# Russell's Paradox and the Excluded-Middle reasoning

1. Jun 12, 2004

### Lama

By tautology x = x means: x is itself, otherwise we cannot talk about x.

Now we can ask if a teotology is also recursive, for example: x = x = x = ...

If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.

So, Russel's paradox is like if by teotology we ask if x is not_x or x = not_x , which is a meaningless question trough an exluded-middle point of view.

Because our one_step_recursion cannot be done, we have no product of tautology and we can conclude that the whole idea of Russel's paradox simply does not hold in an excluded-middle logical reasoning.

By the way, in Russell's paradox x is not the set, but the word "contain".

So I hope that you agree with me that the question: Is "contain" = "do_not_contain", is meaningless through an excluded-middle reasoning.

The set of ALL_sets_that_contain_themselves,
must contain itself as a member of itself, because this is its self identity.

The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself, because this is its self identity.

(remark: the existence of a set is not dependent on its content for example:

The empty set exists as a framework which examines the abstract idea of "emptiness".

In short, only the name of the set depends on its property, but not its own existence as a framework.
)

Therefore there is nothing here that can be found as a state of a paradox.

What do you think?

Last edited: Jun 13, 2004
2. Jun 12, 2004

### ram2048

would help if you laid out the paradox instead of just talking about it and assuming people know what the heck you're talking about :D

3. Jun 12, 2004

### Lama

4. Jun 12, 2004

### Hurkyl

Staff Emeritus
Well, first off, the adverb "not" is only applicable to logical propositions/statements/formulae. Saying "not x" is meaningless unless x is a proposition.

Secondly, "x = not x" is most certainly a meaningful statement. (If, of course, x is a proposition) It's just false.

5. Jun 12, 2004

### Lama

Then please show us how we use the proper formal way to notate x and not_x where x is a general notation for any thing, which is not a logical condition, Thank you.

Secondly, "x = not x" is not a fact in Russell's Paradox, but a sort of a question which its result has logically be checked and determinate by us.

So, first we have to check if this paradox can really exists before we starting our x not_x circular situation, which leads us to conclude that we are in an impossible excluded-middle state.

Last edited: Jun 12, 2004
6. Jun 12, 2004

### wespe

Maybe irrelevant, but there's also a "not" operator in computing which works on numbers represented by bits (one's complement operator).

edit: oops, I just checked with the windows calculator, it turns out it just makes a number negative

Last edited: Jun 12, 2004
7. Jun 12, 2004

### Hurkyl

Staff Emeritus
Well, the first thing is to figure out just what you mean by "not x". As I was composing on my reply, I hit upon something grammatical that may have led you to make statements such as you have been doing.

If we take the statement "y is z", we can break it up into three parts; we have two objects, "y" and "z", and we have a relation, "is".

The negation of this statement is "y is not z". Grammatically, the right way to parse this phrase is that "not" modifies "is". In other words when we break this statement into three parts, we get two objects, "y" and "z", and we have a relation, "is not".

It would be incorrect to interpret the phrase "y is not z" as having two objects, "y" and "not z", being connected by "is".

That is why, symbolically, we write the phrase "y is not z" as something like y != z or $y \neq z$; the relation means "is not".

(maybe things would be clearer if, instead of "is", you use "is equal to"? I think the latter is somewhat more proper)

8. Jun 12, 2004

### Lama

Hurkyl,

x is not_x in an excluded-middle reasoning system, is the reason why we are calling it a paradox.

I say that the situation x is not_x in this case simply does not exist, therefore it is avoided before we can conclude that x is not_x is a paradox in an excluded-middle system.

9. Jun 12, 2004

### wespe

Sorry for interrupting again.
x = not x looks like x = -x
by substituting we get:
x = - ( - ( - (.....- x)
which is like:
+1 * -1 * +1 * -1 * ...... (* denotes multiplication)
Do you think this is relevant?

10. Jun 12, 2004

### master_coda

How is x = -x a paradox? 0 = -0, after all.

11. Jun 12, 2004

### wespe

hmm. do all mathematicians agree -0 = +0? (possibly a stupid question)

But back to the topic: maybe in this context it means x doesn't exist.

12. Jun 12, 2004

### master_coda

Yes, +0 = -0. You could probably come up with a system where that wasn't true, but then zero would not be the additive identity so calling it "zero" would be misleading.

I don't know what's supposed to be so interesting about the statement x = not x. It's just a false statement, like 1 = 2.

13. Jun 12, 2004

### Tom Mattson

Staff Emeritus
I hate to sound like Bill Clinton, but what do you mean by "is"? Do you mean material equivalence? If so, then "x is ~x" is not a paradox. It's just false.

14. Jun 13, 2004

### Lama

Tom Mattson,

'is' equal to '='.

15. Jun 13, 2004

### Tom Mattson

Staff Emeritus
OK, in that case: "x is ~x" is not a paradox. It is a false statement.

16. Jun 13, 2004

### Russell E. Rierson

The set of all sets that are not members of themselves. Seems to boil down to incomplete definitions? Insufficiency of language? exclusion/inclusion ?

So the set of all dogs is not a member of itself, since, it is not a dog.

But the "dog" identity, is an abstract Platonic form, that gives the aspect of "dogness" to all dogs. The identity is self contained.

So a generalization of the set axioms certainly would help, and we can stop flogging the tired ol' ZF horse. Ergo, a merger of symmetry and ZF theory is of paramount importance.

17. Jun 13, 2004

### Lama

Hi Russell E. Rierson,

By tautology x = x means: x is itself, otherwise we cannot talk about x.

Now we can ask if a teotology is also recursive, for example: x = x = x = ...

If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.

So, Russel's paradox is like if by teotology we ask if x is not_x or x = not_x , which is a meaningless question trough an exluded-middle point of view.

Because our one_step_recursion cannot be done, we have no product of tautology and we can conclude that the whole idea of Russel's paradox simply does not hold in an excluded-middle logical reasoning.

By the way, in Russell's paradox x is not the set, but the word "contain".

So I hope that you agree with me that the question: Is "contain" = "do_not_contain", is meaningless through an excluded-middle reasoning.

The set of ALL_sets_that_contain_themselves,
must contain itself as a member of itself, because this is its self identity.

The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself, because this is its self identity.

(remark: the existence of a set is not dependent on its content for example:

The empty set exists as a framework which examines the abstract idea of "emptiness".

In short, only the name of the set depends on its property, but not its own existence as a framework.
)

Therefore there is nothing here that can be found as a state of a paradox.

What do you think?

Last edited: Jun 13, 2004
18. Jun 13, 2004

### master_coda

Do you not understand the difference between a false statement and a meaningless one? "contain" = "do_not_contain" is not a meaningless statement, it is a false one.

And the problem of the set of all sets that do not contain themselves is not one that can be solved by waving your hands and saying "it doesn't contain itself because this is its self identity". The problem is that if the set exists then you can show that the set contains itself if and only if the set does not contain itself, which contradicts the law of the excluded middle.

You can only work around this problem by doing one of two things:

1) You can use a system of logic that does not include the law of the excluded middle. This significantly weakens your system of logic by making the fact that a statement is true almost meaningless.

2) You can use a version of set theory that does consider the set of all sets that do not contain themselves to be a set. This is the approach ZF set theory takes. It avoids the paradox by eliminating certain types of sets.

19. Jun 13, 2004

### Lama

You do not understand my tautology/recursion argument.

Nothing can exist and also contrarict its own existence, therefore Russell's Paradox simply deos not exist and there is no "if" here.

You can wave with your "if" as much as you want, and still Russell's Paradoxs is meaningless exactly like these meaningless questions:

"Is a layer is a honest?" or "Is a honest is a layer?"

"Is black is white?" or "Is white is black?"

There is no false here but only meaningless questions.

"it doesn't contain itself because this is its self identity" and there is no "if" here!

Also you missed the most importent point which is:

Not the set is examined here but the meaningless question:

Is "contain" is "do_not_contain?"

Last edited: Jun 13, 2004
20. Jun 13, 2004

### master_coda

Russel's Paradox does not contradict its own existance, so your argument is invalid. The paradox is just that the existance of the set of all sets that do not contain themselves contradicts the law of the excluded middle. Thus any version of set theory that states that such a set exists is inconsistent.

Your examples you meaningless questions (they actually aren't even questions, not even meaningless ones) have nothing to do with Russel's paradox.