A hidden assumption

[doronshadmi]

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.

Actually what happened is this. I used a symmetrical point of view on definitions that are based on an asymmetric point of view, for example:

From the symmetrical point of view of symmetry\asymmetry complementation a union is superposition of identities determined only by cardinality > 1.

Since I see both a set and a mustiest as particular cases of symmetry\asymmetry complementation, I made a mistake and wrongly used symmetrical union on asymmetrical collection.

You have no chance to do such a mistake because you get union only in terms on asymmetry, where multisets are avoided.

Be aware that a multiset is not based on a set, because a multiset is a collection of indistinguishable objects, where only its cardinality is known.

As for fundamentals, you are unable to understand that a pair and a set of two members are both collections of two distinct objects.

In particular you are unable to say why |{a,b}|=2

...
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.

You do not understand what I said. I spoke about the traditional asymmetrical-only point of view of x<y that thinks that it is a premise, but it does not aware of the fact that x<y is actually a conclusion based on symmetry\asymmetry complementation.

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

Your (current) inability to get superposition does not help much here.

Size and cardinality are not synonyms. Distinguishability does not require order.

About magnitude and size, please see http://en.wikipedia.org/wiki/Magnitude_(mathematics) , cardinality is also understood in terms of magnitude.

As for order, when generalized it enables Distinguishability.

In the case of "non-local", for example, you offered up a definition that could not possibly be satisfied.

Dear jsfisher,

From the invariance point of view there is nothing but sameness.

From the variance point of view there is nothing but difference.

If you try to get the invariance by using only the variance point of view you will get the "content" of { }.

If you try to get the variance by using only the invariance point of view you will get the "content" of { }.

In both cases there is nothing to be researched.

So the researchable cannot be but invariance\variance complementary point of view.

If you are able do that, then and only then you can understand my theory.

 

[jsfisher]

Actually what happened is this. I used a symmetrical point of view on definitions that are based on an asymmetric point of view

Nice attempt to back-pedal, but, no, that's not what happened. You said, quite clearly, a set was the union of its members. No symmetric, no asymmetric points of view, just a bogus statement.

...
In particular you are unable to say why |{a,b}|=2

Oh, sure I can. The a and b each stand for any object. The {a,b} symbolizes a set of the two elements, a and b. The |...| is a cardinality operator (when applied to sets). Therefore, |{a,b}| denotes the cardinality of a set of two elements, a and b, and that would, of course, be 2.

You do not understand what I said. I spoke about the traditional asymmetrical-only point of view of x<y that thinks that it is a premise, but it does not aware of the fact that x<y is actually a conclusion based on symmetry\asymmetry complementation.

No, you do not understand what you said. You said, "Given X < Y..." thereby making it a premise. Don't take it completely out of context, put it somewhere else, then contort the meaning to make it sound like you have some superior knowledge. In the way you presented it, it was a premise, not a conclusion, and any attempt to construe otherwise makes you out as one confused individual.

Your (current) inability to get superposition does not help much here.

Your inability to communicate does not help at all. Language, used properly, is powerful stuff. Just stringing together words with unknown meanings only toggles the discussion with disaligned Hamiltonian restitution.

About magnitude and size, please see http://en.wikipedia.org/wiki/Magnitude_(mathematics) , cardinality is also understood in terms of magnitude.

As for order, when generalized it enables Distinguishability.

Poor reading comprehension again? I said, for example, that distinguishability does not require order, yet you have twisted it 180 degrees from that.

Dear jsfisher,
<snip meaningless stuff>

Care to define any of your terms, or will the gibberish continue?

 

[doronshadmi]

Nice attempt to back-pedal, but, no, that's not what happened. You said, quite clearly, a set was the union of its members. No symmetric, no asymmetric points of view, just a bogus statement.

And it was a mistake because I thought about a collection by looking at it from the symmetrical point of view. Again, you have an immunization against mistakes in this case, because all you can get is a collection of distinct members. You cannot grasp that given X<Y (which is a conclusion based on the asymmetric point of view of Symmetry\Asymmetry complementation) Z is both X and Y (which is a conclusion based on the symmetric point of view of Symmetry\Asymmetry complementation). Your inability to grasp Z as a superposition does not allow you to get non-locality.

Oh, sure I can. The a and b each stand for any object. The {a,b} symbolizes a set of the two elements, a and b. The |...| is a cardinality operator (when applied to sets). Therefore, |{a,b}| denotes the cardinality of a set of two elements, a and b, and that would, of course, be 2.

a and b? are you kidding? The {a,b} symbolizes a set of the two elements?
How do you know that without using the cardinality operator?
What hidden assumption you are using in order to say "a set of the two elements, a and b"? You are the one how wrote:

By the way, do you realize that much of Set Theory does not involve comparisons? Set membership, for example, doesn't involve comparisons.

This is a good example of your inability to understand that comparison stands in the basis of any axiomatic system including ZFC, because:

Axioms are mutually independent self-evident statements that their outcomes are consistent with each other.

Try to define the above without using any comparison.

Try to define anything that its cardinality > 1 without using any comparison.

So you cannot avoid comparison because you are using "a and b" in order to define cardinal 2.

No, you In the way you presented it, it was a premise, not a conclusion,

Given X<Y has nothing to do with premise here. It is a conclusion based on Symmetry\Asymmetry complementation if observed from its asymmetrical point of view. If you continue to force your traditional asymmetric point of view on my definitions, I promise you that all you will get is the "content" of { }.

Your inability to communicate does not help at all.

Your inability to get Symmetry\Asymmetry complementation does not help at all.

Poor reading comprehension again? I said, for example, that distinguishability does not require order, yet you have twisted it 180 degrees from that.

What are you talking about? I explicitly said that order is not a concept that is limited to quantity or size. It can be used as a general concept in order to distinguish between any given collection of at least two elements.

Care to define any of your terms, or will the gibberish continue?

Care to stop forcing you asymmetric-only point of view on my framework?

My mathematical framework is a complementation between VSL and ASL, where Mathematics of the past 2500 years is mostly based on ASL.

Here is a comparison between ASL and VSL, based on Dr. Linda Kreger Silverman's research (please look at the difference between what is called Visual Special Learning and Auditory Sequential Learning ( http://www.visualspatial.org/Articles/intro.pdf , http://www.visualspatial.org/Articles/idvsls.pdf ). ):

You are an ASL-only that has problems with parallel thinking but he is good at serial (step-by-step) thinking.

I am more VSL than ASL.

In my opinion Mathematics will be developed to ASL\VSL framework.

 

[jsfisher]

...gibberish snipped...

Let me know when you able to communicate with well-formed sentences using meaningful words and phrases.

 

[doronshadmi]

Let me know when you able to communicate with well-formed sentences using meaningful words and phrases.

Let me know when you grasp Symmetry\Asymmetry complementation.

 

[jsfisher]

Let me know when you grasp Symmetry\Asymmetry complementation.

You'd need to offer some sort of definition for it, first, since it is of your conception.

 

[jsfisher]

Tell you what...

Let me know when you grasp Symmetry\Asymmetry complementation.

You'd need to offer some sort of definition for it, first, since it is of your conception.

How about I provide some starter definitions, and you fill in the holes or correct them as necessary?


Asymmetry
The property of being not symmetric.

Symmetry
The property of being invariant with respect to some transformation. For example, in normal addition a+b = b+a, so addition is invariant/symmetric under commutation.

Symmetry/Asymmetry Complementation
A mathematical system based on the complementary notions of symmetry and asymmetry with respect to some transformation.


Is that even close to the sort of thing you mean?

Regardless, what is the underlying transformation for your invariant/variant?

 

[RussDill]

ok doronshadmi, there is a simple way to resolve this.

Cantor is wrong, yes?

You claim that the set of reals maps to the set of integers, yes?

Please provide us with a function that maps the set of reals to the set of integers.

 

[jsfisher]

ok doronshadmi, there is a simple way to resolve this.

Cantor is wrong, yes?

You claim that the set of integers maps to the set of real numbers, yes?

Please provide us with a function that maps the set of integers to the set of real numbers.

[pedantic=on]
You meant map the reals to the integers, didn't you. Mapping the integers to the reals is trivial: F = n.
[/pedantic]

 

[doronshadmi]

ok doronshadmi, there is a simple way to resolve this.

Cantor is wrong, yes?

You claim that the set of reals maps to the set of integers, yes?

Please provide us with a function that maps the set of reals to the set of integers.

I do not claim that |R|=|W|.

I claim that anything that is based on the Multitude is incomplete when compared to the non-local atom, which is the invariance itself.

The invariance itself is not and cannot directly understood in terms of a transformation like a+b=b+a because the invariance in its self-state is the total unity (sameness) where no distinct b or distinct a exist.

The invariance is not researchable as a self-state.

The complement of the invariance is the variance.

The variance self state is anything that is totally isolated. This isolation does not enable any distinction like a\b.

The variance is not researchable as a self-state.

The researchable cannot be but invariance\variance complementation.

This complementation enables at least two points of view of itself:

a) The invariant point of view, where the complementation is a collection of superposition of identities.

b) The variant point of view, where the complementation is a collection of distinct identities.

The cardinality of any non-empty collection (whether it is a superposition or not) is the result of invariance\variance complementation and no cardinality has the completeness of the invariance itself.

In that case nothing that is based on the Multitude can be considered as complete, and Contor's notation of a complete collection does not hold.

 

[doronshadmi]

The organic invareance\variance complementation

If you look at this diagram you will find that Fullness and Emptiness are generalized to Non-locality and Locality.

Non-locality is the invariance and Locality is the variance.

A researchable realm is the organic invariance\variance complementation.

 

[ArmillarySphere]

Utter and complete gibberish. A bunch of statements that assert but do not explain, and another post with a picture showing ... nothing at all, really.

If you want to redefine mathematics, try starting from the axioms, and then prove your statements step by step. Otherwise, it's just gibberish piled on gibberish.

 

[doronshadmi]

Utter and complete gibberish. A bunch of statements that assert but do not explain, and another post with a picture showing ... nothing at all, really.

If you want to redefine mathematics, try starting from the axioms, and then prove your statements step by step. Otherwise, it's just gibberish piled on gibberish.

A step-by-step way of thinking is nothing but a particular case of invariance\variance complementation.

A parallel way of thinking is nothing but a particular case of invariance\variance complementation.

Mathematics is not limited to any one of its particular cases, but it is their complementation.

For example, the natural numbers 1 to 5, looks like this when observed from both parallel(symmetric)\serial(asymmetric) points of view:

The invariance is represented by any outer arc, that it is invariant under any superposition\distinction complementation.

From superposition\distinction complementation any given complementation is both global and particular case of the mathematical space, as can be seen in this diagram:

In this example, the number system of natural numbers is represented as both global\local distinct building blocks, but this is nothing but a particular case of distinction.

The general picture is the simultaneity of any possible global\local complementation, where its cardinality does not get the completeness of the invariance itself.


Peano axioms or ZFC (where a collection is based only on distinct objects), are no more than the result of a serial (step-by-step) way of thinking (their results are marked by the orange rectangels of the first diagram).

Actually Mathematics of the past 2500 years is nothing but the result of a serial thinking.

It is about time that the serial paradigm will not be the only player of the mathematical science.

 

[doronshadmi]

What does it mean for a single object to be asymmetric? For example, in what way do you mean the number 8 is asymmetric?

8 is a representation of a cardinal or an ordinal.

In the case of a cardinal 8 is a multitude of superposition\distinction complementation.

In the case of an ordinal 8 is the 8th asymmetric object of an asymmetric object made of 8 distinct objects (a distinct multitude object).

See for example 6 as superposition\distinction complementation:

 

[jsfisher]

...gibberish...

You didn't address my question. Instead, you expanded your undefined terminology pool.

 

[ArmillarySphere]

It's like the Time Cube guy of mathematics...

"when cornered, invent a new term and spin off into a new dimension"

 

[doronshadmi]

You didn't address my question. Instead, you expanded your undefined terminology pool.

Yes I did.

a+b=b+a is not symmetry itself but the result of symmetry\asymmetry complementation.

Symmetry itself cannot be addressed by using the Multitude, because symmetry itself is total unity where no multitude exists.

If you try to understand Symmetry in terms of one of many things (cardinal=1) or many things (cardinal>1), then you are at symmetry\asymmetry complementation that is not symmetry itself.

If you try to understand Symmetry itself by using cardinal=0 as a complement of cardinal≠0, you are not at Symmetry itself because you are thinking in terms of Nothing\Something complementation.

But Symmetry itself is total unity and it is not researchable.

Asymmetry itself is total isolation and it is not researchable.

The researchable is Symmetry\Asymmetry complementation, but then any researchable thing is the complementation's result.

If you unaware of this notion you are unaware of the fact that any given premise is already a notion that is based on the researchable (which is the result of Symmetry\Asymmetry complementation).

In other words, since mathematics is a researchable thing, it must be aware of researchable\non-researchable complementation.

This awareness is not used as the basis of the mathematical science, and one of the wrong results is to understand the multitude (which is a complementation's result) in terms of the completeness of Symmetry at its self state (which is beyond the researchable and no researchable concept like cardinality has its complete self-state).

You have to be aware of the fact that our dialog is possible because of symmetry\asymmetry complementation, and it cannot be done at Symmetry or Asymmetry self-state.

So the fundamental question of a researchable abstract\non-abstract space is:

How the researchable is possible?

I much as I know, this question was not asked for the past 2500 years of mathematical research, and any axiom that I know is based on non-proved assumptions that are already the result of researchable\non-researchable complementation, without being aware of it.

Furthermore, those axioms are tuned to work only on the asymmetrical side of Symmetry\Asymmetry complementation because Mathematics of the past 2500 years is mostly the result of a step-by-step way of thinking, until this particular way of thinking and Mathematics are synonym.

By understanding this it must not be a surprise to be realized of the fact that Mathematics of the past 2500 years is mostly the result of a step-by-step serial thinkers, and the community known as mathematicians are nothing but a community of persons that are qualified in step-by-step way of thinking, which is nothing but the particular case of an asymmetric point of view of Symmetry\Asymmetry complementation.

In that case we must not be surprised to understand why so many people around the globe do not like Mathematics.

If carefully be examined we discover that most of them are not step-by-step thinkers that are actually forced to do mathematics in term of step-by-step thinkers.

There is no conspiracy of step-by-step of thinkers here but a simple (and devastating) ignorance of Researchable\Non-researchable complementation.

As much as I know my suggested Mathematical framework is the first theory that is aware of the Researchable\Non-researchable complementation.


I'll be more than glade to know that I am wrong about what I wrote above.

 

[doronshadmi]

You were doing quite well up until this last bit. The union of all members of a set X is not (in general) the set X.

For example, let X = { {A,B}, {A,C,D} }

The set X has two members. They are {A,B} and {A,C,D}.

The union of all members of set X = {A,B} union {A,C,D}.

{A,B} union {A,C,D} = {A,B,C,D}.

{ {A,B}, {A,C,D} } = set X and therefore set X ≠ {A,B,C,D}.

A set is not the union of all its members.

There is another way to understand the term "the union of all members of X is set X"

You wrote "let X = { {A,B}, {A,C,D} }"

Now you wrote "The union of all members of set X = {A,B} union {A,C,D}."

This is wrong because {A,B} union {A,C,D} is the union of all member of all members od set X.

The right one is:

{{ {A,B}, {A,C,D} } union {{ {A,B}, {A,C,D} } = {{ {A,B}, {A,C,D} }

( http://en.wikipedia.org/wiki/Union_(set_theory) )

Please ahow my mistake .

 

[PixyMisa]

Your slashes are backwards.

 

[ellindsey]

This is wrong because {A,B} union {A,C,D} is the union of all member of all members od set X.

No.

{A,B} U {A,C,D} is {A,B} U {A,C,D}
{A,B} U {A,C,D} is not { A U B U A U C U D }

{A,B} U {A,C,D} is {A,B,C,D}
{ A U B U A U C U D } is invalid unless A,B,C, and D are all sets themselves.

The right one is:

{{ {A,B}, {A,C,D} } union {{ {A,B}, {A,C,D} } = {{ {A,B}, {A,C,D} }

No.

If X = { {A,B}, {A,C,D} }, then what you have just written is X U X.
This isn't the union of all members of set X. This is the union of X with itself.
The union of all members of set X is still {A,B} U {A,C,D}