Deeper than primes

[doronshadmi]

I have said it before in this thread and by pm to Moshe Klein.
I believe Projective Geometry's conception of infinity is much more user friendly than what OM proposes as it is both potential and absolute due to the principle of duality. The problems of infinite measure/numbers are avoided since it is a fundamental non-metric geometry.

Non-locality\Locality is not limited to metric space, as clearly seen in http://www.scribd.com/doc/16542245/OMPT pages 26-30.

OM is both absolute and potential as clearly seen in http://www.scribd.com/doc/16542245/OMPT pages 22-24.

In general, even if there is a transformation between lines and points, the different properties of lines and point are still considered.

You tell the mathematician that he limits by excluding what is already there.
But he could turn around and tell you, you limit by trying to cram in what's not there.

I understand. You want to liberate mathematics from what you see as cages.
Meanwhile the mathematicians aren't seeing them as cages but creations and temporary dwellings.

Consider for a moment that you may have created your own iron ball and chain.

I wish to be consistent about fundamental measurements of the mathematical science, by avoiding properties that are both used and ignored, which is a contradiction.

I gave the example of how Cardinality is the size of the number of members of a given set and also showed that the internal complexity of each member must not be ignored in order to get Cardinality's value, even if we measure only the first level of each member. My argument simply says that by understanding the existence of the internal structure of each member, we have the flexibility to extend Cardinality to include also the internal levels of each member, if we wish to do so.

The one who creates its own iron ball and chain is the one that insists to ignore the internal structure of levels of a given member, by claiming that this is the universal principle of Cardinality, which cannot be changed.

As apparently indicated by your representation a segment of that line does in fact “belong” to one domain and another segment of that line to the other “domain”. The third middle segment apparently “belongs” to nether domain. The point where the domains appear to intersect could belong to either or both depending on how those domains are defined. Your entire concept appears to based upon a simple lack of definition, but only in considerations that you specifically choose not to define.

The fact is simpler than what you think and it goes like this:

This line segment can also be taken as an atom (an atom has no sub-things), and as an atom it both belongs AND does not belong to the given domains.

Your reasoning of a line is strictly limited to a non-atomic view of that line, by divide it to sub-things, and by doing this you prevent from yourself the get the non-local property of the atomic view of that line segment.

Well this returns us to the already exacerbated discussion of closed and open intervals. That a closed interval includes the limits as members of that interval and an open one does not, in no way alters then fact that those are the de facto defined limits of that interval.

The same thing holds by using sub-things in order to define an atomic line segment, which is a contradiction.

You simply avoid any notion that is derived from the relation between a point and a line segment, such that a line segement is an atom.

If a line segment is taken as an atom, then it is obvious that it is only partially limited by locations along it, no matter what name each location has. The names of locations have no impact on the fact that an atomic line segment exists also beyond any given location.

As long as you get a line-segment only in terms of sub-things like points, you simply prevent from yourself to understand the line segment as a non-local atom.

As a result you are closed under collection of localities and can't understand the linkage between Non-locality and Locality.

Well that is part of the problem Apathia, in mathematics as well as physics ‘Non-local’ has specific definitions depending upon the application. Doron’s OM simply uses that term in an indefinite fashion to represent his concept of that “empty tablet upon which everything can be drawn and erased”.

What you call specific definitions is exactly some result of the non-local and local aspects of the atomic state, and indeed these results are drawn and erased according to the linkage between the non-local and local aspects of the atomic state.

I do not see any problem here for the existence and usefulness of your specific definitions.

On the contrary, OM provides the minimal terms that enable these specific definitions that depending upon the application.

By doing this we have the ability to define deeper relations between specific definitions, if we wish to do so.

As usual we have in "non-locality" a term of a number of different usages.
What I'm trying to express myself is non-locality as not a meta-locality (as if seems Doron does), but an abscence of any fixed or absolute locality.

Non-locality is not meta-locality, it is exactly non-local and it has a quality of existence that is different than Locality.

Take for example some law of nature. It is not mata-locality but it is a principle that is invariant of many expressions, where each expression is some local aspect of that low.

If that low is defined as an expression of a deeper low, that from the viewpoint of the non-local deeper low, this less deeper low can be considered as meta-locality w.r.t these expressions (it becomes an intermediate level between non-locality and locality).

 

[Apathia]

Non-locality is not meta-locality, it is exactly non-local and it has a quality of existence that is different than Locality.

Take for example some law of nature. It is not mata-locality but it is a principle that is invariant of many expressions, where each expression is some local aspect of that low.

If that low is defined as an expression of a deeper low, that from the viewpoint of the non-local deeper low, this less deeper low can be considered as meta-locality w.r.t these expressions (it becomes an intermediate level between non-locality and locality).

So instead The Local/Non-Local linkage provides an intermediate level between Locality and Non-Locality where in is the deeper complexity.
It comes to the same thing I'm talking about.

I can understand why you'd like to be able to shun the spatial metaphors inherent in speaking in terms of "Locality" and "Non-Locality" by speaking of "Relation" and "Element" instead.

Perhaps instead of a deeper level of complexity you'd want to speak of the internal relations within each number.
"Internal" "within"
Darn!
It's Philosophy's inevitable pathology.

Anyway, that mass of internal relations becomes a kind of offsite content in reserve for, as you say, those who wish to use it.

 

[doronshadmi]

So instead The Local/Non-Local linkage provides an intermediate level between Locality and Non-Locality where in is the deeper complexity.
It comes to the same thing I'm talking about.

I can understand why you'd like to be able to shun the spatial metaphors inherent in speaking in terms of "Locality" and "Non-Locality" by speaking of "Relation" and "Element" instead.

Perhaps instead of a deeper level of complexity you'd want to speak of the internal relations within each number.
"Internal" "within"
Darn!
It's Philosophy's inevitable pathology.

Anyway, that mass of internal relations becomes a kind of offsite content in reserve for, as you say, those who wish to use it.

Apathia,

I am talking about simple building-blocks that enables non-finite expression of complexity, were we as researches can defined any wished expression that is useful for our purpose.

The two building-blocks are the non-local and local aspects of a one atomic state and emptiness is evacuated space that enables the dance between locality and non-locality, in order to get the discovered\shaped complex realm.

If you ask me then complex systems like us only enable to discover or shape complex realms (abstract or not) but they do not have the power to create it from nothing (at least at this stage of evolution).

Anyway, we have to ask ourselves how complex things exist in the first place.

Locality and nothing is resulted by isolation, which is too weak for the minimal conditions that enable Complexity.

Non-locality and nothing is resulted by connectivity, which is too strong for the minimal conditions that enable Complexity.

The only alternative is Non-locality\Locality Linkage, where nothing is the evacuated space that enables their "dance".

All these tree fundamental conditions are aspects of a one atomic state, that appears as empty (nothing) stage (Non-locality) that has players (locals) on it, which appears as infinitely many complex forms that are based on the linkage between the local and the non-local aspects of the atomic state.

 

[The Man]

The fact is simpler than what you think and it goes like this:

This line segment can also be taken as an atom (an atom has no sub-things), and as an atom it both belongs AND does not belong to the given domains.

Your reasoning of a line is strictly limited to a non-atomic view of that line, by divide it to sub-things, and by doing this you prevent from yourself the get the non-local property of the atomic view of that line segment.

There in lies one of your general misconceptions and self-inconsistent notions, that a line segment is not composed of other line segments that “belong” to the various “domains”. By doing this you limit yourself from considering the aspects of the line segments that are specifically “local” to the given domains. That a given line or line segment can be considered to be comprised of sub-segments in no way restricts one from considering the given line or some given line segment as a whole. However claiming a line or line segment is an “atom” and can not be divided into sub-segments specifically precludes the consideration of, well, sub-segments. The limitations again are entirely yours.

The same thing holds by using sub-things in order to define an atomic line segment, which is a contradiction.

As the restrictions from a line segment being considered as an “atom” are yours, then so too is your own resulting contradiction.

You simply avoid any notion that is derived from the relation between a point and a line segment, such that a line segement is an atom.

No actually we specifically employ the self-consistent relationship of a line segment to the points that define that segment as a whole as well as those sub-segments in each domain, you are the only one avoiding that relationship and self-consistancy.

If a line segment is taken as an atom, then it is obvious that it is only partially limited by locations along it, no matter what name each location has. The names of locations have no impact on the fact that an atomic line segment exists also beyond any given location.

A line segment does not extend beyond the end points that define it. That is why it is, well, a line segment. In your “atomic” consideration a line is always whole and indivisible thus there would be no such thing as a line segment. This fact seems to continue to elude you.

As long as you get a line-segment only in terms of sub-things like points, you simply prevent from yourself to understand the line segment as a non-local atom.

Again you keep attacking a strawman. By your ascription of a line segment as an "atom" you simply limit yourself from considering the line segments in and defined by each of your given “domains”. Again the limitations are entirely yours and results from your “definition” of a line segment as an “atom” which is itself simply self-inconsistent.

As a result you are closed under collection of localities and can't understand the linkage between Non-locality and Locality.

A line segment is “closed under collection of localities”, that is what makes it specifically a line segment and a localized portion of a line.

What you call specific definitions is exactly some result of the non-local and local aspects of the atomic state, and indeed these results are drawn and erased according to the linkage between the non-local and local aspects of the atomic state.

Again going back to your ameba paradigm that simply attempts to consume everything into some amorphous blob.

I do not see any problem here for the existence and usefulness of your specific definitions.

Hence your problems, confusion, misinterpretations and misrepresentations. A line segment is by definition some part, or if you want a localized portion, of a line. If a line is “atomic” and having no parts (thus segments) as per your ascriptions then there can be no such thing as a line segment. This severely limits you from considering or distinguishing the relevant portions of some line. However considering a line segment in no way prevents one from considering a larger line segment of the line or that line itself in its entirety. The problems and limitations are specifically with your “definitions” like “atomic” line segments and your misuse, misinterpretations and misrepresentations of established concepts like cardinality.

On the contrary, OM provides the minimal terms that enable these specific definitions that depending upon the application.

By doing this we have the ability to define deeper relations between specific definitions, if we wish to do so.

It does no such thing, it is simply your self-inconsistent and extremely limiting speculations.

 

[Little 10 Toes]

Doronshadmi, I want to show you the difference between Cardinality (the definition that everyone here uses, mainly the number of members of a set) and your definition (the number of members in a given set, no matter how many sets are involved).

You like to harp about actors and an acting stage. I'm going to expand on that idea.

If you Doronshadmi, had access to a real theater in New York City and wanted to put on a show, you would think something along the lines of "How many people do I need to hire?" Your next train of thought might be "I need actors, managers, sound and light engineers, make-up artists, etc...". You start going over your budget for labor and your schedule, you discover that you need to hire one thousand people. Luckly, you know everyone will work for the same price. If you came up to me (a professional production company) and asked how many Labor Unions do you need to talk to, I would say "You need to talk to 1000 people to produce your show."

"But wait", you reply, "I know I'll need a thousand people to run my show, but I need to know how many labor unions I need to talk to. I don't know if the set designers will build things for my special effects, will the lighting people handle the sound as well, will the costume designers make the costumes, do the background dancers need to hired as dancers that act or actors that dance, do the screenwriters need to be told every stage direction or can they figure it out, and who is going to buy all the materials to build and make everything?"

Raising my voice at you I say, "One thousand people. Why can't you get it? What portion of my answer don't you understand? One thousand people."

"Look Little 10 Toes", you say interupting me. "I know I'll have more than just one person working the spotlights since I'll have two actors on stage at one time. I know that spotlight operator A works lights, and spotlight operator B works lights. Since they both do the same job, I know they're in the same labor union. Here's the idea that I have ..." You procede to describe some great and wonderful scene.

Glaring at you I reply, "Nice scene, but you will still to talk to 1000 people. Why don't you see that I'm right? This is the new way things happen."

*-*-*-*-*-*-*
Let's translate this little story into quasi-math. It's too early for me today to remember correct mathematical notations and spelling so I apologize in advace.

You figured out that the total number of people needed will be 1000. After speaking to other production companies, let's assume you will need 20 labor unions. One of those labor unions will have two people in it (The "spotlight union"). To make it easy for your accountant, you have given each labor union a single letter designation.

Set Show = (Set A, Set B, ... Set T)
The cardinality of set Show = 20 (the number of labor unions needed to produce the show).

Set Spotlight = (Jose, Tyrone)
The cardinality of set Spotlight = 2 (the number of people who operate the spotlights)

If you asked me (the new thinking producer) how many labor unions you would need, and I answer 1000, you'd think I was strange. My answer does not match your question. You are asking me how many sets does it take to complete set Show and not asking me how many people total it takes to produce the show.

Carinality is the number of members of a set.

Cardinality of set Show is 20, cardinality of set Spotlight is 2.

If you want a cardinality of 1000, then you need to have the union of the Labor Union sets.

 

[doronshadmi]

Doronshadmi, I want to show you the difference between Cardinality (the definition that everyone here uses, mainly the number of members of a set) and your definition (the number of members in a given set, no matter how many sets are involved).

You like to harp about actors and an acting stage. I'm going to expand on that idea.

If you Doronshadmi, had access to a real theater in New York City and wanted to put on a show, you would think something along the lines of "How many people do I need to hire?" Your next train of thought might be "I need actors, managers, sound and light engineers, make-up artists, etc...". You start going over your budget for labor and your schedule, you discover that you need to hire one thousand people. Luckly, you know everyone will work for the same price. If you came up to me (a professional production company) and asked how many Labor Unions do you need to talk to, I would say "You need to talk to 1000 people to produce your show."

"But wait", you reply, "I know I'll need a thousand people to run my show, but I need to know how many labor unions I need to talk to. I don't know if the set designers will build things for my special effects, will the lighting people handle the sound as well, will the costume designers make the costumes, do the background dancers need to hired as dancers that act or actors that dance, do the screenwriters need to be told every stage direction or can they figure it out, and who is going to buy all the materials to build and make everything?"

Raising my voice at you I say, "One thousand people. Why can't you get it? What portion of my answer don't you understand? One thousand people."

"Look Little 10 Toes", you say interupting me. "I know I'll have more than just one person working the spotlights since I'll have two actors on stage at one time. I know that spotlight operator A works lights, and spotlight operator B works lights. Since they both do the same job, I know they're in the same labor union. Here's the idea that I have ..." You procede to describe some great and wonderful scene.

Glaring at you I reply, "Nice scene, but you will still to talk to 1000 people. Why don't you see that I'm right? This is the new way things happen."

*-*-*-*-*-*-*
Let's translate this little story into quasi-math. It's too early for me today to remember correct mathematical notations and spelling so I apologize in advace.

You figured out that the total number of people needed will be 1000. After speaking to other production companies, let's assume you will need 20 labor unions. One of those labor unions will have two people in it (The "spotlight union"). To make it easy for your accountant, you have given each labor union a single letter designation.

Set Show = (Set A, Set B, ... Set T)
The cardinality of set Show = 20 (the number of labor unions needed to produce the show).

Set Spotlight = (Jose, Tyrone)
The cardinality of set Spotlight = 2 (the number of people who operate the spotlights)

If you asked me (the new thinking producer) how many labor unions you would need, and I answer 1000, you'd think I was strange. My answer does not match your question. You are asking me how many sets does it take to complete set Show and not asking me how many people total it takes to produce the show.

Carinality is the number of members of a set.

Cardinality of set Show is 20, cardinality of set Spotlight is 2.

If you want a cardinality of 1000, then you need to have the union of the Labor Union sets.

As you clearly said the number of functions (players in my analogy) that are needed to run the show is 1000.

Yet we get this number by distinguish between the functions, where each function has its one internal complexity that is a combination of parts+levels.

Now we can choose to take each function and look at is as a black-box (we do not care about its internal complexity).

As a result we get the traditional Cardinality, which consider the existence of each member as a black-box (its internal complexity is ignored).

By using OM's reasoning we say that if only the first level of the functions is considered we get a partial result of the players of some show.

There is no problem with this approach as long as it is not taken a universal principle that prevents any further extension that may not ignore the internal complexity of each function.

In other words, it is up to us, by OM's reasoning we can ignore (or not) the internal complexity of each function (player), and we may get a finer use of Cardinality, which is used according to our purpose.

From this wider view of Cardinality we are not unconditionally limited to the current traditional black-box view of Cardinality, because parts+levels of each player (function) are available for us according to our purpose.

 

[The Man]

As you clearly said the number of functions (players in my analogy) that are needed to run the show is 1000.

Yet we get this number by distinguish between the functions, where each function has its one internal complexity that is a combination of parts+levels.

Now we can choose to take each function and look at is as a black-box (we do not care about its internal complexity).

As a result we get the traditional Cardinality, which consider the existence of each member as a black-box (its internal complexity is ignored).

By using OM's reasoning we say that if only the first level of the functions is considered we get a partial result of the players of some show.

There is no problem with this approach as long as it is not taken a universal principle that prevents any further extension that may not ignore the internal complexity of each function.

Well since you’re the only one who apparently takes it that way, the problems again simply remain yours.

In other words, it is up to us, by OM's reasoning we can ignore (or not) the internal complexity of each function (player), and we may get a finer use of Cardinality, which is used according to our purpose.

From this wider view of Cardinality we are not unconditionally limited to the current traditional black-box view of Cardinality, because parts+levels of each player (function) are available for us according to our purpose.

Simply the results of the definitions of the set or sets being considered, as has been explained and exemplified to you multiple times already. It is neither a “function” nor a limitation of the concept of cardinality itself, but simply a function and limitation of the specific set or collection being considered. Again the only unconditional (as well as self –inconsistent) limitations are yours, that a line segment cannot itself be comprised of other line segments. OM is simply self-inconsistent and the ability to examine the actual cardinality of any given set is based solely on the definition of that set (what you might refer to as your “stage”). The particular interpretation you have given and resulting limitations of cardinality are yours and are simply based on your misunderstanding and misapplication of that concept. Thus OM is a self-inconsistent solution to a problem that only exists in your mind which results from your own generally inconsistent interpretation of cardinality and the definition of some given set. To try and put it more succinctly the flexibility you claim you are looking for and cannot find entirely within cardinality is not there because you are simply looking in the wrong place, that flexibility is in the definition of the set or sets being considered where it is most applicable, most useful and is given a clear and definitive nature by that or those definition(s) of the set or sets being considered.

 

[doronshadmi]

There in lies one of your general misconceptions and self-inconsistent notions, that a line segment is not composed of other line segments that “belong” to the various “domains”. By doing this you limit yourself from considering the aspects of the line segments that are specifically “local” to the given domains. That a given line or line segment can be considered to be comprised of sub-segments in no way restricts one from considering the given line or some given line segment as a whole. However claiming a line or line segment is an “atom” and can not be divided into sub-segments specifically precludes the consideration of, well, sub-segments. The limitations again are entirely yours.




As the restrictions from a line segment being considered as an “atom” are yours, then so too is your own resulting contradiction.



No actually we specifically employ the self-consistent relationship of a line segment to the points that define that segment as a whole as well as those sub-segments in each domain, you are the only one avoiding that relationship and self-consistancy.



A line segment does not extend beyond the end points that define it. That is why it is, well, a line segment. In your “atomic” consideration a line is always whole and indivisible thus there would be no such thing as a line segment. This fact seems to continue to elude you.



Again you keep attacking a strawman. By your ascription of a line segment as an "atom" you simply limit yourself from considering the line segments in and defined by each of your given “domains”. Again the limitations are entirely yours and results from your “definition” of a line segment as an “atom” which is itself simply self-inconsistent.




A line segment is “closed under collection of localities”, that is what makes it specifically a line segment and a localized portion of a line.





Again going back to your ameba paradigm that simply attempts to consume everything into some amorphous blob.



Hence your problems, confusion, misinterpretations and misrepresentations. A line segment is by definition some part, or if you want a localized portion, of a line. If a line is “atomic” and having no parts (thus segments) as per your ascriptions then there can be no such thing as a line segment. This severely limits you from considering or distinguishing the relevant portions of some line. However considering a line segment in no way prevents one from considering a larger line segment of the line or that line itself in its entirety. The problems and limitations are specifically with your “definitions” like “atomic” line segments and your misuse, misinterpretations and misrepresentations of established concepts like cardinality.



It does no such thing, it is simply your self-inconsistent and extremely limiting speculations.

As about "existence", you have no problem to use the quantifier "there exists", isn't it?


A line segment has 3 options, for example:

There exists domain [ ]

[__] means that the line segment completely belongs to the domain.

[ ]__ means that the line segment completely does not belong to the domain.

[_]_ means that the line segment belongs AND does not belong to the domain.

You choose to ignore the third (non-local) option, so the limitation is entirely yours.

On the contrary a point completely belongs to the domain [.] XOR [ ]. completely does not belong to the domain. Therefore a point cannot but a local element.

Furthermore, the line segment that completely belongs to [__] is still an atom, but its non-local property is not considered.

So is the case if the line segment completely does not belong to the domain.

Let us see again this example:

+------+
|     \|
|      \
|      |\
+------+-\----+
       |  \   |
       |      |
       |      |
       +------+

We can speak about 3 atomic line segments (as The Man does) and in this case the non-local property of a line segment is not considered.

But we also can speak about a one atomic line segment and in this case the non-local property of a line segment is considered.

More details can be seen in http://www.scribd.com/doc/16542245/OMPT pages 22-24.

 

[doronshadmi]

It is neither a “function” nor a limitation of the concept of cardinality itself, but simply a function and limitation of the specific set or collection being considered.

I use the word "function" as a part of the analogy about the functionality of the players on the stage as a part of the production.

In your production the functionality of the players on the stage is considered only as black-boxes, and this is your one and only one option of your show.

In my show, your show is a partial case of a wider show.

Also in your limited show the internal from of each player must be considered in order to get your black-boxes result.

 

[The Man]

As about "existence", you have no problem to use the quantifier "there exists", isn't it?

How does “there exists” quantify anything anyway? Until you can define your intend application of those terms they still remain without meaning in this discussion.

A line segment has 3 options, for example:

So you intend to demonstrate your lack of limitation and self-inconsistency by directly expressing your chosen limitations, based on your obviously self-inconsistent assertion that a line segment cannot itself be comprised of line segments. Brilliant plan did you think of that yourself?

There exists domain [ ]

What do you mean by “exists”? If you mean some domain is defined then simply say so.

[__] means that the line segment completely belongs to the domain.

[ ]__ means that the line segment completely does not belong to the domain.

[_]_ means that the line segment belongs AND does not belong to the domain.

You choose to ignore the third (non-local) option, so the limitation is entirely yours.

As I have already stated several times over on the course of this thread considering a line or line segment as being comprised of other line segments in no way precludes one from considering some given line or line segment as a whole, but you continue to ignore that. Thus the ignorance and misplaced assumption of ignorance remain simply yours.

On the contrary a point completely belongs to the domain [.] XOR [ ]. completely does not belong to the domain. Therefore a point cannot but a local element.

On the contrary to what, your own misguided assumption? Are you talking to voices in your head, you know that you do not need us or this thread if that is all you intend to do. That you simply choose not to consider the “local” line segments in each domain simply demonstrate you limiting yourself and your considerations.

Furthermore, the line segment that completely belongs to [__] is still an atom, but its non-local property is not considered.

So is the case if the line segment completely does not belong to the domain.

Oh so now your indivisible “atom” is devisable into other “atoms” how unsurprisingly self-contradictory.

Let us see again this example:

+------+
|     \|
|      \
|      |\
+------+-\----+
       |  \   |
       |      |
       |      |
       +------+

We can speak about 3 atomic line segments (as The Man does) and in this case the non-local property of a line segment is not considered.

No I just spoke of three line segments; there was nothing “atomic” about them (well at least not in your self-inconsistent abuse of the term). Additionally one could always define a smaller domain that some portion of the smaller line segment would not fall within.

But we also can speak about a one atomic line segment and in this case the non-local property of a line segment is considered.

More details can be seen in http://www.scribd.com/doc/16542245/OMPT pages 22-24.

Oh sure you can speak of an “atomic line segment” all you want, but you are just being self-inconsistent when you do speak of such. If a segment of a line can be considered then nothing precludes the consideration of just some particular segment of that segment which is itself just a segment of that same line as was the original segment being considered.

 

[The Man]

I use the word "function" as a part of the analogy about the functionality of the players on the stage as a part of the production.

In your production the functionality of the players on the stage is considered only as black-boxes, and this is your one and only one option of your show.

In my show, your show is a partial case of a wider show.

Also in your limited show the internal from of each player must be considered in order to get your black-boxes result.

Once again you simply misrepresent the notion of cardinality.

 

[Little 10 Toes]

As you clearly said the number of functions (players in my analogy) that are needed to run the show is 1000.

Yet we get this number by distinguish between the functions, where each function has its one internal complexity that is a combination of parts+levels.

Please note I never mentioned functions anywhere. You are misquoting me. I only say 1000 people and 20 Labor Unions so we don't start going into infinites (like infinite sets or elements).

Your next sentence does not make sense especially with the verb tense. There are no parts+levels. I have a total of 1000 people (elements) that I can divide any way that I desire. I have not formed any sets other than the set Show that has a cardinality of 20 and the set Spotlight has at least 2 members. There are no other sets that have members.

You don't have to use the other 18 other sets. You could have 998 elements in Actors and 2 in Spotlight. Since the other 18 sets have no elements, the cardinality of Show is 2.

Since you have not defined "functions" (well techincally you're using a term specifically for The Man and your line in/out/in of domains, but not for me yet) or "internal complexity" or what you mean by "parts+levels", the rest of your post does not make sense.

Again, I was using a very basic analogy to get my point across.

Please define the above terms and then I'll get back to you.

 

[doronshadmi]

No I just spoke of three line segments; there was nothing “atomic” about them

Look at this:
___ ___ ___ these are 3 segments, where each one of them can be non-local with respect to some domain.

___ ___ ___ are not ___________

Let us see again this example:

+------+
|     \|
|      \
|      |\
+------+-\----+
       |  \   |
       |      |
       |      |
       +------+

You use only ___ ___ ___ and ignore ___________

Furthermore, you do not understand that ___ ___ ___ ≠___________ is a fundamental fact, where in ___ ___ ___ we deal with 3 local states w.r.t the domain above, and in ___________ we deal with a non-local state w.r.t the domain above.

 

[doronshadmi]

Again, I was using a very basic analogy to get my point across.

Please define the above terms and then I'll get back to you.

Since you do not get that Cardinality is a partial case of the existence of complexity, let us use this example:

{} = ___

{{}} = __|__

           |_
{{{}}} = __|__

             |_|_
{{{{}}}} = __|__

                              |_    |_|_
{ {}, {{}}, {{{}}} } = __|____|_____|____

If we ignore the complexity of each member then |{ {}, {{}}, {{{}}} }| = 3

If we do not ignore the complexity of each member then |{ {}, {{}}, {{{}}} }| = 6

 

[The Man]

Look at this:
___ ___ ___ these are 3 segments, where each one of them can be non-local with respect to some domain.

So what, any point can also “be non-local with respect to some domain”, A point simply can not be divided into domains like a line or some multi-dimesional space.

___ ___ ___ are not ___________

All are simply different segments (or domains) of some line or lines

Let us see again this example:

+------+
|     \|
|      \
|      |\
+------+-\----+
       |  \   |
       |      |
       |      |
       +------+

You use only ___ ___ ___ and ignore ___________

Again as the restriction is simply yours, so too is the ignorance.

Furthermore, you do not understand that ___ ___ ___ ≠___________ is a fundamental fact, where in ___ ___ ___ we deal with 3 local states w.r.t the domain above, and in ___________ we deal with a non-local state w.r.t the domain above.

In all cases we are dealing with just segments or domains of a line, “a fundamental fact” you simply continue to ignore. I think part of your confusion stems from your use of two dimensional “domains” in the consideration of a one dimensional element (the line and any segments of that line). This seems to give you the impression that the line “belongs” to the domains, as you put it. If we instead simply consider the one dimensional line itself as the space (as opposed to imbedding the line in some multi dimensional space) then the only domains that can be established in that singular dimensional space are just the line segments themselves. So instead of the line “belonging” to the domains the domains or segments actually belong to or comprise the line or in other words the entire one dimensional space. Imbedding the line in some multi dimensional space only obscures that fact as the only aspects directly relevant to the line are comprised in its singular dimension. Since a point has no dimensions it has no domains or segments and is in fact indivisible as your referenced “atom”. However once a singular (like a line) or multi-dimensional space is considered that space can always be divided into domains which are simply segments in the case of a line.

 

[Little 10 Toes]

(my message snipped)
Since you have not defined "functions" (well techincally you're using a term specifically for The Man and your line in/out/in of domains, but not for me yet) or "internal complexity" or what you mean by "parts+levels", the rest of your post does not make sense.

Again, I was using a very basic analogy to get my point across.

Please define the above terms and then I'll get back to you.

Since you do not get that Cardinality is a partial case of the existence of complexity, let us use this example:
(snip)

Try again.

I did not ask for examples. I did not ask for any "partial case of the existence of complexity". I asked for you to define terms.

 

[doronshadmi]

So what, any point can also “be non-local with respect to some domain”,...

No, it can’t, because it strictly in exactly one and only one relation with respect to some domain, for example:

+------+
|     \|
|      \
|      |\
+------+-\----+
       |  \   |
       |      |
       |      |
       +------+

The point between the two domains can be common XOR non common w.r.t these domains.

If it is considered as common, we actually get a one domain where the point belongs only to that domin.

If the point is non-common, then it does not belong to any domain XOR it belongs to one and only one of the domains.

This is not the case with the line segment if its non-local property is considered w.r.t to the given domain, and as we see, it is both common AND non-common w.r.t the given domain.

 

[doronshadmi]

Try again.

I did not ask for examples. I did not ask for any "partial case of the existence of complexity". I asked for you to define terms.

No, you try again.

I gave you all you need in order to understand why the standard Cardinality is a partial case of Complexity.

Take it or leave it, it is up to you.

 

[doronshadmi]

If we instead simply consider the one dimensional line itself as the space (as opposed to imbedding the line in some multi dimensional space) then the only domains that can be established in that singular dimensional space are just the line segments themselves.

In a one dimensional space there exist lines both in local and non-local states w.r.t each other.

For example: on top of _________ there can be ___ ___ ___ where _________ is non-local w.r.t each ___ , and each ___ is local w.r.t _________

More details about one dimesional relations can be seen in http://www.scribd.com/doc/16542245/OMPT page 23.

These is true for any k>0 dimensional space.

This is not the case in a zero dimensional space, which is local only.

 

[The Man]

No, it can’t, because it strictly in exactly one and only one relation with respect to some domain, for example:

+------+
|     \|
|      \
|      |\
+------+-\----+
       |  \   |
       |      |
       |      |
       +------+

The point between the two domains can be common XOR non common w.r.t these domains.

If it is considered as common, we actually get a one domain where the point belongs only to that domin.

If the point is non-common, then it does not belong to any domain XOR it belongs to one and only one of the domains.

This is not the case with the line segment if its non-local property is considered w.r.t to the given domain, and as we see, it is both common AND non-common w.r.t the given domain.

Obviously you do not understand the meaning of the word “common”. A boundary between domains or sets is the common boundary of both those domains or sets. That it is not included in either of those sets , when both are open sets, simply means that the space as a whole is not connected. Also a singular domain has everything in common with that domain thus “non-common w.r.t the given domain” is simply nonsense. That multiple domains may have a point, points or a boundary in common does not make them a single domain. However it can make the space as a whole connected if it is not the union of disjoint open sets. In the one dimensional space of the line in your diagram the segments in the two given domains are disjoint even if they are represented as closed sets, thus that one dimensional space is not connected by the given domains relevant to that one dimensional space.

In a one dimensional space there exist lines both in local and non-local states w.r.t each other.

For example: on top of _________ there can be ___ ___ ___ where _________ is non-local w.r.t each ___ , and each ___ is local w.r.t _________

More details about one dimesional relations can be seen in http://www.scribd.com/doc/16542245/OMPT page 23.

These is true for any k>0 dimensional space.

This is not the case in a zero dimensional space, which is local only.

“On top of”? What is “on top of” a single dimensional space? Imposing an additional dimension “on top of” a singular dimensional space simply means that you are just not considering a singular dimensional space.