A hidden assumption

[doronshadmi]

most theorems are valid across many related domains.

Please define rigorously the principle that enables that.

Furthermore, please show the unified mathematical theory that enables that.

If you try that, you will find that this principle is exactly
Same_realm(Obsereved, X) http://www.internationalskeptics.com/forums/showpost.php?p=3557678&postcount=1 .

My theory is a generalization of group-theory symmetry that allows D2 to "reduce to" D1, because I directly define the invariance principle, which does not need reflection, rotation or any other manipulation in order to be defined.

The same holds in the case of the variance principle. I directly define it without any manipulation.

From this level of generalization, we get the universal principle of both mathematics and Logic, and this generalization is invariant\variant association where the invariant is the associater (the Same_realm or the non-local) and the variant is the associated (Obsereved ; X or the local).

Please also look at http://www.internationalskeptics.com/forums/showpost.php?p=3557067&postcount=157

If you disagree with me then please explicitly show why it is not the universal principle of both mathematics and logic (please be aware that you cannot use any dichotomy between the variant and the invariant (X vs. XY, in your language), because if you do that, than they cannot share the same realm called the mathematical science).


If you agree with me, then you agree with Hilbert's organic paradigm, but agree is not enough.

Mathematicians all over the word have to combine their talents and work together in order to rigorously develop the unified theory of the mathematical science.

If such a theory already exists, then please explicitly show it.

" most theorems are valid across many related domains." Is a very poor statement for a rigorous basis of such theory.

 

[jsfisher]

1) Please answer to http://www.internationalskeptics.com/forums/showpost.php?p=3555305&postcount=154

What is this? Some sort of childish, "you answer mine then I'll answer yours?" You opened this thread. You made use of terms in ways you have yet to define. The obligation falls to you to take the lead.

Tell you what, though. Define two of the terms in ways we can understand, and then I will discuss with you the difference between how many is a set and how many is in a set.

2) Please define "definition" and "term". It will help me to define the terms above.

Now, you are just being silly. Is this really the path you want to follow? That you will use terms to make your point when your terms have absolutely no meaning to anyone except you?

Start with something easy. Tell us what you mean by "magnitude."

 

[doronshadmi]

What is this? Some sort of childish, "you answer mine then I'll answer yours?" You opened this thread. You made use of terms in ways you have yet to define. The obligation falls to you to take the lead.

Tell you what, though. Define two of the terms in ways we can understand, and then I will discuss with you the difference between how many is a set and how many is in a set.



Now, you are just being silly. Is this really the path you want to follow? That you will use terms to make your point when your terms have absolutely no meaning to anyone except you?

Start with something easy. Tell us what you mean by "magnitude."

Your answers to http://www.internationalskeptics.com/forums/showpost.php?p=3556507&postcount=156 are very important to our dialog (I understand "dialog" as a simultaneous internal\external interaction where each one of us tries to communicate with the other (the external) by discovering his own internal space (the depth of his own notions)). be patient an please believe me in this case .

Thank you.

 

[jsfisher]

Your answers to http://www.internationalskeptics.com/forums/showpost.php?p=3556507&postcount=156 are very important to our dialog (I understand "dialog" as a simultaneous internal\external interaction where each one of us tries to communicate with the other (the external) by discovering his own internal space (the depth of his own notions)). be patient an please believe me in this case .

Thank you.

Communication isn't possible if you persist in using words and phrases that having meaning only to you. You need to either (a) follow standard nomenclature or (b) define your terms. Pick one.

Since option-a seems beyond your abilities, I recommend you proceed with option-b. Start with something easy. Tell us what you mean by "magnitude."

 

[doronshadmi]

Communication isn't possible if you persist in using words and phrases that having meaning only to you. You need to either (a) follow standard nomenclature or (b) define your terms. Pick one.

Since option-a seems beyond your abilities, I recommend you proceed with option-b. Start with something easy. Tell us what you mean by "magnitude."

You know what?

Please formally define "magnitude".

It will help both of us to compare our notions according to something that is accepted by you.

 

[jsfisher]

You know what?

Please formally define "magnitude".

It will help both of us to compare our notions according to something that is accepted by you.

My "notion" of the word, magnitude, is irrelevant. I am not the one who used it in trying to show that Cantor was wrong. It was your term, used in a non-standard way, that is at question, and so the obligation for a definition falls to you.

You wrote this in the opening post:

Cantor is wrong because no non-finite collection has the magnitude of the non-local ur-element ( en.wikipedia.org/wiki/Urelement ) real-line, and as a result any given non-finite collection is open and incomplete when compared with the non-local ur-element real-line's magnitude.

What did you mean when you wrote that?

 

[dogjones]

Your answers to http://www.internationalskeptics.com/forums/showpost.php?p=3556507&postcount=156 are very important to our dialog (I understand "dialog" as a simultaneous internal\external interaction where each one of us tries to communicate with the other (the external) by discovering his own internal space (the depth of his own notions)). be patient an please believe me in this case .

Thank you.

Trrrrrrrrrrrrrrooooooooooolllllllll!

 

[doronshadmi]

My "notion" of the word, magnitude, is irrelevant. I am not the one who used it in trying to show that Cantor was wrong. It was your term, used in a non-standard way, that is at question, and so the obligation for a definition falls to you.

You wrote this in the opening post:



What did you mean when you wrote that?

You did not understand me. Your notion of Magnitude is the accepted (standard) one so all I asked you is to give the formal definition of Magnitude.

A magnitude is the size of a mathematical object.

For example: R set magnitude (size) is the collection of all distinct objects that exist between size 0 and size 1.

In that case R's magnitude is based on the multitude ("infinitely many ...").

I logically showed that a local object is any object that has simultaneously one and only one relation with another object. I call this relation locality.

In that case R is a collection of infinitely many local objects.

Also I logically showed that between any given pair of local objects there is an object that has a simultaneous relation with both of them, I call this simultaneous relation non-locality.

No collection of local objects has the magnitude of a non-local object.

As a result any collection of local objects is incomplete (its magnitude is less than the magnitude of a non-local object).

Non-locality is the invariant object (the non-local) , and no collection of variant (local) objects gets its magnitude.

 

[doronshadmi]

Realy jsfisher,

What prevents from you the simple notion that the universal principle of Logic
is invariant(TT,FF)\variant(FT,TF) association, defined as:

T   variant   F
|             |
|             |            
\--invariant--/

?

 

[jsfisher]

A magnitude is the size of a mathematical object.

As I read this sentence, I was ready to assume you were using "magnitude" to mean "number of elements in" when referring to sets. That would have made your "magnitude" a synonym for "cardinality".

But then I read this:

For example: R set magnitude (size) is the collection of all distinct objects that exist between size 0 and size 1.

I think it safe to assume you are using R to refer to the set of real numbers; however, you are now saying that "magnitude" is a collection. More interestingly, you introduced "size 0" and "size 1" from nowhere. Did you mean (the real numbers) 0 and 1?

Was this a misstatement? Did you mean to say, "the magnitude of the set R is the number of distinct objects that exist between 0 and 1"? That would be consistent with magnitude being used as a synonym for cardinality. The only problem, then, would be that cardinality of R is the number of objects in R. That that is the same as the cardinality of the open interval (0,1) is a provable consequence, but not part of the definition.

With that all said, though, do we have agreement on what you meant by "magnitude"? Namely, magnitude of a set is the number of elements in the set; it is a synonym for cardinality.

...
I logically showed that a local object is any object that has simultaneously one and only one relation with another object. I call this relation locality.

Ok, you need to stop there. Local and non-local are terms you need to define before you proceed. (You might also want to give some thought to the semantics behind "one simultaneous relation".)

 

[doronshadmi]

Did you mean to say, "the magnitude of the set R is the number of distinct objects that exist between 0 and 1"? That would be consistent with magnitude being used as a synonym for cardinality.

As for Magnitude and Size please see http://en.wikipedia.org/wiki/Magnitude_(mathematics)

Ok, you need to stop there. Local and non-local are terms you need to define before you proceed. (You might also want to give some thought to the semantics behind "one simultaneous relation".)

A local object z is any element that satisfies this:

Given any arbitrary interval x<y either z=x or z=y

A non-local object z is any element that satisfies this:

Given any arbitrary interval x<y z is both x and y (this is not a part of the definition but there is no local object w such that x<w<y)

 

[jsfisher]

As for Magnitude and Size please see http://en.wikipedia.org/wiki/Magnitude_(mathematics)

Ok, but, since there is no clear concept of "size" for a set, you need to be precise as to what you mean when speaking of the magnitude of such an object. I'm still guessing cardinality, but you'd need to confirm that.

A local object z is any element that satisfies this:

Given any arbitrary interval x<y either z=x or z=y

Ok, this is also not a set concept, per se, but no matter; however, I am having trouble grasping the utility of this concept. For any completely ordered class, R as an example, all elements would be local. Only in a partially ordered class might there be objects that don't satisfy your "end-point" criterion. I'd be more inclined to call them orphans rather than not local objects, but that's just me.

A non-local object z is any element that satisfies this:

Given any arbitrary interval x<y z is both x and y and there is no local object w such that x<w<y

I would have expected the definitions of local and non-local to be complementary. They aren't. But, more importantly, the condition is impossible to satisfy. Given the nature of a less-than relationship, for no X and Y where X < Y can there be a Z that is both X and Y.

So, the explanation you have provided for "non-local object" is nonsense.

ETA: You have since edited your post, but not in a way that salvages the meaning of "non-local object".

 

[doronshadmi]

Given the nature of a less-than relationship, for no X and Y where X < Y can there be a Z that is both X and Y.

The nature of x or y is their locality.

Given any arbitrary interval x<y (where x or y are local) z is both x and y and therefore non-local.

It is nonsense only of you understand z in terms of a local object like x or y.

Since this is not the case, then z holds.

Furthermore, x<y is observed from a variant (asymmetric and local) point of view, where z (both x and y) is observed form an invariant (symmetric and non-local) point of view.



Form this point of view x and y are in a superposition, for example:

[U]A parallel and simultaneous view of the mathematical universe[/U] 

x     x
y     y
.     .
|     |   
|__=__| superposition (symmetry)

where this is the asymmetrical (and local) point of view.

[U]A serial and linear view of the mathematical universe[/U]

x  <  Y  order (asymmetry) 
.     .
|     |   
|_____|
|

Actually local x or local y cannot be associated without the non-locality between them.

I would have expected the definitions of local and non-local to be complementary.

So the mathematical universe is not less than invariance\variance complementation.


Definition 1: Identity is a property of X or Y, which allows distinguishing among them.

Definition 2: Copy is a duplication of a single identity.

Definition 3: If X or Y have more than a single identity, then X or Y are called uncertain.

Definition 4: If X or Y have more than a single copy, then X or Y are called redundant.

Definition 5: Superposition is Uncertainty and Rerundancy.

An example:

 

[Gate2501]

The nature of x or y is their locality.

Given any arbitrary interval x<y (where x or y are local) z is both x and y and therefore non-local.

It is nonsense only of you understand z in terms of a local object like x or y.

Since this is not the case, then z holds.

Furthermore, x<y is observed from a variant (asymmetric and local) point of view, where z (both x and y) is observed form an invariant (symmetric and non-local) point of view.

Form this point of view x and y are in a superposition, for example:

[U]A parallel and simultaneous view of the mathematical universe[/U] 

x     x
y     y
.     .
|     |   
|__=__| superposition (symmetry)

where this is the asymmetrical (and local) point of view.

[U]A serial and linear view of the mathematical universe[/U]

x  <  Y  order (asymmetry) 
.     .
|     |   
|_____|
|

Actually local x or local y cannot be associated without the non-locality between them.

So the mathematical universe is not less than invariance\variance complementation.

I think that my example better explains the non-locality:

           ,
          /|
         / |                              All aboard!
        /  |
       ----|
    |===========/
    | FAILBOAT /
WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

 

[jsfisher]

The nature of x or y is their locality.

Given any arbitrary interval x<y (where x or y are local) z is both x and y and therefore non-local.

I'd ask for a concrete example, but you have demonstrated in other posts that you are unable to provide concrete examples.

Given that X < Y guarantees that X and Y are distinct, there can be no Z such that Z = X and Z = Y. These are all fundamental properties of the ordering operators. (Equality is transitive and symmetric, so Z = X and Z = Y → X = Y, a contradiction of the premise X < Y.)

You can define "non-local" that way if you like, but your definition insures that the set of all non-local objects is { }.

 

[doronshadmi]

Given that X < Y guarantees that X and Y are distinct, there can be no Z such that Z = X and Z = Y.

Only if Z is local (asymmetric), but it isn't.

If you try to understand Z in terms of asymmerty, then Z is indeed the "content" of {}.

If Z is symmetric, than given any domain, Z is the same if in_the_domain and out_of_the_domain are examined, and it does not matter if both are empty or not.

The same symmetric principle holds also at 2-valued logic

TF
FF
TT

where the difference between F and T is ignored and only the invariance is important.

But it is not enough to get any premise unless Sameness\TF complement each other.


No Z, where Z is asymmetric, has this property.

Furthermore, distinct X or distinct Y cannot be compared and be ordered unless they share the same mathematical space, so the "premise" X < Y is actually a conclusion (which is a particular case) that is based on symmetry\asymmetry complementation, and not vice versa.

If you get this notion, then you made the first step in order to understand my theory.

The notion thet you do not get (yet) is this:

Size (Cardinality) and Distinguishability (Order) are the results of Symmetry\Asymmetry complementation, and not vice versa.

In that case superposition or order are some particular cases of Symmetry\Asymmetry complementation.

Traditional Mathematics of the past 2500 is mostly based on the distinct and it does not aware of the universal principle of Symmetry\Asymmetry complementation.

 

[jsfisher]

Only if Z is local (asymmetric), but it isn't.

Your definition, and your logic, is circular. Your words have no meaning. You have assumed a result then built that result into your premise.

You are back to gibberish.

 

[doronshadmi]

Your definition, and your logic, is circular. Your words have no meaning. You have assumed a result then built that result into your premise.

You are back to gibberish.

On the contrary.

I explicitly showed that the Researchable cannot be but the result of Symmetry\Asymmetry complementation, which is (currently) beyond your Asymmetric-only point of view that is based on a step-by-step way of thinking.

Your (current) inability to be aware of Symmetry\Asymmetry complementation as the universal principle of the Researchable, does not contradict the fact that symmetry is used by you as a hidden-assumption, whether you like it or not.

The difference between you and me, in this case, is that I explicitly define the Symmetry\Asymmetry complementation as the universal principle of the Researchable, were you use it but unaware of it.

 

[jsfisher]

On the contrary.

I explicitly showed that the Researchable cannot be but the result of Symmetry\Asymmetry complementation, which is (currently) beyond your Asymmetric-only point of view that is based on a step-by-step way of thinking.

No, you have simply imagined that to be the case, but are unable to explain what you mean. You cannot prove something in Mathematics using gibberish and hand waving, no matter how impressive either appears to be.

Your (current) inability to be aware of Symmetry\Asymmetry complementation as the universal principle of the Researchable, does not contradict the fact that symmetry is used by you as a hidden-assumption, whether you like it or not.

Again, not a fact, but an assumption on your part. As for you insistence of a "hidden assumption", you're inability to communicate using either conventional terminology, or terminology you can actually define, limits your powers of persuasion considerably.

The difference between you and me, in this case, is that I explicitly define the Symmetry\Asymmetry complementation as the universal principle of the Researchable, were you use it but unaware of it.

(Emphasis added.)
You have explicitly defined very little.

In the case of "non-local", for example, you offered up a definition that could not possibly be satisfied. Then, when challenged on that point, you simply assumed the impossible was possible and then attempted to bury it under load of other terms you have yet to define (or use in a conventional way).

Care to try again with a explanation/definition of "non-local" that is actually satisfiable and doesn't circle back on itself?

 

[jsfisher]

Only if Z is local (asymmetric), but it isn't.

By the way, does this mean you are now redefining "local"? Should we now take local to mean asymmetric? What does it mean for a single object to be asymmetric? For example, in what way do you mean the number 8 is asymmetric?

If you try to understand Z in terms of asymmerty, then Z is indeed the "content" of {}.

If Z is symmetric, than given any domain, Z is the same if in_the_domain and out_of_the_domain are examined, and it does not matter if both are empty or not.

And none of that is relevant to the definition you offered for "non-local".

...
Furthermore, distinct X or distinct Y cannot be compared and be ordered unless they share the same mathematical space, so the "premise" X < Y is actually a conclusion (which is a particular case) that is based on symmetry\asymmetry complementation, and not vice versa.

You need to look up the meaning of "premise". (Well, you really need to look up the meaning of a lot of words, but I'll focus on "premise" for now.) In your definitions for "local" and "non-local" you put "X < Y" as a premise. For you to now claim it is a conclusion in those definitions is complete nonsense.

If you get this notion, then you made the first step in order to understand my theory.

I actually do get your notions. They are exactly like what you had to say about a set being the union of its members. Remember when you were saying that? It was flat out wrong, wasn't it? You were confused, and you refused to accept there could possibly be any error in your thinking, until you were forced through a step-by-step analysis of it. Only then did you have an "Oops!" moment.

Well, this symmetry/asymmetry fixation you have is a lot more convoluted and covered in a lot more detail than the Axiom of Union was, so getting you back to reality is a bit more difficult. However, your own inability to define even you most basic terms for us should give you a clue that your thinking is muddled.

The notion thet you do not get (yet) is this:

Size (Cardinality) and Distinguishability (Order) are the results of Symmetry\Asymmetry complementation, and not vice versa.

Size and cardinality are not synonyms. Distinguishability does not require order. As for this "symmetry\asymmetry complementation", you'd need to provide some sort of definition or explanation of what you mean. Care to try, or do you prefer to continue to babble?