Russell's Paradox and the Excluded-Middle reasoning

[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 itself identity.

The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself, because this is itself 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?

 

[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

 

[Lama]

http://plato.stanford.edu/entries/russell-paradox/

 

[Hurkyl]

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.

 

[Lama]

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

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.

So, please read again my first post and try to understand its tautology/recursion idea, before you air your view about it.

 

[wespe]

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

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

 

[Hurkyl]

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)

 

[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.

So, please read again post #1, thank you.

 

[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)
Looks like a paradox.
Do you think this is relevant?

 

[master_coda]

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)
Looks like a paradox.
Do you think this is relevant?

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

 

[wespe]

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

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.

 

[master_coda]

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.

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.

 

[quantumdude]

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

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.

 

[Lama]

Tom Mattson,

'is' equal to '='.

Please read again #1. It is clearly written there.

 

[quantumdude]

Tom Mattson,

'is' equal to '='.

Please read again #1. It is clearly written there.

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

 

[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, Russell'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 Russell's paradox simply does not exist in our excluded-middle logical reasoning system.

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

So I hope that you agree with me that the question: Is "include = "not include", is meaningless through an excluded-middle reasoning.



What do you think?

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.

 

[Lama]

Hi Russell E. Rierson,

Please read this again:

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 itself identity.

The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself, because this is itself 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?

 

[master_coda]

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 itself identity.

The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself, because this is itself 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?

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 itself 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.

 

[Lama]

The problem is that if the set exists ...

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 itself 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?"

 

[master_coda]

You did 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.

Russel's Paradox does not contradict its own existence, so your argument is invalid. The paradox is just that the existence 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.

 

[Lama]

Also you missed the most important point which is:

Not the set is examined here, but the meaningless question:

Is "contain" is "does_not_contain?"

The set concept is a natural concept here and we can change it by anything that can or cannot contain things.

 

[master_coda]

Also you missed the most important point which is:

Not the set is examined here, but the meaningless question:

Is "contain" is "does_not_contain?"

The set concept is a natural concept here and we can change it by anything that can or cannot contain things.

But this has nothing to do with Russel's paradox. It's just a meaningless question because it makes no grammatical sense.

Unless you mean "is contain = does_not_contain" which is not meaningless either. The answer is just NO. Much like "is true = false" can also be answered NO.

 

[Lama]

It is connected to Russell's paradox.

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

The set of ALL_sets_that_ do_not_contain_themselves,
must not contain itself as a member of itself, because this is itself identity.

Now please omit the green and red words from the above sentences and you have lost the argument here.

It means that the whole "story" here is around the green and the red words.

green word cannot be but a green word.

red word cannot be but a red word.

You can add the set concept or not, but it does not change this tautology/recursion argument.

Therefore Russell's Paradox does not exist.

If by your reasoning "does not exist" = "false", then Russell's Paradox is false.

 

[master_coda]

The set of ALL_sets_that_ do_not_contain_themselves,
must not contain itself as a member of itself, because this is itself identity.

But if the set does not contain itself, then it must be contained in the set of all sets that do not contain themselves. Thus the set must contain itself.

 

[Lama]

Then you ignore again the self identity of x to itself.

 

[master_coda]

Then you ignore again the self identity of x to itself

I don't have to ignore anything. No matter how many defintions you add, and how many different ways you can prove that the set does not contain itself, the contradiction I demonstrated will always exist. Any logical system that is inconsistent cannot be fixed by adding more to the system.

 

[Lama]

Maybe this analogy can help.

Because by the tautology/recursion idea the tautology x = x is like the recursion x = x = x = ... then we can think on a fractal-like information form.

Any part of our fractal is like the whole fractal, and if we have a green fractal it cannot be with any self identity relation with, for example, a red fractal, and vise versa.

Now let us say that a green fractal is the tautology/recusion contain and a red fractal is the tautology/recusion does_not_contain.

It is clear that they cannot be in each other states without first to lose their own existence (self identity).

 

[Hurkyl]

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

The set of ALL_sets_that_ do_not_contain_themselves,
must not contain itself as a member of itself, because this is itself identity.

This is not "self-identity"; this is the "fallacy of composition". In general, the whole does not have the properties of its parts.

"self-identity" says that the set of all sets that contain themselves must be the set of all sets that contain themselves.

 

[Lama]

Maybe this analogy can help.

Because by the tautology/recursion idea the tautology x = x is like the recursion x = x = x = ... then we can think on a fractal-like information form.

Any part of our fractal is like the whole fractal, and if we have a green fractal it cannot be with any self identity relation with, for example, a red fractal, and vise versa.

Now let us say that a green fractal is the tautology/recusion contain and a red fractal is the tautology/recusion does_not_contain.

It is clear that they cannot be in each other states without first to lose their own existence (self identity).

 

[master_coda]

Maybe this analogy can help.

Because by the tautology/recursion idea the tautology x = x is like the recursion x = x = x = ... then we can think on a fractal-like information form.

Any part of our fractal is like the whole fractal, and if we have a green fractal it cannot be with any self identity relation with, for example, a red fractal, and vise versa.

It would be nice if when you used words like tautology, recursion and fractal you used them with the meanings everyone else uses and not your own, but that isn't really a problem here.

If you use the standard ideas of what a set is and what "contains" is in naive set theory, then Russel's paradox is in fact a paradox. If you change the definitions of "set" and "contains" to something else, then it's possible that Russel's paradox is not a problem with the new definitions. However that does not alter the fact that Russel's paradox is a paradox in naive set theory...it just avoids the problem by using a different kind of set theory.

 

[Lama]

Sorry Master_coda,

But again you miss the fine point of my previous post, whit is the word "first".

So here is my analogy again and this time pay attention to this word:

Because by the tautology/recursion idea the tautology x = x is like the recursion x = x = x = ... then we can think on a fractal-like information form.

Any part of our fractal is like the whole fractal, and if we have a green fractal it cannot be with any self identity relation with, for example, a red fractal, and vise versa.

Now let us say that a green fractal is the tautology/recusion contain and a red fractal is the tautology/recursion does_not_contain.

It is clear that they cannot be in each other states without first to lose their own existence (self identity).

 

[Lama]

"self-identity" says that the set of all sets that contain themselves must be the set of all sets that contain themselves.

And also "self-identity" say that the set of all_sets_that_do_not_contain themselves must be the set of all_sets_that_do_not_contain themselve.

 

[master_coda]

Sorry Master_coda,

But again you miss the fine point of my previous post, whit is the word "first".

And you continue to miss the point. This is an entirely new system you've invented, so it has nothing to do with any existing theory. So it proves nothing about how Russel's paradox applies to other theories. The "does not contain" that you are referring to is not the "does not contain" relation used in the construction of Russel's paradox.

 

[Lama]

No set can be its opposite ("contain" , "does_not_contain") without first losing its own existene.

Therefore the circular state of Russell's Paradox does not exist.

 

[master_coda]

No set can be its opposite ("contain" , "does_not_contain") without first losing its own existene.

Therefore the circular state of Russell's Paradox does not exist.

So your argument is that Russel's paradox does not exist because if it did the set that contains all sets that do not contain themselves would not exist?

Normally when you have axioms and you derive a contradiction from those axioms, you fix the problem by changing your axioms, not stating that logic does not exist.