Re: Small Set theory:Revised.

zuhair schrieb:

ah I see let me see. let me restate the axiomatic system again.

-Small Set Theory-

Primitive e

Definition:
x is P_defined <-> Ay (((P[y]->yex)&(~P[y]->(~yex&~y=x))).
x is P_embedded <-> Ay( P[y]<->(yex v y=x) ).
x is P_not embedded <-> Ay ( P[y]<-> yex ).

were P is a formula in one free variable.

Also we can define x is P_defined in the following manner

x is P_defined <-> (x is P_embedded v x is P_not embedded )

i.e.

x is P_defined <-> Ay(( P[y]<->(yex v y=x) )v( P[y]<-> yex )).

Or equivalently
x is P_defined <-> Ay(y=x v ( P[y]<-> yex )).


Axioms:-

Ax.0) Extensionality: As in ZFC.
Ax.1: AxEP ( x is P_defined )
Ax.2: AxAy ( yex & y is P_defined -> ~P[x] )
To be precise, do you mean
AxAy ( (yex & y is P_defined) -> ~P[x] )
or
AxAy ( yex & (y is P_defined) -> ~P[x] )) ?
I assume the former as otherwise all sets contain each other, i.e. your
only primitive e
would be as primitive as can get.
So I assume the former version is intended (I just wanted to make sure
as I will use it below).


Ax.3: AxAy (((P[y]->yex)&(~P[y]->(~yex&~y=x)) & y is not P_defined<-> x
Exist).

Again, the "Exist" predicate looks somewhat tautological when applied
to a variable
that is all-quantified.
Why do you think that this Ax.3 differs from
Ax.3': AxAy (((P[y]->yex)&(~P[y]->(~yex&~y=x)) & y is not P_defined).
or, while we're at it, from the combination of
Ax.3'a: Ay(y is not P_defined)
and
Ax.3'b: AxAy ((P[y]->yex)&(~P[y]->(~yex&~y=x)).
But this (Ax3'a) would imply that the notion of P_defined-ness is
nonsense...

Ax.4: Infinity: as in ZFC.

+/- AC.


Now your argument was that Ax.1 will refute the existance of a univeral
set.

Let V={x|x=x} , i.e. V is the set of all sets , i.e. the universe in
this theory.

You have not shown that V is a set in your theory!
In fact, if you had left out Ax.4, you won't have any set at all, would
you?

Apparently, you want to write "{x|P[x]}" for "some P_defined set".
Somehow, in the current version, I don't find the "For all P there is a
P_defined set" axiom.
Uniqueness and existence are not (yet) immediately clear for each P.
Thus V should be /a/ set such that for all y except possibly V, yeV.
Again, with todays version of axiomatic system, neither existence nor
uniqueness are clear.
If there is a set V containing all but possibly itself, I don't see
that the existence of a V'
containing all but possibly itself is excluded.
Of course, this would imply VeV' and V'eV, but, so what?

In fact, if we drop Infinity axiom and AC for the moment, the finite
part of your set theory
can be modeled by (I think)
- no set at all
- only the empty set {}
- only the two sets {} and {{}}


Let P[y]<-> y = V.

I am not happy with such a P as long as V is not unique.
Fortunately, I will accept uniqueness (but not existence) soon below.



by 1. you thought we should have x is P_defined should exist. But this
is not 1. Axiom actually mean that for every already existing x there
should EXIST a formula P that determine membership in x and non
membership in x according to the definition of x is P_defined given
above. Not for every formula P there should exist x such that x is
P_defined as you thought.

Well, I interpreted your previous versions this way.


There cannot exist x such that V is a member of it. This is a
consequence of Ax.2

Why? I assume I should set
P[y]:<-> Vey
Thus, Ax.2, applied to my special set x, becomes
Ay ( (yex & y is "Vey"_defined) -> ~Vex )
If I want to deduce ~Vex by modus ponens, I need to find a set y such
that
both yex and y is "Vey"_defined.
I might try y=V (and have no other choice as x might claim to be {V}),
but V is not "Vey"_defined
Or is it?
V is "Vey"_defined <-> Ay(y=V v ( Vey<-> yeV )).
I assume that your theory is rich enough to allow at least one more set
y besides V.
Specializing to this y, I obtain
y=V v (Vey <-> yeV)
By assumption, y=V is false and yeV is true.
Hence
Vey.
Thus, if V were "Vey"_defined, V would be an element of all other (i.e.
at least one) sets y
So we better assume that V is not "Vey"_defined.
But if V is not "Vey"_defined, I cannot use Ax.2 to prove that Vex is
false.


Because by Ax.2 any set which include V should not be equal to itself.

OK, I think, now I get what you mean.
You want to use P(y):<-> y=y.
Then
AxAy ( (yex & y is "y=y"_defined) -> ~x=x )
which, using the fact that Ax(x=x), is equivalent to
AxAy ( ~yex v ~(y is "y=y"_defined))
Especially, for arbitrary x and y=V,
~Vex v ~(V is "y=y"_defined)
Since V is "y=y"_defined, we have ~Vex, indeed.

As a consequence, if two sets V,V' are "y=y"_defined, then ~VeV' and
~V'eV.
But if ~V=V' then VeV' and V'eV. Therefore V=V', V is uniquely
determined if it exists.

and thereby not a set, since it violates extensionality.

Granted, I myself used Ax(x=x) quite confidently somewhere above.

Also Ax.3.
which is the existance source of sets in this theory do not allow y
when y is P_defined to be a member of x if x is P_defined.

As mentioned above, I have strong objections to Ax.3.


Therefore Ax in Axiom 1 do not extend to a set that has V as a member,
because simply this set do not exist in this theory.

I think there is no problem with the theorum of Universe, Pairing,
Union, specification, Replacement, power. in this theorum ( after
suitable modefication to suit this axiomatic system of course).

OK, it is for the sake of sanity that the universe V should "in
reality" only be an almost-universe (i.e. not contain itself although
it is a set).
But then probably pairing is also only some "almost-pairing" etc.
I don't believe that the degree of almostness is nicely under control.

I think this theory would meet the objective I wrote before behind
developing this theory. A theory that avoids Russell's paradox with the
most minimum acheivable lose of models.

At least I agree that we are getting closer to something useful.

.

Relevant Pages

  • Re: Small Set theory:Revised.
    ... P_defined set" axiom. ... Uniqueness and existence are not immediately clear for each P. ... There cannot exist x such that V is a member of it. ... In a similar way my example of pairing works. ...
    (sci.math)
  • Re: Small Set theory:Revised.
    ... all quantified as an "essence"of questionable existance. ... This axiom solve Russell's paradox. ... Uniqueness and existence are not immediately clear for each P. ... There cannot exist x such that V is a member of it. ...
    (sci.math)
  • Re: Naive Comprehension.
    ... Axiom schema of comprehension. ... member of itself, ... Then just prove ~xex, for the x that is in action in your proof. ... You haven't proven the existence of singletons, ...
    (sci.logic)
  • Re: ONE
    ... >>> The empty set axiom is superfluous as an axiom ... >>> existence of an empty set is provable from just ... > domain of discourse has at least one member. ...
    (sci.math)
  • Re: ONE
    ... >>> The empty set axiom is superfluous as an axiom for Z ... >>> existence of an empty set is provable from just the ... > domain of discourse has at least one member. ...
    (sci.math)
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