Observations in a premature theory of names.

[zuhair]

Hi all,

I observed the following regarding a theory that I was constructing in
FOL, but I am really not sure of the result.


The language of the theory is FOL with identity,epsilon membership and
the primitive binary relation "name"


Now we define the terms "set" and "ur-element" in the following
manner:


Define: x is a ur-element iff not exist y (y is the name of x)


so in simple words a ur-element is an un-named object.


Define: x is a set iff exist y (y is the name of x)


so sets are named objects.


Now lets have the following axioms:


1)Extensionality1:


for all sets x,y (for all z ( z in x <-> z in y) -> x=y)


2)Extensionality2:


for all #,$,a,b ((# is the name of a & $ is the name of b) ->
(# =$ <-> a=b))


3) Naming: For all x,y ( x is the name of y -> x is a ur-element)


4) Comprehension: if phi is a formula in which x is not free, and
which do not use the relation "name", then all closures of


Exist x ( Exist # ( # is the name of x) &
for all y ( y in x <-> (y is a ur-element & phi) ) )


are axioms.


Theory definition finished/


Now my observation was that if we didn't make the restriction of
excluding the relation "name" from being used in phi, then we can
easily have a Russull like paradox (take the phi to be
Exist x ( y is the name of x & not y in x) , also we'll have all
kinds
of paradoxes similar to Borali-Forti, and lensweiskie's and Cantor's
paradox, all kinds of paradoxes really.


But I noticed that we can block all such paradoxes( that I know off)
by merely forbidding the relation "name" to be used in phi.


Now if we define a relationship of membership e ( not that e here is
not epsilon membership, epsilon membership is denoted by "in")
as following:


Define(e):- x e y <-> Exist # ( # in y & # is the name of x)


Now the strange thing is that we will end up with a set containing all
sets in it I mean the following


Exist x for all y ( y e x )


Proof: Let phi in comprehension be y=y
and well get the set of all ur-elements
which is unique from Extensionality 1
we denote it as V
Now the name of V is also unique (Extensionality 2)
we denote it as @
Now we do have @ in V.
Since all sets have names that are ur-elements
Thus from the definition of "e" we have all sets in V
including V itself since its code(i.e name) is @ which is in V.


I tried to find a paradox involved with the above, I couldn't.


The interesting thing to me is that this theory can also bypass the
blockage on naming in a rather nice manner.


Lets procceed to see what I mean:


Let phi(y)<-> ~y=y
Substitute in comprehension well get the empty set which is unique
denoted as {} and its name is also unique denoted as 0 (so 0 is not
identical to {} here, 0 is the name of {})


Now from comprehension we can substitute a new formula
Let phi(y)<-> y=0
and we get the set {0}
and from the definition of "e" it is obvious that we have
{} e {0}.


and so on we can proceed.


The strange thing about this theory though, is that it has a Boolean
structure, we can have absolute complements, unions and interesections
of any two sets in it.


All these are observations on this theory, which seems interesting to
me.


However this work is still premature, and I don't know how I can get
relations between sets, or powers or infinite unions, all these are
not figured out, but I think if we use typing of objects by adding
another one place predicate symboles To,T1,T2,T3,....(were o,1,2,3,...
are metatheoratical notation(please don confuse the o here with the 0
mentioned above)and we add axioms to the effect of making

x is ur-element <-> To(x)
for all x for all y ( y in x ->Ti(y)) -> Ti+1(x) ) (this is a schema
per each i)


Now we can improve comprehension to typed comprehension in which phi
do not use neither name not any of the Ti in it and the schema would
be


Exist x ( Exist # ( # is the name of x) &
for all y ( y in x <-> ([To(y) or T1(y) or...Tn(y)]& phi) ) )


although this looks like complicating the matter, but I think perhaps
with additional axioms we can arrive at a theory that defines
relations, power, infintary union, etc... and at the same time have
all universal sets like the set of all sets, the set of all sets
equinumerous to a set, the set of all sets ordinally isomorphic to a
set, etc....


The project is still at its infancy.


Any suggestion to how I can improve this theory, any one have an idea
about a similar theories made before that can be of help.


Zuhair

[Abram Demski]

Zuhair,

Have you heard of New Foundations?

http://plato.stanford.edu/entries/quine-nf/

http://en.wikipedia.org/wiki/New_Foundations

The New Foundations system contains a compliment for every set, and
many natural "large" sets, as you say of yours.

What is the idea behind using "names" like you're doing? To me, the
normal concept of name is not extensional-- "one" and "unity" can name
the same entity, yet not be equal. I take it, though, that your point
is not to capture the concept of name formally, but rather it struck
you as a sensible thing to call this formal entity you're using to
construct a set theory. So: I think it would help me if you could give
a bit of intuition about the formal structure you're making here, and
what (if anything) it *means*.

--Abram

--
Abram Demski
http://dragonlogic-ai.blogspot.com/
http://groups.google.com/group/one-logic

[zuhair]

I read a little about new foundations, and I know its Boolean and
contain large sets, yes this theory resembles with that respect,
however this theory premature, I didn't complete it yet.

"name" here is like that of a unique "code", each object is named
uniquely in this set theory by a code or a name, and as you said I
have no attempt to capture "name" as known informally and turn it into
a formal system.

The observations that I wrote here are basic, I saw if we add "name"
or if you want to call it "code" to FOL with identity, and have the
above axiomatic system then we can construct sets of codes without
being involved in an apparent paradox. Of course the first phase of my
plan of the theory is to construct a hierarchy of sets of codes and I
think these sets should be typed (i.e each set of codes or set of sets
of codes ,etc... should have all its members of the same type) though
in my above theory I didn't do that actually.

The second phase is actually to turn all that hierarchy into the sets
we know using a defined relation of membership on the above hierarchy
(like the defined relation "e" that I wrote in the original post) , so
we need something like Deciphering rules.

So in short: Phase 1: construct sets of codes for sets SAFELY (without
being involved in a paradox)

Phase 2: Turn these codes and the sets of codes into true sets that
are related to each other by a "defined" concept of membership, which
is what I call decoding or deciphering.

This is not an easy task at all.

I don't know weather the final result would be something like NF and
its related theories using stratified comprehension, perhaps, Or it
may turn to be other type of theories.

Zuhair

[zuhair]

On Sep 19, 7:18 pm, Abram Demski <abramdem...@gmail.com> wrote:
> Zuhair,
>
> Have you heard of New Foundations?
>
> http://plato.stanford.edu/entries/quine-nf/
>
> http://en.wikipedia.org/wiki/New_Foundations
>
> The New Foundations system contains a compliment for every set, and
> many natural "large" sets, as you say of yours.
>
> What is the idea behind using "names" like you're doing? To me, the
> normal concept of name is not extensional-- "one" and "unity" can name
> the same entity, yet not be equal. I take it, though, that your point
> is not to capture the concept of name formally, but rather it struck
> you as a sensible thing to call this formal entity you're using to
> construct a set theory. So: I think it would help me if you could give
> a bit of intuition about the formal structure you're making here, and
> what (if anything) it *means*.
>
> --Abram

Abram

What is naming after all?

The way how I perceive names is that they are concepts attached to
objects for the purpose of recognizing those objects by them, so when
I hear Abram and image struct my head of you for example.

One can go deeper than that and say that names are wholes of ordered
stringe of symboles by which objects are identified, and in this
manner we will enter the ralem of Mereology, which is a complex
subject that I am not refering to here at all.

Notice the Extensionality 1 works only for sets (i.e it doesn't cover
names) in this theory.


Naming can be unique, or can be not!

In this theory naming is a unique relation, it is actually a bijection
between a subset of ur-elements (unnamed objects) and the class of all
sets in this theory.

So your example of ""one" and "unity" can name

the same entity, yet not be equal"

is not the only case of using "names".

We can axiomatize names such as objects are named uniquely, which is
what i did here, or we can axiomatize then not to be, in both cases
the process is a process of naming!

So this theory use a special case of naming were it is unique, you can
call it "coding".

Zuhair

> Abram Demskihttp://dragonlogic-ai.blogspot.com/http://groups.google.com/group/one-logic- Hide quoted text -
>
> - Show quoted text -

[Abram Demski]

Zuhair,

It's interesting to note that some literature on New Foundations
refers to the singleton set of an object as the "name" of that object.
This is because taking the singleton set of an object changes the
relative type by 1, and is thus necessary for the construction of
certain sets.

Perhaps you could find some nice connection between your system and
New Foundations based on this? I'd have to think about it more.

--Abram

[zuhair]

This post is also posted at Sci. logic, it contains further
observations that I have about this methodology of naming.

Hi all

This topic is a continuation of my earlier post to this usenet titled:
An observation.

see: http://groups.google.com.jm/group/sci.logic/browse_thread/thread/01802a7afc9163d1?hl=en#

So in this topic I'll continue mentioning some of the observations
about this naming methodology and how it overcomes known paradoxes.

( in this thread I reversed the symbolization of membership, so the
primitive membership relation would have the symbol "e",
while the defined membership relation will have the symbol "in" )

The main aim of this methodology of naming is actually to construct a
set theory in First Order Logic, that proves the existence of a
universal set, and the alike large sets, such as equivalence classes
under specific relations , absolute complements, etc..., of course
such a set theory might be useful in delineating theorems concerned
with these sets.

THE MAIN IDEA

The heart of this methodology is to have a DEFINED membership
relation, from primitives of a primitive membership relation and
"naming" relation. Then after that it will be shown how to construct
sets in such a manner that avoids paradoxes, by blocking one of the
relations that defines the defined membership relation.

The Language of the Theory: First order logic with identity and
epsilon membership "e" and the primitive binary relation "name"
and the primitive two place function symbol "ordered pair" symbolized
as "<>".

Define(in):- x in y iff Exist # ( # e y & # is the name of x )

Define: # is a ur-element iff ~Exist $ ( $ is the name of # )

Define: x is a set iff Exist $ ( $ is the name of x )

Axioms:

1)Extensionality1:

for all sets x,y (for all z ( z e x <-> z e y) -> x=y)

2)Extensionality2:

for all #,$,a,b ((# is the name of a & $ is the name of b) ->
(# =$ <-> a=b))

3) Ordered pairs: for all a,b,c,d (<a,b>=<c,d> <-> (a=c & b=d))

4) Naming: For all x,y (x is the name of y -> x is a ur-element)

5) Infinity: For all x,y (<x,y> is a ur-element )

6) Comprehension1: If phi is a formula in which x is not free, and

which do not use the relation "name", then all closures of

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

are axioms.

7) Comprehension2:

If phi is a formula in which neither x nor y are free, and that
doesn't use the primitive binary relation "name", then all closures of

Exist a set x for all y (y e x <->
Exist z (phi(z) & y is the name of
z))).

are axioms.


8) Union:

For all c Exist x for all y (y in x <-> Exist z ( z in c & y in z ))

Theory definition finished/

This theory can prove a lot of things, it can prove pairing, power,
Infinity, null, the set of all sets , the set of all sets bijective to
a set, restricted separation and even restricted relplacement, I think
it might prove all axioms involved in the finite axiomatization of NF
(see Randall Holems article: elementary set theory with a universal
set) except perhaps the axiom of singleton images.

I don't know if this theory can prove a sub-theory of it that is equi-
interpretable with NF or any of its systems.

One thing to say here is that I don't like axiom 8, because it is
against the general philosophy of this theory (the defined membership
relation "in" contain the primitive relation "name" in it).

However what attracts me to this theory is the way how comprehension
evade paradoxes.

For example we can construct the set of all sets (according to
relation "in") in the following manner.

Let phi(y) <-> y=y
substitute in comprehension 2, and you get the set of all names of
sets, which is unique according to extensionality 1
and lets denote it by "V", now the name of V would also be unique and
lets denote it by "@"

Now we have @ e V, since V=V and V is a set.

Now V would have all sets "in" it, i.e:

For every set x : x in V

is a theorem of this theory.

So V is the set of all sets including itself, so we do have
V in V. so V is a universal set.

Now lets try to derive a paradox with that in the ordinary way, lets
take the formula " ~ yey "

Now substitute in comprehension 2, and we'll get the set of all names
of sets that are not epsilon members of themselfs, lets call this set
R (after Russell). Now R itself will not be in itself, since it is a
set of names, R is unique from extensionality 1, so
is its name R' (extensionaltiy 2) , now we do have R' e R

so R={#,$,....,R'}

so we do have R in R, but this is not paradoxical since "in" is not
identical to "e".

I tried to derive a paradox by substituting the formula
x e R & ~x=R' , and we'll arrive at the set R1 which doesn't have R'
as a member of it, but still we have ~ R1 e R1
and so the name of R1 which is R1' is also in R1, but R1 is not equal
to R, also no contradiction. also we'll have R1 in R1 without being
involved in any contradiction.

On the other hand if we allow both primitive relations of
"e" and "name" to be in the same formula of comprehension 1 or
comprehension 2, then all kinds of paradoxes evolve, and we can find a
paradox which mirror's every paradox that we know, for example:

Exist z ( y is the name of z & ~ y e z)
(which is the formula ~ y in y)

This leads to Russell like paradox.

Also we can have formula's mimiking Burali-Forti paradox,
Leinsweiskies paradox , etc....

However the main observation is that all these paradoxes happens when
we have both relations 'e' and 'name' in the same formula, however in
this theory I made a further restriction than only just not allowing
both of these relations to coexist in the same formula, I actually
forbidden all formula's in comprehension to use the primitive binary
relation "name", which is an extra-precaution that I made.

I don't have any proof of weather this theory is equi-consistent to
any of the know theories especially NF and its related theories.

Actually the whole methodology might turn to be inconsistent.

However I saw that these observation are worth mentioning.

Zuhair

[Abram Demski]

Zuhair,

I know this isn't the new part of your presentation, but upon further
reflection the following definitions seem utterly strange to me:

> Define: # is a ur-element iff ~Exist $ ( $ is the name of # )
>
> Define: x is a set iff Exist $ ( $ is the name of x )
>

What an "ur-element" usually means is any thing which *contains* no
other thing. Your definition flips this around, making an ur-element
something which is contained *in* no other thing (since "in" requires
names to stand for the things).

Worse, "set" is totally throwing me off... Normally I'd say a *set* is
something that may contain elements. You define a set to be a named
entity. I don't see what this has to do with set-hood! In your theory,
a named entity *can appear in a set*. So again, it seems like an odd
swap is occurring: where normally a set is something that can
*contain*, in your theory, it's something that can *be contained*.

So, I'm pretty confused at why you define those terms that way...

Also, why do you want pairs to be nameless?

Other than that, it seems interesting. Avoiding paradoxes is always nice!

--Abram

[zuhair]

On Sep 28, 9:21 pm, Abram Demski <abramdem...@gmail.com> wrote:
> Zuhair,
>
> I know this isn't the new part of your presentation, but upon further
> reflection the following definitions seem utterly strange to me:
>
> > Define: # is a ur-element iff ~Exist $ ( $ is the name of # )
>
> > Define: x is a set iff Exist $ ( $ is the name of x )
>
> What an "ur-element" usually means is any thing which *contains* no
> other thing. Your definition flips this around, making an ur-element
> something which is contained *in* no other thing (since "in" requires
> names to stand for the things).
>
> Worse, "set" is totally throwing me off... Normally I'd say a *set* is
> something that may contain elements. You define a set to be a named
> entity. I don't see what this has to do with set-hood! In your theory,
> a named entity *can appear in a set*. So again, it seems like an odd
> swap is occurring: where normally a set is something that can
> *contain*, in your theory, it's something that can *be contained*.
>
> So, I'm pretty confused at why you define those terms that way...
>
> Also, why do you want pairs to be nameless?
>
> Other than that, it seems interesting. Avoiding paradoxes is always nice!
>
> --Abram

Well of course you know that the methodology that this theory uses is
not traditional at all, and of course I do agree with you that the
notion of "ur-element" in this theory is not the same as it is in
other theories, but however I named them ur-elements because they bear
SOME resemblance to ur-elements in other theories.

The main thing that make an object a ur-element is that it is "non
extensional", that is the main aspect of ur-elements in most set
theories that have them, although some set theories might keep ur-
elements as extensional objects and define them by breaching the
uniquness of Quines atom as non unique singletons that are in
themselfs ...etc, altogether we can say that ur-elements are objects
in which their identity is not determined by the identity of other
objects related to it by relation of membership.

In this theory we only have sets as classes, so we don't have proper
classes, now it is understandable why I define sets in the manner I
did, because the main objective of this theory is to have a universal
set, and this can be done by saying that sets are named objects and
then we construct the set of all names, which of course has a name
(because its a set!), and by that manner we'll have the set of all
sets if we define the membership "in" , in the manner that I set
forth.

So what is not a set here is also called a ur-element. It is contained
by the primitive membership relation epsilon "e" in other objects, but
of course not by the defined membership relation "in", which makes it
lack one of the properties that ur-elements in other set theories
have.

Perhaps it would have been better if I called them "non extensional"
elements,
or "sub-elements" or something to that effect, than calling them ur-
elements, but still I do see that denoting them as ur-elements bearing
in mind that this is only approximate way of denoting them and not
exact way, have no harm at all.

So all in all, I can say that ur-elements here share the basic feature
that ur-elements has in other theories, that is_ it make no sense to
speak of objects being members in them, i.e their membership is
irrelevant to the role they play in the theory.

Let me recall your passage:

---Worse, "set" is totally throwing me off... Normally I'd say a *set*

is
something that may contain elements. You define a set to be a named
entity. I don't see what this has to do with set-hood! In your theory,
a named entity *can appear in a set*. So again, it seems like an odd
swap is occurring: where normally a set is something that can
*contain*, in your theory, it's something that can *be contained*.

----

Well actually I think here you made a small mistake, you are confusing
a "set" with a "class".

A class is something that may contain elements.

A set is a class that is contained in another class.

(however this is the approach of NBG and Morse-Kelley, but not of
Ackermann's in which we can have proper classes contained in other
classes, but still set can be define even in Ackermann's as classes
that are contained in a specific class V, were V is a primitive
constant).

So all in all sets can be defined in the following manner:

Sets are classes that are contained in an arbitrary class, or in a
specific class.

In this theory, Sets are also defined in this manner, since all named
objects would be contained by the defined membership relation "in" in
the set of all sets, and all sets are either empty or do contain other
sets "in" them.

So sets here do share the common informal concept of sets in other
theories.

Zuhair

[Abram Demski]

Zuhair,

That is a satisfying explanation of your definitions. Thank you.

> Well actually I think here you made a small mistake, you are confusing
> a "set" with a "class".

Indeed that is what was throwing me off. I make this "mistake" on
purpose, because I think of classes as a "hack"... my opinion is that
classes are "really sets," so to speak, in the same way one might say
that a system which has truth values "true" and "false" but which also
claims some sentences "have no truth value" is really a 3-valued
system, so that "has no truth value" is a third truth value. So, in
any case, I didn't make the jump to thinking about sets *as opposed to
classes* when I saw your definition.

--Abram

[zuhair]

On Oct 1, 9:38 am, Abram Demski <abramdem...@gmail.com> wrote:
> Zuhair,
>

> That is a satisfying explanation of your definitions. Thank you.
>
> > Well actually I think here you made a small mistake, you are confusing
> > a "set" with a "class".
>
> Indeed that is what was throwing me off. I make this "mistake" on
> purpose, because I think of classes as a "hack"... my opinion is that
> classes are "really sets," so to speak, in the same way one might say
> that a system which has truth values "true" and "false" but which also
> claims some sentences "have no truth value" is really a 3-valued
> system, so that "has no truth value" is a third truth value. So, in
> any case, I didn't make the jump to thinking about sets *as opposed to
> classes* when I saw your definition.
>
> --Abram

Thanks a lot Abram for explaining to me your thoughts about classes, I
had the same feeling about classes and sets actually, anyhow.

There is something in your post that I forgot to address which is
this:

You wrote:
Also, why do you want pairs to be nameless?

My answer to that, is that because of the restrictions that I made on
both comprehension axiom schemas that do not allow me to use the
relation name,
this restriction though it make us avoid paradoxes but yet it is
handicapping in another sense, this made me define ordered pairs as ur-
elements, however there is many stuff I actually missed to write for
example there should be an axiom which state that every ordered pair
<a,b> names the set {{a},{a,b}} were a and b are ur-elements, not only
that another axiom I should make which states that
every ordered pair <{a},{b}> names the set {{{a}},{{a},{b}}} were a
and b are ur-elements.
Also I want to make a further restriction that is
for all x for all y ( y e x -> y is a ur-element ).

So we don't have a set {a} that has epsilon membership in {{a}} for
example i.e we can't have {a} e {{a}}.

so when I write {{a}} I mean {a} in {{a}} and not {a} e {{a}}

Anyhow the main reason that made me define ordered pairs as ur-
elements is that I can build the set of all natural numbers in this
way.

Define (Successor):- x=Successor (y) iff x=<y,y>
Now define the predicate R :- R(x) iff (0 e x & Az ( z e x ->
Successor(z) e x ))

Now take the formula For all x ( R(x) -> y e x )
Now substitute this formula in comprehension 1 and you get the set of
all natural numbers

so here we have 0=0 , 1=<0,0> , 2= <1,1> , 3=<2,2> , ......

So axiomatizing ordered pairs to be ur-elements proves infinity i.e.
proves the existence of the set of all natural numbers.And the two
axioms that I've mentioned above should code for Cartesian products
and the axiom of singleton images that is present in Randall Holmes's
finite axiomatization of NFU.

Anyhow given all the above axioms that should be added I think that
Randall Holmes's theory would be equi-interpretable to a subtheory of
this theory, but
I don't know about weather the other direction is provable?

Best Regards Abram

Zuhair