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From: zuhair on 15 Feb 2010 20:43
On Feb 15, 7:58 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Aatu Koskensilta <aatu.koskensi...(a)uta.fi> writes:
> > zuhair <zaljo...(a)gmail.com> writes:
>
> >> Phi(y) is non self comparing formula if non of its sub-formulas is a
> >> formula having one free variable occurring twice in it.
>
> > In presence of equality every formula is logically equivalent to a
> > non-self-comparing formula. Just replace
>
> >   xRx
>
> >  with
>
> >   y=x & xRy
>
> > and add an existential quantifier to bind y.
>
> Ah, it seems I misread your definition. Apologies for that.

Yea, the formula "y=x & x R y" has y free in it, so it is a non self
comparing
formula, because it has two free variables x and y, while the formula
Exist y ( y=x & x R y ) is a self comparing formula , since only x is
free in it and x has two occurrences in it and since any formula is a
sub-formula of itself.

Zuhair



>
> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: zuhair on 15 Feb 2010 22:00
On Feb 15, 12:08 am, zuhair <zaljo...(a)gmail.com> wrote:
> Hi all,
>
> The paradoxes of set theory: Russell's, Burali-Forti's,
> Cantor's, Lesniewski's, and even Holmes paradox,
>
> I noticed that all such paradoxes occur with formulas that are
> binary relations having One free variable in them,i.e formulas in
> which a free variable occur in two places having no other free
> variables in it, so either the formula is atomic with ONE free
> variable and a primitive binary relation, or the formula having ONE
> free variable occurring twice and the other variables bound,
> for example
>
> y=y
> y e y
> ~ y e y
> Exist x ( x e y & ~ y e x )
> Exist x,z ( y e x & x e z & z e y )
> Exist x ( x e y & ~ Exist c ( c e x & c e y ) )
>
> all such formulas are binary relation formulas of y with itself , i.e.
> of
> the general form y R y. So they are formulas that compare y with
> itself.
>
> It is not a surprise that such formulas would produce paradoxes,
> since the class that is defined after such formulas might have this
> relation with itself.
>
> I think these formulas are not suitable for constructing sets, and
> they'd better be avoided.
>
> So any formula Phi(y) having a sub-formula that is a binary relation
> of One free variable with itself  is to be named as
> a " self comparing formula", and it is to be avoided, so
> Phi(y) is non self comparing formula if non of its sub-formulas is
> self comparing formula.
>
> So we may define the following theory T in FOL with identity and e:
>
> T is the set of all sentences entailed (from FOL with identity and
> epsilon membership) by the axiom schema outlined below the following
> definition:
>
> Define (set):
>  x is a set iff for all y ( for all z ( z e y iff z e x) -> y=x )
>  or equivalently:
>  x is a set iff ~ exist y ( ~ y=x & for all z ( z e y iff z e x) )
>
> Axiom Schema of Comprehension: If Phi is a non comparing
> formula in which at least y is free, and in which x is not free, then
> all closures of
>
> Exist a set x for all y ( y e x iff Phi )
>
> are axioms.
>
> Theory definition finished/
>
> My personal guess is that this theory must be inconsistent!
> Theorems of this theory include the existence of a set of all sets,
> the set of all non sets, also the existence of an empty set (use the
> predicate (y=a & ~y=a)), also pairing,union,power,also the set of all
> supersets of a set, all are theorems here, separation using non
> comparing formula, replacement using non comparing formulas.
>
> I don't know if infinity can be proved in this theory? and what is the
> stance of this theory with choice?
>
> Anyhow I do think that this theory is mostly inconsistent, and it
> would be a good exercise to prove its inconsistency.
>
> Zuhair

The rational beyond this method is that most paradoxes have the
following general form:

For all x ( x R x <-> ~ x R x )

So it would be clear that any formula in which x occur free twice and
in which there
is no free variable other than x, then such formula defines a binary
relation of x with
itself, and of course such formula would be risky, since it might lead
to a paradox,
since the general form of paradoxes involves binary relations of x
with itself as
depicted above, so these formulas are to be avoided, so any formula in
which all its
sub-formulas are not binary relations of a variable with itself is to
be called
a "non self comparing" formula and it can be used in comprehension as
outlined in
the head post.

Zuhair

From: spudnik on 16 Feb 2010 12:50
most of Russell;'s lagubbrious paradoxes are perilinguistic,
lacking the element (or variable) of time; are they not?

thus:
of course, there is a base-one;
what is it's digital counter, by induction on base-ten?
in factorial base, it has n digits; eh?

> In base 1, the factorial n! has n! digits.
> [OK I realize there's no "base 1"...]

thus:
sea-level is not rising, globally --
http://www.21stcenturysciencetech.com/Articles%202007/MornerInterview.pdf
-- and warming is mostly equatorial. however,
there is massive loss of soil, and that might change *relative* sea-
level,
in some locations, as well as dysplace some sea!

thus quoth:
Let’s take a look at the complexity of polar bear life. First, the
polar bear has been around for about 250,000 years, having survived
both an Ice Age, and the last Interglacial period (130,000 years ago),
when there was virtually no ice at the North Pole. Clearly, polar
bears have adapted to the changing environment, as evidenced by their
presence today.
(This fact alone makes the polar bear smarter than Al Gore and the
other global warming alarmists. Perhaps the polar bear survived the
last Interglacial because it did not have computer climate models that
said polar bears should not have survived!)
http://www.21stcenturysciencetech.com/Articles%202007/GW_polarbears.pdf
http://www.21stcenturysciencetech.com/Global_Warming.html

thus:
the photographic record that I saw,
in some rather eclectic compendium of Einsteinmania,
seemed to show quite a "bending" effect, I must say;
not that the usual interpretation is correct, though.

Nude Scientist said:
> > "Enter another piece of luck for Einstein. We now know that the light-
> > bending effect was actually too small for Eddington to have discerned

--Another Flower for Einstein:
http://www.21stcenturysciencetech.com/articles/spring01/Electrodynamics.html

--les OEuvres!
http://wlym.com

--Stop Cheeny, Rice & the ICC in Sudan;
no more Anglo-american quagmires!
http://larouchepub.com/pr/2010/100204rice
From: zuhair on 17 Feb 2010 23:00
On Feb 15, 12:08 am, zuhair <zaljo...(a)gmail.com> wrote:
> Hi all,
>
> The paradoxes of set theory: Russell's, Burali-Forti's,
> Cantor's, Lesniewski's, and even Holmes paradox,
>
> I noticed that all such paradoxes occur with formulas that are
> binary relations having One free variable in them,i.e formulas in
> which a free variable occur in two places having no other free
> variables in it, so either the formula is atomic with ONE free
> variable and a primitive binary relation, or the formula having ONE
> free variable occurring twice and the other variables bound,
> for example
>
> y=y
> y e y
> ~ y e y
> Exist x ( x e y & ~ y e x )
> Exist x,z ( y e x & x e z & z e y )
> Exist x ( x e y & ~ Exist c ( c e x & c e y ) )
>
> all such formulas are binary relation formulas of y with itself , i.e.
> of
> the general form y R y. So they are formulas that compare y with
> itself.
>
> It is not a surprise that such formulas would produce paradoxes,
> since the class that is defined after such formulas might have this
> relation with itself.
>
> I think these formulas are not suitable for constructing sets, and
> they'd better be avoided.
>
> So any formula Phi(y) having a sub-formula that is a binary relation
> of One free variable with itself  is to be named as
> a " self comparing formula", and it is to be avoided, so
> Phi(y) is non self comparing formula if non of its sub-formulas is
> self comparing formula.
>
> So we may define the following theory T in FOL with identity and e:
>
> T is the set of all sentences entailed (from FOL with identity and
> epsilon membership) by the axiom schema outlined below the following
> definition:
>
> Define (set):
>  x is a set iff for all y ( for all z ( z e y iff z e x) -> y=x )
>  or equivalently:
>  x is a set iff ~ exist y ( ~ y=x & for all z ( z e y iff z e x) )
>
> Axiom Schema of Comprehension: If Phi is a non comparing
> formula in which at least y is free, and in which x is not free, then
> all closures of
>
> Exist a set x for all y ( y e x iff Phi )
>
> are axioms.
>
> Theory definition finished/

It is inconsistent!

Let Phi be Exist v Exist u ( y=u & v=u & ~v e u )

Thus Russell's paradox is produced.

Zuhair
>
> My personal guess is that this theory must be inconsistent!
> Theorems of this theory include the existence of a set of all sets,
> the set of all non sets, also the existence of an empty set (use the
> predicate (y=a & ~y=a)), also pairing,union,power,also the set of all
> supersets of a set, all are theorems here, separation using non
> comparing formula, replacement using non comparing formulas.
>
> I don't know if infinity can be proved in this theory? and what is the
> stance of this theory with choice?
>
> Anyhow I do think that this theory is mostly inconsistent, and it
> would be a good exercise to prove its inconsistency.
>
> Zuhair

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