Russell's Paradox and the Excluded-Middle reasoning

[PFanalog57]

The set of dogs cannot be called "the set of dogs" if it has nothing to to with dogs.

"Dogness" is the abstract proprty of anything which is related to dogs, and this abstract property is the self identity of "the set of all dogs".

"DOGNESS" IS NOT ANY PARTICULAR DOG.

Perhaps "dogness" can be viewed as a form of constraint forcing the members included in the set of dogs to this defining aspect.

The set of all dogs is a subset of the set of all mammals...

Eventually, the set that includes "everything" is reached by removing nested constraints.

 

[PFanalog57]

Your quote from that link seems to say exactly the opposite...

Perhaps you can demonstrate[with logical symbolism] how the composition fallacy is not a form of modus ponens error?

 

[Lama]

Consider the set which contains the empty set, that set is not empty...

A set is only a framework where we can examine our ideas, and its own existence does not depend on the properties of its contents.

Only its name (identity) is denpend on the properties of its contents.

Again you use a non-abstract approech of the set concept.

As for "Dogness", I use this world as the most geneal concept of anything that is realed to dogs, but also does not have to be a dog at all.

If you have another word instead of "Dogness" to what I wrote above, then I'll be glad to get it from you.

 

[Lama]

Understand this simple fact - you cannot assume that a set must share any properties with the things it contains. If you do make this assumption, one of two things will happen:

1) your system of logic will be inconsistent

2) your system of logic will be unable to make any deductions about sets that do not share any properties with their elements, and will therefore be much, much weaker than existing logical systems

1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.

2) Please show us some mathematics where sets with no names (identities) are involved.

 

[master_coda]

1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.

2) Please show us some mathematic where sets with no names (identiteis) are involved.

What does that have to do with anything? Of course you need to know about the properties of the contents of the set. My point was that just because a property holds true for every element in a set, you cannot then conclude that the property also holds true for the set itself.

 

[matt grime]

perhaps, doron (i can't be bothered to work out which psuedonym you'll end up using in the next post), you should learn from this that attempting to apply mathematical ideas in nonmathematical ways is fraught with danger and doesn't tell you maths is flawed, which is you opinion apparently, but that your attempts to misuse it are. One need only view your opinion about dogness to see this.

 

[Lama]

you cannot then conclude that the property also holds true for the set itself.

Identity of a set has nothing to do with 'true' or 'false'.


The identity of a set is based on the most abstract basis of its contents, which gives it its name.

Therefore no set can contrarict its own identity (name).

In the case of Russell's paradox the identity of a set is forced to stay invariant, when its most abstract basis of its identity is chaneged.

This is exactly as if we say: Black is White or vise vera.

 

[Lama]

perhaps, doron (i can't be bothered to work out which psuedonym you'll end up using in the next post), you should learn from this that attempting to apply mathematical ideas in nonmathematical ways is fraught with danger and doesn't tell you maths is flawed, which is you opinion apparently, but that your attempts to misuse it are. One need only view your opinion about dogness to see this.

Please read post #63

 

[master_coda]

Identity of a set has nothing to do with 'true' or 'false'.


The identity of a set is based on the most abstract basis of its contents, which gives it its name.

Therefore no set can contrarict its own identity (name).

In the case of Russell's paradox the identity of a set is forced to stay invariant, when its most abstract basis of its identity is chaneged.

This is exactly as if we say: Black is White or vise vera.

"No set can contradict its own name" isn't just something you can just assert. It doesn't even make sense. Your "proof" is nothing more than you saying "I made up a rule about sets, and Russel's paradox violates it, so the paradox must be wrong".

What are you going to do next? Tell us that 0 = 1 and so obviously x/0 = x?

 

[Lama]

Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict itself identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?

Can you be master-coda which is not master-coda?

 

[master_coda]

Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict itself identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?

Can you be master-coda which is not master-coda?

If you have A = "set of all sets that contain themselves" then all you know is that if a set is in A, then that set must contain itself. The definition does not say that A must also contain itself.

If B = "set of all even numbers" you cannot assume that B must itself be an even number. If C = "set of all sets that contain themselves" you cannot assume that C is a set that contains itself. If D = "set of all sets that do not contain themselves" then you cannot assume that D must not contain itself.


And, even if you were to add an axiom to your system that said "the set of all sets that do not contain themselves does not contain itself", Russel's paradox still applies. No amount of whining that the set must not contain itself will change that.

 

[Hurkyl]

In this case A cannot be in a state that do not satisfies the properties of its parts, and this it exactly what I say.

Then you have to prove it. You have to prove that A satisfies the condition required to be a member of A.

The set of all dogs is not a dog itself but it has a common property with each dog that included in it, which we can call it "Dogness".

Can you show one example of "dogness" that is shared by "the set of all dogs"?

1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.

"The set of all dogs" gets is name precisely because:

(1) every dog is contained in the "set of all dogs"
(2) everything in the "set of all dogs" is a dog

There is no reason to think that the "set of all dogs" should have any properties of dogs. The relation here is:

A thing is a member of the "set of all dogs" if and only if that thing is a dog.


But, finally, "the set of all dogs" doesn't need to have a name; it can be discerned completely by the two properties I listed above.

In fact, pay attention to the fact that the name "the set of all dogs" is not really a name at all; it is a phrase stating what objects are in the set!


Speaking loosely, the "identity" of a set is entirely determined by the identities of its contents. This is stated in the axioms of ZFC by "A and B are the same set if and only they have the same elements".

By the way, to say that a set is not its members is a non-abstract and trivial thing to say.

You have this exactly backwards. A set is an abstract thing, whose "identity" is given entirely by the concept of "membership". The fact that you are unwilling (unable?) to separate the two ideas "properties of a set" and "properties of the elements of a set" is a very strong indicator of non-abstract thinking.

 

[Lama]

Master coda, think simple.

If by some operation or by avoiding some operation, a set contradict its own identity, then and only then we can conclude that a set loses its own identity.

And when a set loses its own identity, we cannot talk about its lost identity, as if it is still available for us.

 

[master_coda]

Master coda, think simple.

If by some operation or by avoiding some operation, a set contradict its own identity, then and only then we can conclude that a set loses its own identity.

And when a set loses its own identity, we cannot talk about its lost identity, as if it is still available for us.

Again, these are nothing more than rules you have just made up.

 

[Lama]

Hurkyl,

From your last post we can clearly see that you do not read all the posts that I write, and as a result most of what you write is based of luck of information.

But maybe this is the nature of your work, to read pieces of information and jump from thread to thread like a butterfly.

In short, please read ,for example, post #63.

 

[master_coda]

Hurkyl,

From your last post we can clearly see that you do not read all the posts that I write, and as a result most of what you write is based of luck of information.

But maybe this is the nature of your work, to read pieces of information and jump from thread to thread like a butterfly.

In short, please read ,for example, post #63.

Ha! You critizing people for lack of understanding is just funny. You don't even know the basic principles of logic, you don't know how to define things, and you can barely produce coherent English.

Perhaps you should consider the possibility that we all haven't fallen over ourselves worshipping your brilliance because you ideas make no sense, and not because we are stupid.

 

[Lama]

Again, these are nothing more than rules you have just made up.

No you use these simple rules to keep the identity of somthing.

These are the most simple rules that for example keeping you for not be me.

 

[Lama]

and you can barely produce coherent English.

My language is Hebrew.

..and not because we are stupid.

I think that Hurkyl, Matt Grime and you dear master coda are very intelegent and wise persons.

I simply have another point of view on the foundations of the language of Mathematics and to what directions it has to be developed.

 

[Hurkyl]

Self-identity says "a thing is itself"; nothing more, nothing less.

In the case where we're considering the thing "The set of all sets that contain themselves", self identity says:


"The set of all sets that contain themselves" is "the set of all sets that contain themselves".

Notice that this is a different statement than

"The set of all sets that contain themselves" is a "set that contains itself".


The other examples I mentioned are intended to show this. The following three statements are all false:

"The set of all dogs" is a "dog"
"The set of everything worth less than a dollar" is "worth less than a dollar"
The set of everything blue is blue.


The point is, "self-identity" cannot be used (by itself) to prove:

"The set of all T" is a "T".

 

[Hurkyl]

Perhaps you can demonstrate[with logical symbolism] how the composition fallacy is not a form of modus ponens error?

Any example would do.

Here's one: \forall x \in A: P(x) therefore P(A).

 

[Lama]

The point is, "self-identity" cannot be used (by itself) to prove:

Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict itself identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?

Can you be Hurkyl which is not Hurkyl?

"The set of all sets that contain themselves"

A set with no identity is only the green part.

Some identity (name) is the red part.

We need both green and red parts to make Math.

And the red part cannot contradict itself during Math operations.

 

[Hurkyl]

The problem is that you're not being consistent about when you use the green part.

Self identity says:

The set of all sets which contain themselves is The set of all sets which contain themselves.

Furthermore, we have the tautology

A set which contains itself is a set which contains itself.

However, you are saying

The set of all sets which contain themselves is a set which contains itself.

You can either have the green on both sides of "is", or on neither side; it is (usually) incorrect to mix and match.

 

[Lama]

Sorry Hurkyl,

But you have changed what I wrote.

The green part is: The set of...

The red part is: all_sets_that...

 

[PFanalog57]

Any example would do.

Here's one: \forall x \in A: P(x) therefore P(A).

If A then B


If each stick is breakable, then the whole bundle of sticks is breakable.

The fallacy of composition has a faulty premise.

 

[Lama]

If each stick is breakable, then the whole bundle of sticks is breakable.

It is like a self similarity of a fractal.

 

[Lama]

What I suggest is very simple:

1) The language of Mathematics is based on two systems: The relative and the absolute.

2) The absolute system is a finite or infinitely many elements with no unique identity.

3) The relative system is a finite or infinitely many possible unique names.

4) If a possible unique name is related to some absolute element, it determines its identity.

5) There are two basic types of operations on an element with a unique identity:

a) An operation that changes its identity.

b) An operation that does not change its identity.

6) In an excluded-middle reasoning, an absolute element can have simultaneously a one and only one unique name (identity).


If the absolute element is a set under an excluded-middle reasoning, then:

1) Its identity depends on the most abstract property of its content; therefore it cannot contradict the most abstract property of the content.

2) This identity cannot be changed under any operation, unless the most abstract property of the content is changed.

3) If the identity of a set is changed under an operation, its previous identity is not related to it anymore.

 

[master_coda]

Those rules you proposed are not any use when you still have a fundamentally flawed concept of "identity"; you still believe that any property that describes the contents of a set must describe the set itself.

 

[Lama]

Those rules you proposed are not any use when you still have a fundamentally flawed concept of "identity"; you still believe that any property that describes the contents of a set must describe the set itself.

An identity of a set cannot be but with a relation with the most abstract proprty of the content.

It is very fundamental.

By "most abstract proprty" I mean thet when this proprty is omited, the set losing its identity, and return to the state of a non-unique absolute element.

The idea of the relative/absolute relations can be used as a fundamental idea in more then one aspect of the Langauge of Mathematics.

For eample, please read this: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

 

[master_coda]

An identity of a set cannot be but with a relation with the most abstract proprty of the content.

It is very fundamental.

By "most abstract proprty" I mean thet when this proprty is omited, the set losing its identity, and return to the state of a non-unique absolute element.

The idea of the relative/absolute relations can be used as a fundamental idea in more then one aspect of the Langauge of Mathematics.

For eample, please read this: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

You seem to be treating abstact ideas as if they are phsyical objects that can be changed or destroyed.

For example you talk about a set as if it is something that can be made to go away; as if by changing the properties of the set, you somehow make the original abstraction disappear.

A set which exists cannot be somehow made to not exist. You can construct a new set that has weaker properties (the equivalent of "removing" the properties or "identity" of the original set) but that does not make the original set no longer exist.

 

[matt grime]

none of the things you suggest mathematics must be, or do, is informed, lama/doron/shmesh/etc, and just demonstrates your complete lack of understanding, and you total ignorance of the world you claim to talk about. Are you even aware of topoi where the 'excluded middle' fails to be true? No, you aren't. Mathematics is far richer than you can even begin to understand, and the repeated demonstrations of your ignorance of it are not particulalry endearining.

You are also inconsitent in the extreme. One need only look at you belief in dichotomic options to see that.