Deeper than primes

[epix]

epix, please read about Universal quantification ( http://en.wikipedia.org/wiki/Universal_quantification ), before you air your view about this subject.

The term "universal quantifier" is a misnomer. The symbol cannot exist on its own and since there is no such thing as "for some x in P," for example, then the variability assumed in quantities doesn't materialize. Just find and copy/ paste a statement, such as "not forall x there exists..."

Objects belong to a particular set or they don't. If you define a set, you define its membership in such away that there doesn't exist at least one member that doesn't satisfy the definition of the membership. If you consider incrementing the size of the set by at least one member, you need to know if the property of such object satisfies the definition of the set. That means the subject of the possible increment either qualifies or not. So I treat A upside down as a qualifier. "Forall x in S" means that for every x that satisfies the definition of S, there exist... or whatever follows.

Tell you something: Once I stack with the general definition of a spacial point, applied it and saw sh-t. Then I redefined the point as a square whose side approaches zero. Having done so, that stuff I wanted to see showed up right before my eyes. If you parrot everything what the theoretician say, nothing gets done -- ever.

 

[doronshadmi]

The term "universal quantifier" is a misnomer. The symbol cannot exist on its own and since there is no such thing as "for some x in P," for example, then the variability assumed in quantities doesn't materialize. Just find and copy/ paste a statement, such as "not forall x there exists..."

Objects belong to a particular set or they don't. If you define a set, you define its membership in such away that there doesn't exist at least one member that doesn't satisfy the definition of the membership. If you consider incrementing the size of the set by at least one member, you need to know if the property of such object satisfies the definition of the set. That means the subject of the possible increment either qualifies or not. So I treat A upside down as a qualifier. "Forall x in S" means that for every x that satisfies the definition of S, there exist... or whatever follows.

Tell you something: Once I stack with the general definition of a spacial point, applied it and saw sh-t. Then I redefined the point as a square whose side approaches zero. Having done so, that stuff I wanted to see showed up right before my eyes. If you parrot everything what the theoretician say, nothing gets done -- ever.

Do you understand the co-existence of "the general definition" (non-locality, or the rule) with "a spacial point" (locality, or a particular expression)?

Do you understand the co-existence of "Forall x" (localities, or the particular expressions of some rule) "in S" (non-locality, or the rule) means that for every x (localities, or the particular expressions of some rule) that satisfies the definition of S (non-locality, or the rule),...?

Do you "Just find and copy/ paste a statement," as the only source of your notions?

Please to not ignore:

http://www.internationalskeptics.com/forums/showpost.php?p=7111406&postcount=15199

http://www.internationalskeptics.com/forums/showpost.php?p=7111434&postcount=15200

http://www.internationalskeptics.com/forums/showpost.php?p=7111142&postcount=15196

Please use your logic in order to deal with the following:

"The Ocean is still the Ocean even if it is both calm (at the bottom level) AND not calm (at the surface level)"

 

[epix]

Do you understand the co-existence of "the general definition" (non-locality, or the rule) with "a spacial point" (locality, or a particular expression)?

Do you understand the co-existence of "Forall x" (localities, or the particular expressions of some rule) "in S" (non-locality, or the rule) means that for every x (localities, or the particular expressions of some rule) that satisfies the definition of S (non-locality, or the rule),...?

Of course I don't and am not eager to learn. Those are your coined terms describing ghostly relationships that lead you toward making phantom claims, such as the one concerning the relation between a set and its power set, which made jsfisher to switch to bold font repeatedly. You can keep your translation of Forall x as "localities, or the particular expression of some rule" and apply it to your chamber mathematics, and I will go by 'Forall apples that are McIntosh there is basket B such as . . .'

Btw, why would be x a locality, which I think means that x has a predetermined position somewhere? If you organize integers as ... -2, -1, 0 ,1, 2, ... the way they use to sit on the x-axis, you never achieve any bijection between them and the infinite set of naturals, and your glasses will fall into obscurity.

 

[The Man]

The Man, you simply can't comprehend that "1-dimensional element" , "line", "unclosed interval", etc... is the same magnitude that enable smaller magnitudes to be distinguished from each other.

Doron “you simply can't comprehend that” your “magnitude” is a dimension.

You get only the level of the smaller magnitudes, without being aware of the magnitude that actually enables the distinction of their elements.

Again stop simply trying to posit aspects of your own failed reasoning onto others.

Once again you demonstrate your weak logic, which can't get the co-existence of the non-local and the local as fundamental agents of Unity, which enable the existence of finite or infinite complexity.

Doron you claim…

Remove the 1-dimensional element and there is no pair of distinct 0-dimensional elements.

Remove the pair of the distinct 0-dimensional elements, and the size of the 1-dimensional elements is unknown.

That’s not just “co-existence” as you would simply like to portray it, but specifically a mutual dependence.

Please read also http://www.internationalskeptics.com/forums/showpost.php?p=7111214&postcount=15197 .

I already have.

Without this fundamental understanding your science is no more than the surface level of the ocean of knowledge.

Doron you just calling it “fundamental understanding” still doesn’t make it science nor any kind of knowledge but just some fanciful musings you like to call “fundamental understanding”.

For example, by your surface-only reasoning you can't comprehend the following:

The Ocean is still the Ocean even if it is both calm (at the bottom level) AND not calm (at the surface level).

Again stop simply trying to posit aspects of your own failed reasoning onto others.

Also being “not calm (at the surface level)” or any level and “calm” at some other level certainly doesn’t disqualify “The Ocean” or an ocean from being “The Ocean” or an ocean. So your statement is a complete non-sequitur.

 

[The Man]

Here is some concrete example of The Man's "reasoning":

"the set of all natural numbers is closed under an integer operation of succession"

Actually it is just a consequence of that “all” qualifier when applied to the natural numbers however given your assertions, I wouldn’t expect you to understand that.

The proposition is wrong simply because any collection is actually open if succession is its inherent property, and this openness it true whether the succession is immediate, in the case of integers, or not immediate, in the case of rational or irrational numbers.

Fine then show what natural number is not a member of the set of all natural numbers. Otherwise stop simply babbling about what you would just like to be wrong.

So, next time if some professional mathematician asks you to show some object that is not a member of its set, please ask him\her to show you the complete infinite set of these members, without ignoring the following facts:

Doron you are claiming a succeeding natural number that is not in the set of all natural numbers and nothing you can ask of anyone will assuage your responsibility to demonstrate your claim.

1) One example at a time of some object that belongs to some infinite set, does not prove the completeness of that set.

Lists still aren’t sets Doron. The fact that no member is missing proves the completeness of any set. Once again you are deliberately misusing the word “completeness” to infer the inclusion of elements that you claim yourself not to be members of the given set.

2) Any infinite set exists exactly because succession (immediate or not) is its inherent property.

And it is an operation under which that the set is closed, otherwise a successor to a member of the set need not be a member of the set. We have been over this many times already Doron.

3) Succession among objects is possible only if there is an unclosed interval among them, otherwise there is not room for Succession, in the first place.

3 succeeds 2 in the closed interval [2,3]. So once again you are just demonstrably wrong.

4) Without the interval among objects, there are no distinct objects in the first place, and the only possible set is the empty set.

You’ve got that backwards, as usual and perhaps deliberately, without distinct objects there can be no intervals and {0} or {0,0,0,0…} would still be possible with “no distinct objects”. Perhaps you are simply confusing (again possibly deliberately) “no distinct objects” with just ‘no objects’.

 

[doronshadmi]

Of course I don't and am not eager to learn. Those are your coined terms describing ghostly relationships that lead you toward making phantom claims, such as the one concerning the relation between a set and its power set, which made jsfisher to switch to bold font repeatedly. You can keep your translation of Forall x as "localities, or the particular expression of some rule" and apply it to your chamber mathematics, and I will go by 'Forall apples that are McIntosh there is basket B such as . . .

In other words, you simply ignore http://www.internationalskeptics.com/forums/showpost.php?p=7111925&postcount=15202 and its links.

Btw, why would be x a locality, which I think means that x has a predetermined position somewhere? If you organize integers as ... -2, -1, 0 ,1, 2, ... the way they use to sit on the x-axis, you never achieve any bijection between them and the infinite set of naturals, and your glasses will fall into obscurity.

btw:

0 ↔ 1
-1 ↔ 2
1 ↔ 3
-2 ↔ 4
2 ↔ 5
-3 ↔ 6
3 ↔ 7
...

Locality is not only a location along the real line, but it is also interpreted as some particular expression (for example :some integer, or some natural number number), which obeys some general rule (non-locality) (for example: the definition of the set of integers, or the definition of the set of natural numbers).

 

[epix]

So, next time if some professional mathematician asks you to show some object that is not a member of its set, please ask him\her to show you the complete infinite set of these members, without ignoring the following facts:

1) One example at a time of some object that belongs to some infinite set, does not prove the completeness of that set.

x not in S doesn't relate to the concept of completness. Basket labeled Fruit is full (complete) even though there is one x=potato in it among all apples and oranges. x not in S doesn't mean that x is not a part of S, but that it doesn't belong to S.

If a mathematician asks you to find pi in the set of natural numbers, why would you insist on seeing "the whole set?" Only in the case the set was misdefined. Pi is not a member of the set of natural numbers and if the set was defined as N, then pi is not simply there.

I think that you initially claimed that forall x in N, there exists x, such as x not in N. The response to your claim was "show me a natural number that is not a part of N." Since x=x, the missing member is the one that is included in the set. Conversely any included member is the one that is not included in the set.


reality ain't no fiction
we prove it through contradiction
thin and rare don't describe dense
we prove it through common sense

 

[doronshadmi]

Doron “you simply can't comprehend that” your “magnitude” is a dimension.

You simply can't comprehend that magnitude is first of all a degree of existence from emptiness (the has no predecessor) to fullness (that has no successor).

Again stop simply trying to posit aspects of your own failed reasoning onto others.

This is exactly what you are doing all along this thread, by trying to posit aspects of your surface-only reasoning on others.

Doron you claim…

That’s not just “co-existence” as you would simply like to portray it, but specifically a mutual dependence.

Define it as you wish, it does not change the fact that a point (for example) can't be stretched into a line and still be considered as a point, and a line can't be reduced in to a point and still be considered as a line.

Doron you just calling it “fundamental understanding” still doesn’t make it science nor any kind of knowledge but just some fanciful musings you like to call “fundamental understanding”.

Your inability to get this fundamental understanding does not change the fact that this is a fundamental understating.

Again stop simply trying to posit aspects of your own failed reasoning onto others.

Again this is exactly what you are doing all along this thread, by trying to posit aspects of your surface-only reasoning on others.

Also being “not calm (at the surface level)” or any level and “calm” at some other level certainly doesn’t disqualify “The Ocean” or an ocean from being “The Ocean” or an ocean.

Great. By using this understating, try to grasp this:

The Line is still the Line even if it is both in AND not in a given domain (which is a property that no point has).

 

[epix]

Originally Posted by epix
Btw, why would be x a locality, which I think means that x has a predetermined position somewhere? If you organize integers as ... -2, -1, 0 ,1, 2, ... the way they use to sit on the x-axis, you never achieve any bijection between them and the infinite set of naturals, and your glasses will fall into obscurity.

Locality is not only a location along the real line, but it is also interpreted as some particular expression (for example :some integer, or some natural number number), which obeys some general rule (non-locality) (for example: the definition of the set of integers, or the definition of the set of natural numbers).

That's why I don't want any part of your constructs. I stick with

GENERAL is to BROAD as SPECIFIC is to NARROW

instead.

Btw, I don't see any equivalency between the meaning of the word "rule" and the meaning of "definition," as it transpires from your example. Doronetics is too advanced a concept for me, coz the innards of my head have been terminally stereotyped by counting potatoes in the kitchen. 1 potato, 2 potato, ...

 

[doronshadmi]

Actually it is just a consequence of that “all” qualifier when applied to the natural numbers however given your assertions, I wouldn’t expect you to understand that.

No, it is just a consequence of your misleading understanding of the real nature of infinite collections, which are opened under succession.

Fine then show what natural number is not a member of the set of all natural numbers.

Once again, no amount of particular examples of members that belong to some infinite set, define it as a complete set, because any infinite set is opened under succession.

Doron you are claiming a succeeding natural number that is not in the set of all natural numbers

No I do not claim such nonsense.

Once again, no amount of particular examples of members that belong to some infinite set, define it as a complete set, because any infinite set is opened under succession.

Lists still aren’t sets Doron. The fact that no member is missing proves the completeness of any set.

I am talking exactly about sets.

Once again, no amount of particular examples of members that belong to some infinite set, define it as a complete set, because any infinite set is opened under succession.

And it is an operation under which that the set is closed, otherwise a successor to a member of the set need not be a member of the set.

Opened does not mean that the result of the succession does not belong to its given set.

On the contrary, Opened means that succession is an inherent property of any given infinite set, which makes it incomplete.

3 succeeds 2 in the closed interval [2,3]. So once again you are just demonstrably wrong.

The Man, the interval is between 2 and 3.

Take it, and you don't have the distinct pair [2,3].

You’ve got that backwards, as usual and perhaps deliberately, without distinct objects there can be no intervals and {0} or {0,0,0,0…} would still be possible with “no distinct objects”. Perhaps you are simply confusing (again possibly deliberately) “no distinct objects” with just ‘no objects’.

I made a mistake here, the right one is:

"4) Without the interval among objects, there are no distinct objects in the first place, and the biggest possible set has no more than one member."

By the definition of Set {0} = {0,0,0,0…}.

By the definition of Multiset {0} ≠ {0,0,0,0…}, but then {0,0,0,0…} has an interval that is based on Redundancy among more than one object.

 

[doronshadmi]

I think that you initially claimed that forall x in N, there exists x, such as x not in N.

Wrong.

If you claim that some infinite set is incomplete, the a professional mathematician asks you to show some object that is not a member of its set.

 

[epix]

Define it as you wish, it does not change the fact that a point (for example) can't be stretched into a line and still be considered as a point, and a line can't be reduced in to a point and still be considered as a line.

Can you copy/paste a statement that claims such a thing? Before you do, be advised that "dragging a point" is not equivalent to "stretching a point." Dragging a point -- a 0-dim. object used to define location -- is equivalent to moving your finger whilst counting potatoes. Stretching a point means stretching your finger. That, of course, takes some time.

 

[doronshadmi]

That's why I don't want any part of your constructs. I stick with

GENERAL is to BROAD as SPECIFIC is to NARROW

instead.

There is no instead here. Since BROAD and NARROW are in co-existence, the BROAD is the rule and the NARROW is some expression of it.

Btw, I don't see any equivalency between the meaning of the word "rule" and the meaning of "definition," as it transpires from your example. Doronetics is too advanced a concept for me, coz the innards of my head have been terminally stereotyped by counting potatoes in the kitchen. 1 potato, 2 potato, ...

Yet the rule is "potato" and "1 potato, 2 potato, ..." are particular expressions of it.

 

[doronshadmi]

Can you copy/paste a statement that claims such a thing?

Again you demonstrate your inability to determine your notions, if they can't be copy/paste from already agreed notions.

New notions are not born by only copy/paste already agreed notions. Do you understand that simple notion?

Before you do, be advised that "dragging a point" is not equivalent to "stretching a point." Dragging a point -- a 0-dim.

How many points do you have if you dragging a point?

object used to define location -- is equivalent to moving your finger whilst counting potatoes.

In order to be able to do that, there must be an interval among the potatoes, otherwise, there is no more than one potato to be counted.

Stretching a point means stretching your finger.

No, the finger is the rule that is non-local with respect to the counted (local) objects, where no amount of the counted objects (particular expressions) is the counting finger (the rule).

 

[epix]

There is no instead here. Since BROAD and NARROW are in co-existence, the BROAD is the rule and the NARROW is some expression of it.

Why it isn't the other way around: NARROW is the rule and BROAD is some expression of it? Have your choice been guided by your own private definition of "co-existence?"

Yet the rule is "potato" and "1 potato, 2 potato, ..." are particular expressions of it.

So a potato is the rule and the multitude of potatoes is a particular expression of it. Are you sure you got that right? See, potatoes are vegetables available in the PRODUCE section of the market. Since PRODUCE is 86% similar word to PRODUCT, the notion of a potato being a rule is just one lazy step away and takes on itself the form of PRODUCT RULE:
http://www.math.hmc.edu/calculus/tutorials/prodrule/

You may have experienced a corrupted subliminal association leading toward your statement that potato is the rule.

 

[epix]

Again you demonstrate your inability to determine your notions, if they can't be copy/paste from already agreed notions.

New notions are not born by only copy/paste already agreed notions. Do you understand that simple notion?

I do understand that you conceive a patently false ideas and deliver them the way that it looks as if someone else thought up that nonsense. Then you pick up a fight with your own invention but make it look as if you emerged victorious over your opponent -- an opponent who doesn't exist.

How many points do you have if you dragging a point?

Let's see . . . if I drag a thing, then I drag one thing. That means I drag 1 point.

In order to be able to do that, there must be an interval among the potatoes, otherwise, there is no more than one potato to be counted.

Even if you encounter a couple of potatoes that touch each other resembling co-joined twins, you still can space them apart and start anew: 1 potato, 2 potato, . . .

No, the finger is the rule that is non-local with respect to the counted (local) objects, where no amount of the counted objects (particular expressions) is the counting finger (the rule).

I see it differently. The finger F changes its position X as you point and count (FX) the objects O:

F1 --> 1st O
F2 --> 2nd O
F3 --> 3rd O

and so on. Basically your moving finger creates a set of positions and bijects the objects to be counted.

 

[doronshadmi]

I do understand that you conceive a patently false ideas

Wrong, your ideas are false exactly because you do not understand of the result of the co-existence of locality with non-locality.

Let's see . . . if I drag a thing, then I drag one thing. That means I drag 1 point.

Do you get a line by dragging a one point?

Even if you encounter a couple of potatoes that touch each other resembling co-joined twins, you still can space them apart and start anew: 1 potato, 2 potato, . . .

Option 1: If they touch each other, they are not considered anymore as different objects (we get a one object to be counted).

Option 2: The potatoes are distinct exactly because they can't be reduced into a point, so in this case the interval is the impossibility of any potato to become a point.

The two options are in co-existence, as seen for example in:

I see it differently. The finger F changes its position X as you point and count (FX) the objects O:

F1 --> 1st O
F2 --> 2nd O
F3 --> 3rd O

and so on. Basically your moving finger creates a set of positions and bijects the objects to be counted.

In order to do that any counted object must be distinguished from the other objects, please show how it is possible without the co-existence of locality and non-locality.

 

[The Man]

You simply can't comprehend that magnitude is first of all a degree of existence from emptiness (the has no predecessor) to fullness (that has no successor).

Your preferred nonsense doesn’t make your “magnitude” any less of a dimension.

This is exactly what you are doing all along this thread, by trying to posit aspects of your surface-only reasoning on others.

This “surface-only reasoning” nonsense is just your own personal ascription Doron, stop simply trying to pawn it off onto others.

Define it as you wish, it does not change the fact that a point (for example) can't be stretched into a line and still be considered as a point, and a line can't be reduced in to a point and still be considered as a line.

Doron a point is defined as having no dimensions while a line is defined as having only one. So those definitions are exactly what means a zero dimensional point can’t be a one dimensional line. While your “Define it as you wish” stance seems to still deliberately ignore that fact.

Your inability to get this fundamental understanding does not change the fact that this is a fundamental understating.

Your inability to get that you again calling it “fundamental understanding” still doesn’t make it science nor any kind of knowledge but just some fanciful musings that you still like to call “fundamental understanding”.

Again this is exactly what you are doing all along this thread, by trying to posit aspects of your surface-only reasoning on others.

This “surface-only reasoning” nonsense is just your own personal ascription Doron, stop simply trying to pawn it off onto others.

Great. By using this understating, try to grasp this:

The Line is still the Line even if it is both in AND not in a given domain (which is a property that no point has).

Since you insist, but I expect what you assert to be "Great" migth just be grating for you. Just as your ‘some part calm and some other part not calm’ non-sequitur about an ocean has nothing to do with what defines an ocean as being an ocean. Your ‘in and not in a given domain’ assertion about your “line” has nothing to do with what defines a line as being a line. The obvious difference between the former and the later (which again is most likely deliberate) is that in the former you explicitly assert that one part of the ocean is calm while some other part is not. However in the latter you make no such explicit assertion. One could presume that such an assertion was implicit but that would be inconsistent with your assertion of your indivisible line. As a result you are simply, and as already stated many times before, using some criteria for determining that your indivisible line is “not in a given domain” that is something other than the negation of your criteria for determining that your indivisible line is “in a given domain”. Thus again and as usual you are simply being inconsistent with yourself. So which is it Doron, is your line in fact indivisible or is your criteria for determining that your indivisible line is “not in a given domain” just something other than the negation of your criteria for determining that your indivisible line is “in a given domain”. Either way, and as always, you simply remain the staunchest opponent of just your own notions.

 

[The Man]

No, it is just a consequence of your misleading understanding of the real nature of infinite collections, which are opened under succession.

Doron, changing your phrasing from “uncovered” to “opened under succession” doesn’t help you and only shows you don’t know what your talking about. All sets are complete. By definition they have only and all the members they are define to have.

Once again, no amount of particular examples of members that belong to some infinite set, define it as a complete set, because any infinite set is opened under succession.

Once again Doron it is the lack of missing members that define any set as complete and by definition a set is missing no members. As jsfisher has pointed out many times you continue to try to argue with just definitions and are apparently still deliberately confusing a set with a list.

No I do not claim such nonsense.

You claim all kinds of such nonsense. However if you do claim that any integer successor to any member of the set of all natural numbers is also a member of that set then that set is closed under an integer operation of succession.

Once again, no amount of particular examples of members that belong to some infinite set, define it as a complete set, because any infinite set is opened under succession.

Once again it is not about the “amount of particular examples of members” (that again would be a list) it is the lack of missing defined members (which again by definition there are none). Stop deliberately confusing a list with a set.

I am talking exactly about sets.

Then stop deliberately confusing them with lists. A set is complete by its definition, while a list may not be complete and in some cases can not be completed.

Once again, no amount of particular examples of members that belong to some infinite set, define it as a complete set, because any infinite set is opened under succession.

Once again it is not about the “amount of particular examples of members” (that again would be a list) it is the lack of missing defined members (which again by definition there are none). Stop deliberately confusing a list with a set.

Opened does not mean that the result of the succession does not belong to its given set.

“opened under succession” does. Do try to learn the meaning of words and phrases instead of just substituting your own demonstrable lack of any particular meaning.

On the contrary, Opened means that succession is an inherent property of any given infinite set, which makes it incomplete.

As already demonstrated succession is an aspect of the elements of some sets (specifically well ordered sets), even finite sets.

The Man, the interval is between 2 and 3.

Doron, the set from the closed interval [2,3] in the integers (or naturals) includes only “2 and 3” while the set from the open interval (2,3) in the integers (or naturals) is the empty set.

Take it, and you don't have the distinct pair [2,3].

“Take it”? To where? Or from where?. It isn’t part of that interval or the set from that interval. However, as already well established before you simply want to take elements that you assert yourself are not members of a set to claim that set as “incomplete” because it, by your own assertions, lacks those elements as members. As always you remain the staunchest opponent of just your own notions.

I made a mistake here, the right one is:

"4) Without the interval among objects, there are no distinct objects in the first place, and the biggest possible set has no more than one member."

By the definition of Set {0} = {0,0,0,0…}.

Your “definition” is just a false assertion of an equality. Your propensity for just replacing one mistake with some other still doesn’t help you, it never has and it never will.

By the definition of Multiset {0} ≠ {0,0,0,0…}, but then {0,0,0,0…} has an interval that is based on Redundancy among more than one object.

So we can add “interval” and “Multiset” to the list of words you evidently just don’t want to understand.

 

[doronshadmi]

Doron a point is defined as having no dimensions while a line is defined as having only one. So those definitions are exactly what means a zero dimensional point can’t be a one dimensional line. While your “Define it as you wish” stance seems to still deliberately ignore that fact.

You still do not get that your "mutually dependence" of points and lines do not change the fact that a point and a line can't be transformed into each other, and this is exactly the reason of why they are in co-existence, which is a fact that you simply can't comprehend.

Your inability to get that you again calling it “fundamental understanding” still doesn’t make it science nor any kind of knowledge but just some fanciful musings that you still like to call “fundamental understanding”.

In your realm there is no co-existence of points and lines, because you get a line as a collection of distinct points.

So once again, please demonstrate the existence of distinct points along a line, by ignoring the existence of the line between the points.

This “surface-only reasoning” nonsense is just your own personal ascription Doron, stop simply trying to pawn it off onto others.

Stop imply trying to pawn your points-only reasoning on others.

Since you insist, but I expect what you assert to be "Great" migth just be grating for you. Just as your ‘some part calm and some other part not calm’ non-sequitur about an ocean has nothing to do with what defines an ocean as being an ocean.

You still do not get the difference between a rule and some collection of its particular expressions.

In both cases, the rule is called Ocean or Line, where some particular expressions of the rule Ocean are both being calm AND not calm, exactly as some particular expressions of the rule Line is being both in AND not in a given domain.

You surface-serial-point-only reasoning simply can't comprehend it.

Your ‘in and not in a given domain’ assertion about your “line” has nothing to do with what defines a line as being a line. The obvious difference between the former and the later (which again is most likely deliberate) is that in the former you explicitly assert that one part of the ocean is calm while some other part is not. However in the latter you make no such explicit assertion. One could presume that such an assertion was implicit but that would be inconsistent with your assertion of your indivisible line. As a result you are simply, and as already stated many times before, using some criteria for determining that your indivisible line is “not in a given domain” that is something other than the negation of your criteria for determining that your indivisible line is “in a given domain”. Thus again and as usual you are simply being inconsistent with yourself. So which is it Doron, is your line in fact indivisible or is your criteria for determining that your indivisible line is “not in a given domain” just something other than the negation of your criteria for determining that your indivisible line is “in a given domain”. Either way, and as always, you simply remain the staunchest opponent of just your own notions.

Furthermore, your weak reasoning unable to comprehend the generalization of Ocean and Line analogies into Rule\Rule's expression(s).