the set {x|phi} is
to be imagined as the fictitious container
that allows every phi object to enter into
it and that do not allow any non phi
object to enter into it.

Axioms of set theory lay out further
characteristics of those fictitious containers
and set rules of defining them from predicates.
The case of undefinable sets is to be
understood as fictitious containers that
have a selection rule but we cannot
describe it, so they are to be understood
as selective containers but describing this
selection is beyond the descriptive tools
of the theory in question. Do these fictitious
containers have a kind of existence that
is separate from human ideation?

I don't know!

Zuhair

[smn]

You can beg the question of whether your interpretation of sets (or
of all of mathemmatics for that matter ) is fictitous by referring to
them as abstract entities. Philosophy people are still arguing that
one with really not very logical arguments (sometimes).Regards ,smn

[porky_...@my-deja.com]

Yeah. As I was reading the original post, I thought "we may ask the
same questions about, say, natural numbers". Or any other mathematical
construct. Like, the way natural numbers are defined (Peano axioms),
the collection of natural numbers is infinite. We don't know if in
"reality" (whatever it means) there exists arbitrarily large
collections of whatever objects. So, we don't know how the concept of
natural numbers corresponds to reality. IMHO, it is "ideation". Like,
probably, everything else in math.

If we use ZF axioms, there is not universal set, "there is no
universe", as Halmos said. Again, IMHO, it's not such a big deal. We
don't strive to create a perfect model of the universe. Rather we need
something that serves our needs, and, in practice, we always consider
"universe of discourse".

PPJ.

[Zuhair]

> one with really not very logical arguments (sometimes).Regards ,smn- Hide quoted text -
>
> - Show quoted text -

Well by fictitious I meant what can be grasped by human imagery, it
doesn't entail
non existence in an independent manner from human imagery.

Zuhair

[Zuhair]

An obvious example of selectiveness that
we cannot describe is choice.

Zuhair

[Zuhair]

On Dec 15, 5:10 am, "porky_pig...@my-deja.com" <porky_pig...@my-

> PPJ.- Hide quoted text -

>
> - Show quoted text -

Agreed. Still that doesn't entail the non existence of the universe
in the real world, it could be the case that ZF is speaking about
part of it, I can say that there do not exist a set of all Von Neumann
ordinals in the real world, I can say that for sure, but weather the
set of all sets exist or not in the real world this is something that
I don't have an answer for.

Anyhow I agree with you no big deal.

Zuhair

[porky_...@my-deja.com]

(that's "whether", sorry, I'm not a spelling Nazi.)

> set of all sets exist or not in the real world this is something that
> I don't have an answer for.
>
> Anyhow I agree with you no big deal.
>
> Zuhair

Sometime I think of a set as a container, sometime I think of a set as
a list. So, we can list things. (OK, to be rigorous, to be able to
list anything, we need "Any set can be well-ordered", which is
equivalent to AC", let's not bother with those details). So, in
principle, at least, we can think of list of all sets. At least I have
no problems thinking of that. Now the question is whether such a
collection is a set. And now by "set" we don't mean just any arbitrary
container, but container subject to a number of restrictions (e.g., ZF
set of axioms). Apparently it's not, but that does not make it "non-
existent".

That's why I like NBG set of axioms better than ZF. ZF simply shoves
those bad boys aside and says "you do not exist". NBG says "you
certainly exist, but since you don't meet our criteria of being a set,
we therefore shall call you a "class". Well, to be precise, it says
"An arbitrary collection of object is a class, and furthermore, if
that collection meets some additional requirement, it has a right to
call itself a "set".

Note: that's the tendency I've noticed in some books who are based on
ZF. "What about those collections that are not sets? Or, relax, they
can't exists, forget about them!". But I *can* think of collection of
all cardinals, there's nothing wrong with that, damn it! So why should
I pretend that collection does not exist? So, IMHO, NGB approach is
more aligned with our intuition than ZF.

Regards,

PPJ.

[quasi]

On Thu, 15 Dec 2011 14:21:18 -0800 (PST), "porky_...@my-deja.com"
<porky_...@my-deja.com> wrote:

>Sometime I think of a set as a container, sometime I think of
>a set as a list. So, we can list things. (OK, to be rigorous,
>to be able to list anything, we need "Any set can be
>well-ordered", which is equivalent to AC", let's not bother
>with those details). So, in principle, at least, we can think
>of list of all sets. At least I have no problems thinking of
>that. Now the question is whether such a collection is a set.
>And now by "set" we don't mean just any arbitrary
>container, but container subject to a number of restrictions
>(e.g., ZF set of axioms). Apparently it's not, but that does

>not make it "non-existent".

>
>That's why I like NBG set of axioms better than ZF. ZF simply
>shoves those bad boys aside and says "you do not exist". NBG
>says "you certainly exist, but since you don't meet our
>criteria of being a set, we therefore shall call you a "class".
>Well, to be precise, it says "An arbitrary collection of
>object is a class, and furthermore, if that collection meets
>some additional requirement, it has a right to call itself a
>"set".
>
>Note: that's the tendency I've noticed in some books who are
>based on ZF. "What about those collections that are not sets?
>Or, relax, they can't exists, forget about them!". But I *can*
>think of collection of all cardinals, there's nothing wrong
>with that, damn it! So why should I pretend that collection
>does not exist? So, IMHO, NGB approach is more aligned with
>our intuition than ZF.

I agree!

While, I'm not knowledgeable about the axiomatic foundations,
and in particular, I have only vague notions of the features
of ZF and NBG, I would prefer a system less restrictive than
ZF with regard to which "collections" are formally allowable.

I also know very little about Category Theory, but I know
that in Category theory, "collections" such as the category
of all groups (with morphisms as group homomorphisms) _is_
allowed.

The awkwardness of not being able, in ZF, to regard the
collection of all groups as an object (capable of being an
element of another object) appears as a deficiency.

quasi

[quasi]

Ok, I'll (foolishly?) take a cut at this foundations stuff.

Given my half-remembered mild acquaintance with the standard
theory, I'll almost certainly copy parts of it, at least
the parts that I think I need.

Also, given my ignorance of the obstacles, I'll surely miss
my goal, either falling short or falling into some well known
traps (paradoxes leading to contradictions).

Still, it's an honest attempt, I'm not trolling, and I'll
surely give up the quest if I'm convinced that I've set up a
system which is inherently and irreparably inconsistent.

But what is the goal?

Basically, the goal is to allow larger collections than sets,
and to allow them as bona fide objects within some hierarchy
of objects. I definitely want there to exist objects whose
elements are, for example, "all sets", or "all groups", but
hopefully, it can go further, achieving much greater
generality with regard to the construction and allowability
of objects.

Ok, so here's my super-naive first (and maybe last) attempt ...

Undefined terms:

* universe
* object
* element of
* condition

Axioms:

* There exists a unique object U called the universe.

* Some objects are elements of other objects. Write
a [[ b if a is an element of b, or equivalently b ]] a.

* All objects except U are elements of U.

* Two objects are equal iff they have the same elements,
that is, a = b iff x [[ a => x [[b and x [[ b => x [[ a.

* There does not exist an infinite descending chain

a_1 ]] a_2 ]] a_3 ]] ...

of objects.

* Given objects a,b there exist objects

a U b, a /\ b, a \ b, a x b

defined in the usual way.

* A condition is a statement which for each object in U
other than U, is either true or false. Any such statement
is a condition.

* If C is a condition, there exists a unique object

{x | C(x)}

whose elements are all objects other than U which
satisfy the condition.

Ok, that's it.

I suspect my proposed system is defeated by some kind of
Russell-like paradox. If not, maybe it's inconsistent in some
other way? If so, please kill it quickly and painlessly.

If not obviously inconsistent, to what extent, if any, is it
different than the standard approach? I know I've adopted
(co-opted) many of the standard notions, but have I achieved
the greater generality expressed in my stated goal?

I apologize in advance for my naivety, and I know that I may
be inviting serious self-embarassment and potentially harsh
criticism, so if that happens, I'll accept it as something
that I brought on myself.

quasi

[Rupert]

On 15 Dez., 23:21, "porky_pig...@my-deja.com" <porky_pig...@my-

In the modal-structural interpretation advocated by Geoffry Hellman in
"Mathematics Without Numbers", sets are possible structures. It is not
possible that all the cardinals could exist at once. Given any
possible collection of cardinals, there exists another possible
cardinal not in the collection.

[Zuhair]

On Dec 16, 1:21 am, "porky_pig...@my-deja.com" <porky_pig...@my-

This is a very important point, and I mentioned this point many times
in this Usenet: our perception of a *collection* is different from
the concept of *set* as mentioned in set theory. A set is better
be seen as a container and not as an aggregate of objects
as commonly perceived, so what Russell paradox is saying is that
there do not exist a container of all containers that do not allow
themselves to be in themselves, however that should not be taken
to mean that there is no aggregate of those containers, NBG
doesn't resolve Russell's paradox fundamentally, it is only
a technical trick, it divides containers in two kinds those that
are allowed to be contained in some container and those
that are not allowed by any container to enter into them,
the first kind is taken to represent *sets* the other kind
is taken to represent *proper class*, although I personally
don't like this terminology but I shall abide by it for the
sake of familiarity to ease discussion, so the trick is
to stipulate a construction scheme that defines containers
of containers that are sets that fulfill a certain property,
this partially avoids Russell's paradox, but still at large
the paradox is there, still I can imagine an aggregate
of all those containers (i.e. classes) that do not
allow themselves into themselves, and this is not
even touched by NBG, so in reality NBG is not
greatly different from ZF as regarding this fundemental
point. To resolve that at a fundamental level we need
to make a clear fundamental distinction between
classes(i.e. selective containers) and Aggregates!
Aggregates are not containers, informally speaking
an aggregate of objects A and B is an object that
*is* composed of A and B, it is not a container
that allows only A and B to be contained in it,
no it is *the* objects A and B themselves seen
as one object, this is something quite different
from the set concept, so according to this
informal record there cannot be an aggregate
of no objects, so there cannot exist an empty
aggregate, and also an aggregate of one object
is that object itself, those consequences are
clearly different from what is happening with
sets. An aggregate have the same kind
of existence its elements has, so if
specific persons x and y are husband and wife
they constitute a binary aggregate this aggregate
has the same physical existence of each of x and y
Unlike sets this might not be necessarily the
case. Now aggregates of multiple objects
are always different from their elements,
so for example you can have an aggregate
of all containers that do not allow themselves
in themselves, in other words, an aggregate
of all classes not in themselves, and clearly
this aggregate itself is not a container! (unless
the theory has one class only)
so it is not a class, so the paradox disappear.
Many times I think that the "Universe of Discourse"
is better be thought of as an aggregate rather than
a class or a set. But the basic point is there
one must differentiate between these two concepts
intuitively that of a container which is the concept
that underlies sets and classes, and that of
an aggregate which is the concept that best
grasps the word "collection".

Zuhair

[Zuhair]

Well that depends on the definition of cardinal, a well founded
cardinal
certainly follows what you said, but in NFU this is not the case, you
can indeed have a set of all cardinals there.

Zuhair

[Zuhair]

On Dec 14, 10:53 pm, Zuhair <zaljo...@gmail.com> wrote:

See: http://zaljohar.tripod.com/index.html

Zuhair

[LudovicoVan]

[Cross-post and follow-up to alt.philosophy.]

"Zuhair" <zalj...@gmail.com> wrote in message
news:e587459e-bece-44bf...@d10g2000vbk.googlegroups.com...

> Do these fictitious containers
> have a kind of existence that
> is separate from human ideation?

As you ask about existence, the question becomes philosophical. The answer
would be: _no_, as an "observable" wants an "observer" (i.e. the very idea
of it). In fact, we can question the status of a reality external to our
conscience, but there can be no doubt that what we can *conceive* is in
our/the mind: reality, as such, is maybe experienced, definitely unknowable.

Oranges, tables, cars, mathematical sets, parliamentary groups, satellites,
shoes, electromagnetism, oceans, computers, pizza, trees, global debt, no
taxation on financial transactions, Margaret Thatcher, the movies, war and
peace, summer, pudding, you, Mao and I, the moon, and eventually *every
thing*: they only exist in our imagination, are expressed through language,
take the form of a culture. Cosmos is out, in the silence.

-LV

[smn]

On Dec 15, 10:31 pm, quasi <qu...@null.set> wrote:
> On Thu, 15 Dec 2011 18:55:22 -0500, quasi <qu...@null.set> wrote:

> quasi- Hide quoted text -

>
> - Show quoted text -

Hi Quasi ,You certainly want U not empty ,say aeU (so a not=U ) ,then
{a}={x:x=a} has exactly one element ,namely a .You certainly want {a}
not =U so acording to you {a}eU .
You certainly want that if A,B eU then also A union B eU .Then C=U\
{a} does not belong to U
since otherwise C union {a} =U would belong to U ,a contradiction But
C is not = to U either and you said all objects except U belong to U .

What you want is the Kelly -Moore system (cf appendix to
Kelly ,general Topology ) which is a popular system for
mathemtics .Say and object A is a set iff AeX for some object X and
{x:Cx}=Y is the object satisfying xeY iff x is a set and Cx . U=
{x:x=x} is the universe .xeU iff x is a set and Russell={x:not(xex)}
which is a subcollection of U but not and element of U .All objects
are subcollections .Regards smn

[Zuhair]

On Dec 16, 9:31 am, quasi <qu...@null.set> wrote:
> On Thu, 15 Dec 2011 18:55:22 -0500, quasi <qu...@null.set> wrote:

Thanks quasi for this system, the basic difference between your
approach and the standard one lies in the definition of a condition,
it opens the room for some statements to be neither true
nor false of some elements of U, i.e. some statments might
not be conditions, and this introduces intuitionist logic arguments,
for example you said that a x b is defined in the usual manner
I take that to mean a set of all Kuratowski ordered pairs of
elements
of a and b, so your theory must include an theorem of pairing, now
your theory have Boolean union as you stated, and a\a
would be the empty set, accordingly we can construct the
Von Neumann ordinals! now take the condition C(x) to be
"x is a Von Neumann ordinal" and you will get
Burali-Forti's paradox since clearly
U is not the set of all Von Neumann ordinals. However
you can always defend your position by saying that
the statement "x is a Von Neumann ordinal" is not a condition
so there might exist an object in U that is neither
a Von Neumann ordinal nor is not a Von Neumann ordinal
which as I said introduces intuitionistic logic into play.
You need to specify the language of your theory, if it is
first order logic, then it is inconsistent. if it is intuitionistic
logic then that is something else.

Zuhair

[quasi]

Thanks for the comments.

When I get a chance, I'll try to revise it, incorporating
some of the points you made.

quasi

[quasi]

On Sat, 17 Dec 2011 00:14:34 -0800 (PST), Zuhair <zalj...@gmail.com>
wrote:

>On Dec 16, 9:31 am, quasi <qu...@null.set> wrote:
>> On Thu, 15 Dec 2011 18:55:22 -0500, quasi <qu...@null.set> wrote:

>Thanks quasi for this system, the basic difference between
>your approach and the standard one lies in the definition of

>a condition, it gives the room for some conditions to be
>neither true nor false of some elements of U and this

>introduces intuitionist logic arguments, for example you said
>that a x b is defined in the usual manner I take that to mean
>a set of all Kuratowski ordered pairs of elements of a and b,
>so your theory must include an theorem of pairing, now your
>theory have Boolean union as you stated, and a\a would be the
>empty set, accordingly we can construct the Von Neumann
>ordinals! now take the condition C(x) to be "x is a Von
>Neumann ordinal" and you will get Burali-Forti's paradox
>since clearly U is not the set of all Von Neumann ordinals.
>However you can always defend your position by saying that
>the statement "x is a Von Neumann ordinal" is not a
>condition so there might exist an object in U that is
>neither a Von Neumann ordinal nor is not a Von Neumann
>ordinal which as I said introduces intuitionistic logic into
>play. You need to specify the language of your theory, if it
>is first order logic, then it is inconsistent. if it is
>intuitionistic logic then that is something else.

Hmmm ...

I'm pretty sure I want first order logic, but if as you say,
that would lead to an inconsistent theory, then some revision
will be necessary.

I know I'm missing some basics with regard to the formalities
of language required to even correctly specify a theory, but
hopefully, my understanding of the details of those
formalities will improve over time.

I may take another cut at it.

In the meantime, I'll try to digest the issues you raised.

Thanks.

quasi