Witt

Stanford Encyclopedia of Philosophy
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Russell’s Paradox

Russell’s paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox prompted much work in logic, set theory and the philosophy and foundations of mathematics during the early part of the twentieth century.
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If S={x:~(x e x)}, then, (S e S) <-> ~(S e S) is the paradox.

"If S is not a member of itself, then by definition it must be a member of itself", is not true. That is, Fy -> (y e {x:Fx}), is invalid.

~(S e S) <-> (S e S), is false.

i.e. ~({x:~(x e x)} e {x:~(x e x)}) <-> ({x:~(x e x)} e {x:~(x e x)}), is false.


Proof:

If we grant first order predicate logic, and add Russell's contextual definition of Classes, determined by some predicate, then the antinomy does not occur.

D1. G{x:Fx} = Ey(Ax(x e y <-> Fx) & Gy) Df.

1. z e {x:Fx} <-> Ey(Ax(x e y <-> Fx) & z e y)
1a. z e {x:Fx} <-> Ey(Ax(x e y <-> Fx) & Fz)
1b. z e {x:Fx} <-> (EyAx(x e y <-> Fx) & Fz) is valid

2. z e {x:Fx} -> Fz is valid.
And,
3. Fz -> (z e {x:Fx}) is invalid.
But,
3a. EyAx(x e y <-> Fx) -> (Fz <-> (z e {x:Fx})) is valid.

i.e. the class {x:Fx} exists, implies, (z satisfies F) iff (z is a member of the class {x:Fx})...is valid.


4. EyAx(x e y <-> Fx) is invalid

e.g. ~EyAx((x e y) <-> ~(x e x)) is a theorem.


Proof:

5. ~EyAx((x e y) <-> ~(x e x))

5a. Ax((x e y) <-> ~(x e x)) -> ((y e y) <-> ~(y e y))
(when x has the value y)

5b. Ax((x e y) <-> ~(x e x)) -> (contradiction)
..(p <-> ~p) is contradictory.

5c. ~Ax((x e y) <-> ~(x e x))

5d. Ay(~Ax((x e y) <-> ~(x e x)))

5e. ~EyAx((x e y) <-> ~(x e x))

QED.


The Russell class is not a member of itself!

Proof:

6. ({x:~(x e x)} e {x:~(x e x)})
6a. <-> Ey(Ax((x e y) <-> ~(x e x)) & (y e y)), by D1.
6b. <-> contradiction, by 5c.
Therefore
6c. ~({x:~(x e x)} e {x:~(x e x)}), is a theorem.

The answer to Russell's question 'Is the class of those classes that are not members of themselves, a member of itself or not?' is NO it is not.

This proof preceeds any axioms of set theory. Only the axioms of first order predicate logic are used, and D1.

Is anyone here interested in this stuff?

Witt

Demosthenes

I'm interested in set theory. I'm trying to follow your arguments but it's difficult due to the formatting. I'm not familiar with the notation "e"

ComestibleVenom

I think it's that backward's 3, tranlatable roughly as "X belongs to the set of Y".

John Page

Witt:

As you know, I agree with your conclusion . My reasoning here, which boils down to concluding a thing (set, whatever) cannot be a predicate and the thing itself, QED.

What interests me is how we decide this (why do we think what we think). If Russell's Antinomy had not arisen, what implications would that have had for how the mind categorizes and perceives the world? IMO, Russell's viewpoint (mental perspective) was that categories were intrinsic, that numbers and classes existed independently of the mind. This last statement appears true of most mathematicians I meet, who seem imbued with the idea that math is pure and numbers are eternal. I take the view that numbers happen, they are mental entities alone.

I think the Antinomy and its consideration tells us the mind can take any view as to truth and, because this is so, the coherence requirement becomes of paramount importance. If we apply or invent a rule or axiom for a formal system, it is coherence that forces us to examine the validity of the rule or axiom.

Of course, we can still timker with identity....

Cheers, John

Witt

Demosthenes:
I'm interested in set theory. I'm trying to follow your arguments but it's difficult due to the formatting. I'm not familiar with the notation "e"


Hi Demosthenes,

I agree that the formatting is a pain, but..

"e" is the membership relation.

(x e y) means, x is a member of y.

{x:Fx} is the class determined by the predicate F.

e.g. {x: x is red} means the class of x's such that it is red.
i.e. the class of red things.

{x:Fx} is defined by W. V. Quine as (the y: Ax(x e y <-> Fx)) See: Methods of Logic (1982), 48, page 300.

D1, G{x:Fx} <-> Ey(Ax(x e y <-> Fx) & Gy), conforms to Quine's meaning.

G{x:Fx}, means, the class of x's such that Fx..has the predicate G.

x e {x:Fx}, means, x is a member of the class of x's such that Fx is true.

Witt

Witt

Re: Russell's Paradox Resolved

Witt :
The answer to Russell's question 'Is the class of those classes that are not members of themselves, a member of itself or not?' is NO it is not.

This proof preceeds any axioms of set theory. Only the axioms of first order predicate logic are used, and D1.
-------------------------------------------------------

John:
As you know, I agree with your conclusion . My reasoning here, which boils down to concluding a thing (set, whatever) cannot be a predicate and the thing itself, QED.

It seems you are denying that, x is a member of x, has sense.
Russell, Quine, and Carnap, invoke some form of Russell's type theory, in order to deny unstratified entities, Do you?

I do not.
Type theory is a convenient fiction.

It seems to me that the universal set is indeed a member of itself.
In which case we have proof of ..Ex(x is a member of x).

John:
What interests me is how we decide this (why do we think what we think). If Russell's Antinomy had not arisen, what implications would that have had for how the mind categorizes and perceives the world?

Imo, the mind does not, without directions, percieve 'logical types' at all.

They are a convenience of language.

We categorize things by descriptive predicates.

It is certain to me that there are no 'types' in the world.
We concoct all types of things, don't we.
The world presents to us, sensations and that is all.
Mind does the rest...we make stories, which have no physical conterpart, imo.
Convenience, brevity, that is the issue.

John: IMO, Russell's viewpoint (mental perspective) was that categories were intrinsic, that numbers and classes existed independently of the mind. This last statement appears true of most mathematicians I meet, who seem imbued with the idea that math is pure and numbers are eternal. I take the view that numbers happen, they are mental entities alone.

Numbers are designed entities, they have designation.

It does seem that mathematical minds lean toward platonism.
It also seems clear that nominalism is inadequate to produce any mathematics, i.e. they cannot produce: classes, sets, or numbers.
(their logic is restricted to first order logic, imo)

John: I think the Antinomy and its consideration tells us the mind can take any view as to truth and, because this is so, the coherence requirement becomes of paramount importance.

Yes, comprehension and coherence are needed.
When our language demonstrates a contradiction, we must, back up the truck.. to where the logical error was.

John: If we apply or invent a rule or axiom for a formal system, it is coherence that forces us to examine the validity of the rule or axiom.

Agreed, we don't want any system that is based on contradiction.
And, contradiction is incoherent.

I think more than coherence is needed

John: Of course, we can still timker with identity....

Yes, many find it entertaining to tinker about such things, but, don't ask why..!?!?

Witt

John Page

The key issue here is the process of stratification. How do things become stratified in the mind?

I think types are a convenient fiction (not just type theory). We assume or endow a shared identity upon objects so I concur with the "brevity" comment you make later in your post.
How about "When our mind senses an inconsistency, we must back up the truck to identify its source."?
I disagree. The universal set is a concept contained within a mind. I we thus limit the scope of meaning of the term "universal set" it should be clear that it is not a member of itself.
Me, a name, I call myself....(apologies to Sound of Music).

Cheers, John

ex-xian

This is cool! I've had two logic classes and I'm taking a class in analytic philosphy along with a mathematical foundations class this fall. So I'm going to save this post and revisit later.

Witt

Witt:
It seems you are denying that, x is a member of x, has sense.
Russell, Quine, and Carnap, invoke some form of Russell's type theory, in order to deny unstratified entities, Do you?

I do not.
Type theory is a convenient fiction.
--------------------------------------------

John:
The key issue here is the process of stratification. How do things become stratified in the mind?

Things are descriptions of: physical objects, classes of physical objects, classes of classes..etc.

Some predicates are non-typical in the sense that they apply to different levels of objects. For example Existence and Identity are universal predicates in that they apply to all levels of objects.

x=x or E!x, has sense for: physical objects, classes of physical objects, classes of classes of physical objects..etc. and when x is a proposition or a predicate or a description of any of the above.

Russell denied the existence of all unstratified predicates, including ~(x e x), to avoid his paradox.

But, if x is the class of teaspoons, then ~(x e x), is true.
Quine allows unstratified predications but he denies their corresponing class. EyAx(x e y <-> Fx) is valid for all stratified F, for him.
Quine has also shown that some unstratified formulas do have existent extensions.

It is true that language, simplified by the imposition of type theory, has a logic that is easier to deal with.
For example, Carnap's simple language A (Intro to Symbolic Logic, 1962), where he identifies all predicates with their extension..ie. with their class.

John:
I think types are a convenient fiction (not just type theory). We assume or endow a shared identity upon objects so I concur with the "brevity" comment you make later in your post.

We do assume some sense of stratification in our objects.
E.g. x is blue, does not seem to have sense if x is a number or a proposition.

Witt :
Yes, comprehension and coherence are needed.
When our language demonstrates a contradiction, we must, back up the truck.. to where the logical error was.

John:
How about "When our mind senses an inconsistency, we must back up the truck to identify its source."?

Most definitely, when we appear to have a contradiction, we surely have done something logically wrong.

Impossible happenings cannot occur.

Witt :
It seems to me that the universal set is indeed a member of itself.
In which case we have proof of ..Ex(x is a member of x).

John:
I disagree. The universal set is a concept contained within a mind. I we thus limit the scope of meaning of the term "universal set" it should be clear that it is not a member of itself.

Why is it clear to you?

V={x:x=x}, or, V={x:E!x}.

V e V <-> Ey(Ax(x e y <-> x=x) & y e y)
V e V <-> Ey(Ax(x e y) & y e y)
V e V <-> EyAx(x e y).
V e V <-> E!(V)

Many systems maintain 1. EyAx(x e y) and 2. EyAx~(x e y) axiomatically.
That is to say, the univeral set exists, and the null set exists.

John:
Me, a name, I call myself....(apologies to Sound of Music).

Your singing is pretty bad, but, I hear your thought. (ha ha)

Witt

John Page

Hi Witt:
As we continue to circle like logical vultures around the object, its form and representation:
Yes, but also "things" are physical objects etc. Things can be used to describe other things, but if there are only descriptions one sinks into idealism. IMO our minds employ categories to concoct complex descriptions and outside the mind these categories are meaningless.
I disagree. There are no universal predicates. First, we can define non-existent objects (both physically non-existent and mental non-existent). Second, identity is a necessary fiction of the mind, referencing the concept that points in spacetime are unique. Identity is true for all mental objects, though, otherwise the means of perceiving them as separate mental objects would not exist!
Hmmmmmm. But there is no such thing as a universal proposition therefore the truth of x=x only has sense as an assumption. e.g. The house is red. Somebody paints the house blue. The house is not red.

This is why I propose that what is really happening can best be described as r0x = r1x. No two x's are absolutely identical, even idealized x's (classes, descriptions etc.) which must be two separate mental entities. It is the mental process of comparison of r0x and r1x that makes it appear that x=x.
Depends what you mean by both x's. A description of a teaspoon is not its class. Similarly, the class "teaspoon" is not a description of "class teaspoon".
Carnap didn't go far enough, the language notation must unambiguously show that all identities are unique - identifying a predicate with only its class assumes class membership a priori.
V (the mental concept of the universal set) is merely a member of the set U (the mental concept of the set of all mental concepts). Now, if we use A to represent "all things" we can see that A contains U contains V.

Furthermore, because our minds only deal with mental concepts, A is in reality the mental concept of the universal set which is exactly what we were claiming that V was!!. Hopefully this demonstrates how "the set of all sets is a member of itself" is an illusion.

If there is an issue it is with U (the mental concept of the set of all mental concepts). Is there self-membership here. I do not regard this as a problem because these mental concepts do not have to conform to the system of logic that is being used to analze them. "I do not exist", for example. . That mental concepts can appear (to any particular observer) incoherent marries with my real-world experience.

Relativism. If you wish to adopt a closed model of U then U is a member of U and you can explain Russell's Antinomy any way you wish (you can choose from any Axiom of Choice!!). If on the other hand, one admits that one doesn't know all mental concepts then the subset of U known to your mind is not, in fact, the set of all mental concepts. QED.

Footnote: You could invent the concept of the Universal Mind that knows all mental concepts. My response would be that the concept of a Universal Mind belongs to your set of mental concepts (that you may intersubjectively share).

Cheers, John