The simplest way to understand sets.

[zuhair]

Perhaps the simplest way to understand sets, is to informally imagine
them as "containers" the sizes of which depends on field of the study
we desire these sets to play a role into.

So a Nucleus theory in FOL with identity and membership would be the
following:

Define(set):- x is a set <-> Exist y ( x e y ).
Define(proper class):- x is a proper class <-> ~ x is a set.

Axioms:

1.Extensionality: for all z ( z e x <-> z e y ) ->x=y

2.Comprehension: if phi is a formula in which x is not free, then all
closures of

Exist x for all y ( y e x <-> y is a set phi )

are axioms.

3. Pairing: For all sets r,s
Exist a set x for all y (y e x <-> (y=r or y=s))

4.Size Limitation:
Exist x for all y ( y is a set <-> y sub-numerous to x)

5.Proper classes: All proper class are of the same size.

For all proper classes x,y ( x equi-numerous to y ).

were sub-numerous and equi-numerous are defined as:

y sub-numerous to x <-> Exist f ( f:y-->x , f is injective ).

x equi-numerous to y <-> Exist f ( f:x-->y , f is bijective ).

Theory definition finished/

One can define the one place predicate "maximal set" in the following
manner:

x is a maximal set <->
for all y ( y is a set <-> y sub-numerous to x)

If one want to extend this theory, then one can add power and union in
the following manner:

Power: For all x ((x is a set & ~ x is maximal set) ->
Exist a set x' (~ x' is maximal set &
for all y ( y e x' <-> y subclass x ))).

of course here x' is the power set of x.

Similarly

Union:For all x
((x is a set & ~ x is maximal set) ->
(Union(x) is a set & ~ Union (x) is maximal set)).

were Union(x) is defined as the class of exactly all members of
members of x, which is provable to exist from comprehension.

This theory is very simple, it imply global choice also!

However one can extend this theory in a piece-meal manner Like for
example adding the axiom : the class of all countable ordinals a
maximal set, etc,etc...

Actually the Nucleus theory (i.e. the one with the five axiom schemes
above without) do prove: the theorems of the empty class is a set,
Separation and Replacements on sets, union of sets, power of sets
smaller than Omega, also it proves that Omega exist.

This theory + Extensions of power and union depicted above, is one
step higher than Morse-Kelley's, the existence of an inaccessible
set.

Zuhair

[John Jones]

A container doesn't have a representation in mathematics, mostly because
inside and outside doesn't.

[zuhair]

y inside x <-> y e x
y outside x <-> (~ y e x & ~y=x )

so ideally y should neither be inside itself, nor outside itself.

so "inside" is not the negation of "outside", nor is
"outside" the negation of "inside".

Zuhair

[John Jones]

That's a drawing, not a syntactical procedure. There's no connection
between the ideas of inside and outside there, except visually.

Again, you need three determinants to describe inside and outside.

And to for "it" to be inside or outside "itself" requires explanation of
the term "itself". It looks as though it is both object and reference to
object. This means that even if "itself" is shown, it need not be
represented.

[zuhair]

I am thinking day after day, that "sets" is about drawing, and not
just only syntax.

There's no connection
> between the ideas of inside and outside there, except visually.
>
> Again, you need three determinants to describe inside and outside.
>
> And to for "it" to be inside or outside "itself" requires explanation of
> the term "itself". It looks as though it is both object and reference to
> object. This means that even if "itself" is shown, it need not be
> represented.

Well itself is shown through identity theory! weather this is using
the second order well known definition of Leibniz, or
a first order identity theory.

[zuhair]

I wrote:
>
> This theory + Extensions of power and union depicted above, is one
> step higher than Morse-Kelley's, the existence of an inaccessible
> set.

which is incorrect.

the correction statement is:

This theory + power and union as depicted above + the axiom that the
set of all finite ordinals is not a maximal set, is one step higher
than Morse-Kelley's, since it proves the existence of an inaccessible
set, something that Morse-Kelley's alone don't(unless we add axioms of
inaccessible cardinals to it).

>
> Zuhair

[Jan Burse]

Hi Zuhair

zuhair schrieb:

> Well itself is shown through identity theory! weather this is using
> the second order well known definition of Leibniz, or
> a first order identity theory.

Did you ever experiment with dropping identity from
your set theory? Lets not motivate it for the moment
philosophically, but just for the sake of insight into
mathematical reducibility.

Lets assumes to use a standard intended interpretation
of membership "in", and then derives "=" from it.
Very easily we have:

x = y :<=> Az(z in x <-> z in y) (Def)

Interestingly the following can readily be derived from
the above definition:

x = x (Thm)
x = y -> y = x (Thm)
x = y & y = z -> x = z (Thm)

Unfortunately the definition does not yet yield full
congruence. We have only concruence in the second
argument of the membership relation:

x = y & z in x -> z in y (Thm)

Therefore the following axiom is needed:

x = y & x in z -> y in z (Axm)

Logically we have then for formulas A that are only
built on the basis of membership "in" the substitution
rule. A meta result which holds in any signature with
equality and congruence:

x = y -> (A(x) -> A(y)) (Meta)

Now lets turn to John Jones: When somebody casts a
mathematical theory, and uses some notions like
membership "e", identify "=" etc.. than he puts these
notions into relation.

But the goal is then not necessary to show that these
notions are minimal or whatever. The goal is often to
reach some theorem. A critical minimalist program is
an other direction of thought:

Reaching a theorem on the basis of some notions:

Showing the entailment that from an
axiomatic theory for the notions the
theorem follows

Theory1 |- Theorem
(Example some limitation of
size theorem in set theory)

Reducing notions to other notions:

Showing the entailment that from one
axiomatic theory for the first notions
the other axiomatic theory for the second
notions follows, and vice versa:

Theory1 -||- Theory2
(Example = is not needed
in set theory)

From the above it can be seen that both questions
can be independently pursuit. Any result from the
critical minimalist program can be immmediately
transferred, since we have:

Theory2 |- Theorem
(Limitation of size in set
theory without = (sic!))

The critical minimalist program is then of interest
when the theorem has not yet been settled. For
example switching to a theory2 could open up other
paths for obtaining a theorem.

Also there is the interesting branch of reverse
mathematics now, which is also related to a
certain critical minimalist view. When I am right,
the program is to reduce the background theory as
much and them obtain results of the form:

Theory0 |- (Theorem1 <-> Theorem2)

Which gives immediately:

Theory0+Theorem1 -||- Theory0+Theorem2

Bye

[zuhair]

hmmm, I shall read that carefully

Thanks

[John Jones]

Yes. the drawing is the only representation of a set. But a drawing
conveys nothing that is syntactic.

>
> There's no connection
>> between the ideas of inside and outside there, except visually.
>>
>> Again, you need three determinants to describe inside and outside.
>>
>> And to for "it" to be inside or outside "itself" requires explanation of
>> the term "itself". It looks as though it is both object and reference to
>> object. This means that even if "itself" is shown, it need not be
>> represented.
>
> Well itself is shown through identity theory! weather this is using
> the second order well known definition of Leibniz, or
> a first order identity theory.

You will have to point out any relevant distinctions. Because an object
that points itself out through "self-identity" looks like a contradiction.

[John Jones]

Jan Burse wrote:
> Hi Zuhair
>
> zuhair schrieb:
>
>> Well itself is shown through identity theory! weather this is using
>> the second order well known definition of Leibniz, or
>> a first order identity theory.
>
> Did you ever experiment with dropping identity from
> your set theory?

He can't, because "inside" and "outside" are relationships to something
that is only itself.

> Lets not motivate it for the moment
> philosophically, but just for the sake of insight into
> mathematical reducibility.
>
> Lets assumes to use a standard intended interpretation
> of membership "in", and then derives "=" from it.

How can you derive "in" from "+=" ? There is no syntactic manipulation
to bring them together.

[zuhair]

On Nov 21, 9:13 pm, John Jones <jonescard...@btinternet.com> wrote:
> Jan Burse wrote:
> > Hi Zuhair
>
> > zuhair schrieb:
>
> >> Well itself is shown through identity theory! weather this is using
> >> the second order well known definition of Leibniz, or
> >> a first order identity theory.
>
> > Did you ever experiment with dropping identity from
> > your set theory?
>
> He can't, because "inside" and "outside" are relationships to something
> that is only itself.

I think Jan Burse was speaking about my theories in general and not
only this one, however as regards this point specifically you are
right John, since inside and outside are relationships defined in
reference of "itself", then we cannot drop identity from my theory

we have

inside x
outside x
x itself.

From identity theory every x is identical to itself, this is the first
axiom of identity theory actually which is the axiom of reflexiveness
of identity relation.

For all x ( x=x )

Zuhair

[zuhair]

It looks like circularity and not contradiction, but yet it is neither
circular(cyclical) not contradictory.

Zuhair

[Jan Burse]

John Jones:

>> He can't, because "inside" and "outside" are relationships to something
>> that is only itself.

Depends how you define "can". Usually when a term is defined
in other terms predating the term itself, this can count as
dropping. Because the term is only a shorthand for the
predating term.

So when I do a definition:

x = y :<=> Az(z in x <-> z in y) (Def)

Then "in" predates "=". And "=" is just a short hand for some
formula using "in". Which means that we can drop "=", everything
that is said by using "=", can also be said by using "in".

This is the nature of non recursive definitions. They can
be expanded and are only here for convenience.

I would say its possible to drop equality from set theory,
we can formulate everything alone by the use of membership.
Lets make a test. Zuhair defined:

y inside x :<=> x in y
y outside x :<=> not y in x and not x=y.

When we use the above cast of set theory where "=" is a
definition, then the second definition is equal to
saying:

y outside x :<=> not y in x and exists z(z in x xor z in y)

Nice, isn't it.

Bye

[debaser]

musatov wrote to John Jones, et. al.:

On Nov 22, 6:49 pm, Jan Burse <janbu...@fastmail.fm> wrote:
> John Jones:
>
> >> He can't, because "inside" and "outside" are relationships to something
> >> that is only itself.
>
> Depends how you define "can".

'May' is the term I strongly prefer. Even in the non-positive case
'may not' is more precise.

Usually when a term is defined 'may' is more precise:

> in other terms predating the term itself, this can count as
> dropping. Because the term is only a shorthand for the
> predating term.
>
> So when I do a definition:
>
>         x = y :<=> Az(z in x <-> z in y)        (Def)
>
> Then "in" predates "=". And "=" is just a short hand for some
> formula using "in". Which means that we can drop "=", everything
> that is said by using "=", can also be said by using "in".
>
> This is the nature of non recursive definitions. They can
> be expanded and are only here for convenience.
>
> I would say its possible to drop equality from set theory,
> we can formulate everything alone by the use of membership.
> Lets make a test. Zuhair defined:
>
>         y inside x :<=> x in y
>         y outside x :<=> not y in x and not x=y.
>
> When we use the above cast of set theory where "=" is a
> definition, then the second definition is equal to
> saying:
>
>          y outside x :<=> not y in x and exists z(z in x xor z in y)
>
> Nice, isn't it.
>
> Bye

-------------------------------------------

please notice illustrations:

*example 1*

-----------
Equation 1
-----------
x
i * --- = a
y

-----------
Equation 2
-----------
y
i * --- = b
x

-----------
Equation 3
-----------

i
y * --- = c
x

-----------
Equation 4
-----------

x
y * --- = d
i

-----------
Equation 5
-----------

i
x * --- = e
y

-----------
Equation 6
-----------

y
x * --- = f
i

let (a + c + e) = n

let (b + d + f) = p

(a + c + e) - (b + d + f) = n - p

(a + c + e) + (b + d + f) = n + p

-----------------------------------------------
in coding, perhaps as you referenced inside-out
-----------------------------------------------

<mode></code> \
\
=inside
/
</mode><code> /

<code></mode> \
\
= outside
/
/
</code><mode>

-----------------------------

<mode></code> \ / <code></mode>
\ /
outside=in
/ \
</mode><code> / \ </code><mode>

-----------------------------

</code><mode> \ / </mode><code>
\ /
inside=out
/ \
<code></mode> / \ <mode></code>

--------------------------------

Musatov

[John Jones]

Jan Burse wrote:
> John Jones:
>>> He can't, because "inside" and "outside" are relationships to something
>>> that is only itself.
>
> Depends how you define "can". Usually when a term is defined
> in other terms predating the term itself, this can count as
> dropping. Because the term is only a shorthand for the
> predating term.
>
> So when I do a definition:
>
> x = y :<=> Az(z in x <-> z in y) (Def)
>
> Then "in" predates "=".

It can't pre-date it. The Peano axioms and the syntax of arithmetic
don't allow a sequential distinction between "in" and "=".

> And "=" is just a short hand for some
> formula using "in". Which means that we can drop "=", everything
> that is said by using "=", can also be said by using "in".
>
> This is the nature of non recursive definitions. They can
> be expanded and are only here for convenience.

None of that makes sense. Your reasons aren't properly given.

>
> I would say its possible to drop equality from set theory,
> we can formulate everything alone by the use of membership.

You can't use the property of membership to define a set.

> Lets make a test. Zuhair defined:
>
> y inside x :<=> x in y
> y outside x :<=> not y in x and not x=y.
>
> When we use the above cast of set theory where "=" is a
> definition, then the second definition is equal to
> saying:
>
> y outside x :<=> not y in x and exists z(z in x xor z in y)
>
> Nice, isn't it.
>

Maybe so. I wouldn't know.

[Virgil]

In article <hesrk5$sil$1...@news.eternal-september.org>,
John Jones <jonesc...@btinternet.com> wrote:

> Jan Burse wrote:
> > John Jones:
> >>> He can't, because "inside" and "outside" are relationships to something
> >>> that is only itself.
> >
> > Depends how you define "can". Usually when a term is defined
> > in other terms predating the term itself, this can count as
> > dropping. Because the term is only a shorthand for the
> > predating term.
> >
> > So when I do a definition:
> >
> > x = y :<=> Az(z in x <-> z in y) (Def)
> >
> > Then "in" predates "=".
>
> It can't pre-date it. The Peano axioms and the syntax of arithmetic
> don't allow a sequential distinction between "in" and "=".

It can if the definition predates the Peano axioms and arithmetic!

>
>
> > And "=" is just a short hand for some
> > formula using "in". Which means that we can drop "=", everything
> > that is said by using "=", can also be said by using "in".
> >
> > This is the nature of non recursive definitions. They can
> > be expanded and are only here for convenience.
>
> None of that makes sense.

It makes sense to me.

> Your reasons aren't properly given.

Your objections aren't properly given.

>
> >
> > I would say its possible to drop equality from set theory,
> > we can formulate everything alone by the use of membership.
>
> You can't use the property of membership to define a set.

Maybe you can't, but your limits do not bind others.

[John Jones]

Virgil wrote:
> In article <hesrk5$sil$1...@news.eternal-september.org>,
> John Jones <jonesc...@btinternet.com> wrote:
>
>> Jan Burse wrote:
>>> John Jones:
>>>>> He can't, because "inside" and "outside" are relationships to something
>>>>> that is only itself.
>>> Depends how you define "can". Usually when a term is defined
>>> in other terms predating the term itself, this can count as
>>> dropping. Because the term is only a shorthand for the
>>> predating term.
>>>
>>> So when I do a definition:
>>>
>>> x = y :<=> Az(z in x <-> z in y) (Def)
>>>
>>> Then "in" predates "=".
>> It can't pre-date it. The Peano axioms and the syntax of arithmetic
>> don't allow a sequential distinction between "in" and "=".
>
> It can if the definition predates the Peano axioms and arithmetic!

That's like saying "a banana comes before an orange" because we define
it so. But here, defining doesn't make anything so!

>> You can't use the property of membership to define a set.
>
> Maybe you can't, but your limits do not bind others.
>>> Lets make a test.

It's not grammatically possible. You can't talk about a "set" and then
look for something that can define it.

[Virgil]

In article <heuqek$s7d$1...@news.eternal-september.org>,
John Jones <jonesc...@btinternet.com> wrote:

> Virgil wrote:
> > In article <hesrk5$sil$1...@news.eternal-september.org>,
> > John Jones <jonesc...@btinternet.com> wrote:
> >
> >> Jan Burse wrote:
> >>> John Jones:
> >>>>> He can't, because "inside" and "outside" are relationships to something
> >>>>> that is only itself.
> >>> Depends how you define "can". Usually when a term is defined
> >>> in other terms predating the term itself, this can count as
> >>> dropping. Because the term is only a shorthand for the
> >>> predating term.
> >>>
> >>> So when I do a definition:
> >>>
> >>> x = y :<=> Az(z in x <-> z in y) (Def)
> >>>
> >>> Then "in" predates "=".
> >> It can't pre-date it. The Peano axioms and the syntax of arithmetic
> >> don't allow a sequential distinction between "in" and "=".
> >
> > It can if the definition predates the Peano axioms and arithmetic!
>
> That's like saying "a banana comes before an orange" because we define
> it so. But here, defining doesn't make anything so!

No symbol has any intrinsic meaning. And both "=" and "in" are symbols.
If one assigns a meaning to "in" first then "=" can be defined in terms
of "in", and if one defines "=" first then "in" can be defined in terms
of "=". And one of them must come first.

>
>
> >> You can't use the property of membership to define a set.
> >
> > Maybe you can't, but your limits do not bind others.
> >>> Lets make a test.
>
> It's not grammatically possible. You can't talk about a "set" and then
> look for something that can define it.

One can certainly talk about undefined 'words' before giving them
definitions.