A hidden assumption

[jsfisher]

I do not think that a collection of distinct members exists without a collector.

The "Collector" (which is a better concept, in this case) enables the association between distinct elements.

All you are doing is substituting one word for another. Renaming things doesn't add anything new to the discussion.

...
As for a set, please define a set that its cardinality > 1 without any association between its members.

You haven't defined what you mean by 'association between its members', yet.

Let me address it this way, an association is like the memory of a set that enables to simultaneously collect distinct members. This memory is not one of the members that are collected but it is an underlying property that without it any element is totally separated from any other element

Gibberish.

(I have changed "member" to "element" because without this memory no two things can be considered as members of a one object, known as set).

Parenthetical gibberish.

...<string/bead analogy>...
By using Logic, I have found that the invariant property of Symmetry is the string-like element, where the variant property of Asymmetry is a bead-like element.

You have not yet shown us how you relate symmetry to set. Nor have you explained symmetry of what. Ditto for asymmetry. You cannot jump off into 2-valued logic discussions if you won't tell us what the values represent.

 

[jsfisher]

Doronshadmi, here's a general principle: if you can't produce any particular cases from your supposed "generalization", then the generalization is probably wrong.

On the contrary, any particular case that you wish can be produced by Association(placeholder_A,placeholder_B)

For example: let Association be +, let placeholder_A be 3, let placeholder_B be 5.

So +(3,5)=8

The embedded quotation should have been associated with ben_m, not me.

 

[doronshadmi]

What property does your new concept ("collector") have that our old concept ("set") does not have?

Since the collector is the non-local atom, then no collection of infinitely many local elements gets its magnitude.

As a result anything that is based on "infinitely many ..." is incomplete.

In other words, you have missed my previuos post.

OK, I think I finally understand.

Again you have missed my theory by trying to translate it to your familiar mathematical material.

Please look at (1,1) and (1,-1).

Again you ignored the collector ( notated by "(" and ")" ) and looked only on the collected 1,1 or 1,-1.

( ) is non-locality where 1 or -1 are local.

No matter how many local objects you have ( ) is always beyond them.

Please do not look at that as "playing with notation" but try to get the abstract notion here.

 

[doronshadmi]

Except that this is not what you did with "associations" earlier---what the heck happened to "association" telling you whether or not a result "was valuable" or not?

Please see http://www.internationalskeptics.com/forums/showpost.php?p=3550350&postcount=113

As for Association(placeholder_A,placeholder_B), this is the universal principle that enables the reseachable, whether it is valuable or not.

 

[jsfisher]

What property does your new concept ("collector") have that our old concept ("set") does not have?

Since the collector is the non-local atom, then no collection of infinitely many local elements gets its magnitude.

As a result anything that is based on "infinitely many ..." is incomplete.

You still haven't defined your terms.

You seem to be trying unify the concepts of set and element of set into a single concept. I don't know why you'd do this. Moreover, it does not interfere with the notion of cardinality (which is where you want to end up).

Let's do that, though. So I don't tread on your vocabulary, I'll call it a 'bundle'. A bundle consists of a set as the first element of the bundle and all the elements of the set as the remaining elements of the bundle.

What would that add to the discussion?

 

[doronshadmi]

jsfisher,Please reply to this part.

Furthermore, ZFC axioms cannot fully work if they are not collected. Please try to deduce anything from them by totally avoiding any association between them (after all each axiom is independent of any other axiom, so it can be done. The name of the collector in the case of axioms is known as "Consistency".)

If you do that you will find that ZFC is a very weak theory.

Maybe it will help you to understand my (non-local) notion.

 

[ben m]

Again you have missed my theory by trying to translate it to your familiar mathematical material.

Please look at (1,1) and (1,-1).

Again you ignored the collector ( notated by "(" and ")" ) and looked only on the collected 1,1 or 1,-1.

No, you *are* playing with notation. (1,1) is not a set of 1 and 1, but rather one of the four possible points in the D2 dihedral group. I have to label the points somehow, and (1,1) makes it easy to picture the geometric version of the D2 group. Would you prefer I called the points "cat", "dog", "cow", and "chicken"? No, I call them (1,1), (1,-1), (-1,-1), and (-1,1), the same way that everyone represents points in the x-y plane. There's no "collector" because they aren't sets or associations. They're just four objects with four names. You're going to have to get used to it.

You're objecting to "familiar mathematical material"? Were dihedral groups already familiar to you this morning? Remember how long math has been around, Doron---a lot of things that you work out laboriously are likely to turn out to be old hat. This happened to Ramanujan, actually. He didn't have access to books when he did his famous solo proofs in India. He wrote down many beautiful theorems; some were new but it turned out that many of them had been known in Europe for 100 years.. Despite being a genius, Ramanujan couldn't tell whether he was doing new work or redoing old work---are you, Doron, saying that you can tell? I didn't think so. Learn the familiar material. Learn how to compare your work to it.

 

[doronshadmi]

You still haven't defined your terms.

You seem to be trying unify the concepts of set and element of set into a single concept. I don't know why you'd do this. Moreover, it does not interfere with the notion of cardinality (which is where you want to end up).

Let's do that, though. So I don't tread on your vocabulary, I'll call it a 'bundle'. A bundle consists of a set as the first element of the bundle and all the elements of the set as the remaining elements of the bundle.

What would that add to the discussion?

The cardinality of a bundle is based on local elements.

Let X be a bundle and let x be an object that is related to it.

x is local if it cannot be simultaneously in more than one relation with X, for example:

x is in X , x is out X , x is not in AND not out X

Let A be x is in X

Let B be x is out X

Let C be x is not in AND not out X

The logical connective between A and B, A and C or B and C cannot be but XOR, so x cannot be simultaneously in more than a one relation with X.

x is non-local ( represented as ___ ) if it is related simultaneously to A and B, A and C or B and C , for example:

A AND B

A AND C

B AND C

In other words, if x can be related simultaneously to X by more than a one relation, then x is non-local, otherwise it is local.

No infinitely many local elements has the magnitude of the non-local element.

 

[jsfisher]

jsfisher,Please reply to this part.

Maybe it will help you to understand my (non-local) notion.

I cannot respond to your question. You continue to use terms you have not defined.

How about this, though. Can you provide any sort of concrete example of how a set might be constructed in your framework? I'll even go first with an example in naive set theory and another in ZFC. Ok?

Naive:
A set is a well-defined collection of distinct items according to some membership function. I'll invent 'whole numbers less than 5' as my membership function.

{1,2,3,4} is therefore a set under the naive set theory model.


ZFC
Axiom of empty set: {} is a set.
Axiom of paring: {} is a set, then so is {{}}
Axiom of paring: {} and {{}} are sets, then so is { {}. {{}} }

{ {}, {{}} } is therefore a set under the ZFC model.

 

[doronshadmi]

No, you *are* playing with notation. (1,1) is not a set of 1 and 1, but rather one of the four possible points in the D2 dihedral group. I have to label the points somehow, and (1,1) makes it easy to picture the geometric version of the D2 group. Would you prefer I called the points "cat", "dog", "cow", and "chicken"? No, I call them (1,1), (1,-1), (-1,-1), and (-1,1), the same way that everyone represents points in the x-y plane. There's no "collector" because they aren't sets or associations. They're just four objects with four names. You're going to have to get used to it.

You're objecting to "familiar mathematical material"? Were dihedral groups already familiar to you this morning? Remember how long math has been around, Doron---a lot of things that you work out laboriously are likely to turn out to be old hat. This happened to Ramanujan, actually. He didn't have access to books when he did his famous solo proofs in India. He wrote down many beautiful theorems; some were new but it turned out that many of them had been known in Europe for 100 years.. Despite being a genius, Ramanujan couldn't tell whether he was doing new work or redoing old work---are you, Doron, saying that you can tell? I didn't think so. Learn the familiar material. Learn how to compare your work to it.

Dear ben_m, thank you for your fine reply.

You still mix between particular results and my generalization.

My generalization has nothing to do with any particular result. It simply defines the must have term of any researchable thing, whether it is a set or not.

In order to get point (1,1) on a plane, you have no choice but to collect two values.

It cannot be done if ( ) (the collector) is avoided.

This is the beautiful thing about my generalization.

When you get it then any given particular mathematical branch is unified under a single principle that enables to translate notions between different mathematical branches.

This is my answer to Hilbert's organic paradigm of the mathematical science.

Quoting again Hilbert’s famous Paris 1900 lecture:

“…The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.”

 

[jsfisher]

The cardinality of a bundle is based on local elements.

Define 'local' in this context.

Let X be a bundle and let x be an object that is related to it.

Define 'related' in this context. It seems to have no meaning here.

x is local if it cannot be simultaneously in more than one relation with X, for example:

x is in X , x is out X , x is not in AND not out X

Ok, so you will be looking exclusively at the membership relation between x and X, is that right?

Is 'not in AND not out' permitted in your scheme? Why not 'in AND out' as well?

Let A be x is in X

Let B be x is out X

Let C be x is not in AND not out X

The logical connective between A and B, A and C or B and C cannot be but XOR, so x cannot be simultaneously in more than a one relation with X.

Define 'logical connective'.

x is non-local ( represented as ___ ) if it is related simultaneously to A and B, A and C or B and C , for example:

A AND B

A AND C

B AND C

There's that 'related' word again. What does it mean here?

Doesn't "A AND C" mean "x is in X AND (x is not in AND not out X)" which reduces to "x not out X"

In other words, if x can be related simultaneously to X by more than a one relation, then x is non-local, otherwise it is local.

No infinitely many local elements has the magnitude of the non-local element.

This final statement of yours doesn't follow from anything else.

 

[doronshadmi]

I cannot respond to your question. You continue to use terms you have not defined.

How about this, though. Can you provide any sort of concrete example of how a set might be constructed in your framework? I'll even go first with an example in naive set theory and another in ZFC. Ok?

Naive:
A set is a well-defined collection of distinct items according to some membership function. I'll invent 'whole numbers less than 5' as my membership function.

{1,2,3,4} is therefore a set under the naive set theory model.


ZFC
Axiom of empty set: {} is a set.
Axiom of paring: {} is a set, then so is {{}}
Axiom of paring: {} and {{}} are sets, then so is { {}. {{}} }

{ {}, {{}} } is therefore a set under the ZFC model.

Ok. Let us use ZFC.

Axiom of empty set: {} is a set.

Let us say that you are unable to associate (you are eable to use only one axiom and cannot use anything that is related to another axiom) between the axiom of the empty set and the axiom of paring.

 

[doronshadmi]

Define 'local' in this context.



Define 'related' in this context. It seems to have no meaning here.



Ok, so you will be looking exclusively at the membership relation between x and X, is that right?

Is 'not in AND not out' permitted in your scheme? Why not 'in AND out' as well?



Define 'logical connective'.



There's that 'related' word again. What does it mean here?

Doesn't "A AND C" mean "x is in X AND (x is not in AND not out X)" which reduces to "x not out X"



This final statement of yours doesn't follow from anything else.

Let us try this version:

Let X be { } and let x be an object that is related to it.

x is local if it cannot be simultaneously in more than one relation with X, for example:

x is in {x} , x is out { }x , x is not in AND not out { x

Let A be {x}

Let B be { }x

Let C be { x

The logical connective between A and B, A and C or B and C cannot be but XOR, so x cannot be simultaneously in more than a one relation with X ( notated by { } ).

x is non-local if it is related simultaneously to A and B, A and C or B and C , for example:

A AND B ( represented as {_}_ )

A AND C ( represented as {_ )

B AND C ( represented as { _ )

In other words, if x can be related simultaneously to X ( represented as
{ } ) by more than a one relation, then x is non-local, otherwise it is local.

 

[jsfisher]

In order to get point (1,1) on a plane, you have no choice but to collect two values.

This seems to be the crux of at least some of the misunderstanding.

No act of "collection" is required for a point in a plane to exist. A point's Cartesian coordinates (or polar coordinates or whatever) can be used to identify a particular point, but that is all it is, and identification, and nothing more.

Cartesian coordinates do not collect points and sets do not collect elements.

 

[jsfisher]

Let us try this version:...

You didn't define your terms; you just shifted your notation a bit.

 

[jsfisher]

Ok. Let us use ZFC.

Axiom of empty set: {} is a set.

But ZFC is not your set theory framework. Yours has associations and relations and XORs and NXORs and symmetry and asymmetry. You've showed none of that.

ETA: The challenge was to take your undefined/ill-defined concepts and put them into something of a complete example. I was hoping your example would help communicate that which you seem unable to explain in words.

Let us say that you are unable to associate (you are eable to use only one axiom and cannot use anything that is related to another axiom) between the axiom of the empty set and the axiom of paring.

Define 'associate' as you use the term.

 

[doronshadmi]

This seems to be the crux of at least some of the misunderstanding.

No act of "collection" is required for a point in a plane to exist. A point's Cartesian coordinates (or polar coordinates or whatever) can be used to identify a particular point, but that is all it is, and identification, and nothing more.

Cartesian coordinates do not collect points and sets do not collect elements.

There is no "act" here. My theory is timeless.

About the case of a point on a plane.

The plane is not a collection of points, but it is a non-local ur-element (an atom) notated as ( ) where a point is some local ur-element that is related to it but it is not a part of it , exactly as two axioms are related to each other without defining each other.

 

[doronshadmi]

But ZFC is not your set theory framework. Yours has associations and relations and XORs and NXORs and symmetry and asymmetry. You've showed none of that.

ETA: The challenge was to take your undefined/ill-defined concepts and put them into something of a complete example. I was hoping your example would help communicate that which you seem unable to explain in words.



Define 'associate' as you use the term.

jsfisher,

If you will look for my theory under the traditional point of view, you will not get it.

I asked you to do a simple thought experiment, by trying to isolate each given ZFC axiom in such a way that avoids any "flow of information" between them.

If you do that, I think that you will do the first step in order to understand my theory.

 

[jsfisher]

There is no "act" here. My theory is timeless.

Ok, but you were the one that said, "You have no choice but to collect...." Maybe you are being colloquial, but you often describe things in terms of verbs.

About the case of a point on a plane.

The plane is not a collection of point, but it is a non-local ur-element (an atom) notated as ( ) where a point is some local ur-element that is related to it but it is not a part of it , exactly as two axioms are related to each other without defining each other.

Define 'non-local'. Pick a term other then 'ur-element' (since that was is already taken) then define it. Define 'local'. Define 'related'.

Also, are you really saying a point is related to the plane, but not part of it?

 

[jsfisher]

jsfisher,

If you will look for my theory under the traditional point of view, you will not get it.

I asked you to do a simple thought experiment, by trying to isolate each given ZFC axiom in such a way that avoids any "flow of information" between them.

If you do that, I think that you will do the first step in order to understand my theory.

You are using terminology that you think has meaning, but does not. Information does not flow between ZFC axioms. (By the way, did you notice that you used another verb form to describe something?) The axioms simply exist. { {}, {{}} } is a set withing the ZFC framework simply because it is consistent with the axioms, not because of any flow of information.

The best way for you to get others to understand your theory is for you to describe it using defined terms. You continue to not define things.