A collection of infinitely many distinct elements

[ddt]

We have, let's say, at least two (there can be infinitely many) poles (as Apathia suggested) .

Now I say to myself:"How it is possible to define in the first place, that there is a thing that is a pole of another thing?

My informal research:

1) No pole is the cause of the other, otherwise there is a hierarchy of cause\result and no polarity can be found.

2) These poles cannot be considered as poles if not both of them are simultaneously considered.

3) There must be a point of view that enables this simultaneous consideration, and it is not any one of the poles, otherwise we do not have a simultaneous consideration of the poles.

4) If one of the poles is the point of view of the other, then a non-simultaneous consideration of the poles is used.

5) A non-simultaneous consideration is a serial consideration of the poles.

6) Let Serial be non-simultaneous consideration.

7) Let Parallel be simultaneous consideration.

8) The parallel point of view is not polar. In that case it is considered as the middle between the poles.

9) The middle is not any of the poles, but without it there cannot be a parallel point of view, and no pole can be found.

10) The middle is not "a one of many" thing (I call it non-local, symmetry, invariant).

11) Each pole is "a one of many" thing (I call it local, asymmetry, variant).

[...]

Let us stop here, check it and try to formalize it, before we continue.

I specifically cut off your list at this point, as it's there still about poles, and the rest goes off far too quick into depth. Stick to the poles first. List examples how it should work. Lots of them. Like you did with the stick. What are those poles? What is their function? Examples clarify that.

That might also be instructive to see if you really need another logic. Devising another logic is not for the faint of heart. It's really easier if you can stick with classical logic and build a theory on top of that. The sticks example showed, at least to me, the beginning of a theory that just builds on top of classical logic.

Each pole is "a one of many" thing (I call it local, asymmetry, variant)

And stick to one word for each notion - say, local/non-local, it's the one you've been using most often anyway. And what do you mean with "a one of many"? The phrase never jived with me.

 

[jsfisher]

I agree with ddt's comment. Let's not rush to the finish line. Spend some time exploring your first few notions. Can they be framed in a more rigorous model? Are they consistent? Do they imply things you hadn't considered?

One other thing, though:

6) Let Serial be non-simultaneous consideration.
7) Let Parallel be simultaneous consideration.
13) I call the middle Relator, function, mapping, connective.
14) I call a pole Related, object.

Resist the temptation to rename things. The new names tend to insulate the discussion from the original concept rather than enhance understanding.

 

[doronshadmi]

And what do you mean with "a one of many"? The phrase never jived with me.

An informal representation of non-local\local relation:

| = "a one of many" thing.

____ = not "a one of many" thing.


An example of the relation between | ___ :


_|__|__|_|_

As you can see no matter how many | there are (where each | is "a one of many" thing) , they cannot be _____ which is not "a one of many" thing.

When I thought about that in terms of Set, I have found that ____ is equivalent to the notion that is represented by the outer "{" "}", where the outer "{" "}" is the set's non-local property, that no amount of its members (where each member is equivalent to | ) can be.

My notion is that completeness is non-local by nature, that no amout of local things has.

The cardinality of {} (represented also by ___) is 0 (where {} is non-locality).

The cardinality of {{}} (represented also by _|_) is 1 (where the inner {} is local).


If these notions can be addressed by classical logic, that will be nice.

 

[Apathia]

No pole is the cause of the other, otherwise there is a hierarchy of cause\result and no polarity can be found.

You don't actually need to posit First Cause or First Causes. Simply let them cohere. It's not turtles all the way down but a self-coherent system. (You wanted organic didn't you?) Then what you have is the essential Taoist framework. Take it to Taoist enthusiast boards on the web. Call it Taoist Mathematics, if you like. I think you'll find very positive reception for a change.

 

[doronshadmi]

You don't actually need to posit First Cause or First Causes. Simply let them cohere. It's not turtles all the way down but a self-coherent system. (You wanted organic didn't you?) Then what you have is the essential Taoist framework. Take it to Taoist enthusiast boards on the web. Call it Taoist Mathematics, if you like. I think you'll find very positive reception for a change.

2) ... poles cannot be considered as poles if not both of them are simultaneously considered.

 

[ddt]

An informal representation of non-local\local relation:

| = "a one of many" thing.

____ = not "a one of many" thing.


An example of the relation between | ___ :


_|__|__|_|_

As you can see no matter how many | there are (where each | is "a one of many" thing) , they cannot be _____ which is not "a one of many" thing.

I'm not completely clear on that picture yet.
- Is the horizontal line one "not one of many" thing?
- What does it mean that the vertical lines touch/cross the horizontal line - is that some deliberate connection?

By contrast, I do "get" why you use the word "non-local". That is, saying both "x in A" and "x out A" hold is like saying "it's here and it's also there".

When I thought about that in terms of Set, I have found that ____ is equivalent to the notion that is represented by the outer "{" "}", where the outer "{" "}" is the set's non-local property, that no amount of its members (where each member is equivalent to | ) can be.

But the { } in set notation is just a notational device; it's in no way comparable to the members of the set.

My notion is that completeness is non-local by nature, that no amout of local things has.

The cardinality of {} (represented also by ___) is 0 (where {} is non-locality).

The cardinality of {{}} (represented also by _|_) is 1 (where the inner {} is local).

If these notions can be addressed by classical logic, that will be nice.

You can't say a set is represented by ____, as you're building something else than (classical) set theory. You may draw analogies, but that's it. Cardinality is out for the moment.

Is "completeness" fundamental to the idea of poles? If not, then we'll disregard it first.

Let's first get clear about the poles.

 

[Apathia]

2) ... poles cannot be considered as poles if not both of them are simultaneously considered.

Can't be considered at all unless in reference to each other.
2 is a necessary qualification.
Good going, Doron. This is a nice departure from your previous style of communicating that was getting nowhere.

 

[jsfisher]

The cardinality of {}...is 0 (where {} is non-locality).

The cardinality of {{}}...is 1 (where the inner {} is local).


If these notions can be addressed by classical logic, that will be nice.

An issue I see with this is you have two, distinguishable null sets, at least as presented. As itself, the null set is non-local, but as a member of another set, it become local.

I'd think this would wreck havoc on the notion of membership.

 

[doronshadmi]

An issue I see with this is you have two, distinguishable null sets, at least as presented. As itself, the null set is non-local, but as a member of another set, it become local.

I'd think this would wreck havoc on the notion of membership.

The outer "{" "}" represents a notion that can be found whether the set is empty or not.

This is the whole idea, when some notation is used, it is there because it represents some notion, so when you look at some notion that is represented as {{}}, then you look at it without ignoring any part of it (whole\part(s) approach is used here).

In that case {{}} is not another version of the empty set, but this is a version (out of infinitely many versions) of a non-empty set.

The empty set has exactly one version, represented by {}.

By using the whole\part(s) relation the outer "{" "}" always represents the non-local aspect of the concept of set, whether it is empty or not.

The notion is this:

1) there exists {}, which is the non-local aspect of what is called Set.

2) If it refers to itself, then what is called set is its own member, and we get a non-empty set, but also in this case the non-local aspect of set holds and notated by the outer "{" "}", otherwise {{}} does not hold (if the non-local aspect of the set is ignored as inseparable property of set) and all we can get is the empty set.

3) Please be aware that if non-locality refers to itself, then and only then locality (notated by, at least the inner "{" "}" of {{}} "gets on stage".

4) It does not mean that locality is based on non-locality, it only means that locality manifests itself at the presence of the non-local.

5) By (2) and (3) it does not mean that non-locality is based on locality, it only means that non-locality manifests itself at the presence of the local.

 

[doronshadmi]

ddt,

Let us use your example. You wrote:

"Another example of the theory might be the interpretation: "person x lives in country A" resp. "person x lives outside country A". For a person with two homes in different countries, both statements are true."

By this example, the person is the non-local and each address is the local.

Let us change your example without changing the notion.

Instead of country A , not country A and a person, let us think of city A , city B , road C.

Road C connects between city A , city B, or in other words, it is both in city A and in city B.

City A is not city B (and vice versa) and no one of them is road C.

If we translate this example to set, then the outer "{" "}" is road C and city A, city B are members of the set {A,B}.

In that case road C is the non-local aspect of {A,B} and city A , city B are the local aspect of {A,B}.

Please be aware of the fact that the existence of road C, city A, city B do not depend on each other, but they manifest themselves as non-local,local at the presence of each other.

As this is done, we know that road C is non-local and city A or city B are locals.


Now let's do it more abstract.

A = the set of even numbers.

B = the set of prime numbers.

In that case 2 is the non-local aspect of A,B sets, where A,B are the local aspect of this notion (the whole\parts notion).

(By the way, the difference between a set that it is its own member and a set that it is not its own member, is equivalent to the difference between the empty set and a non-empty set. Please see (2) at http://www.internationalskeptics.com/forums/showpost.php?p=3789199&postcount=909 ).

 

[Reality Check]

The outer "{" "}" represents a notion that can be found whether the set is empty or not.

This is the whole idea, when some notation is used, it is there because it represents some notion, so when you look at some notion that is represented as {{}}, then you look at it without ignoring any part of it (whole\part(s) approach is used here).

In that case {{}} is not another version of the empty set, but this is a version (out of infinitely many versions) of a non-empty set.

The empty set has exactly one version, represented by {}.
...snip...

What you stated was:

The cardinality of {}...is 0 (where {} is non-locality).
The cardinality of {{}}...is 1 (where the inner {} is local).

The first statement assigns {} the property of non-locality.
The second statement assigns {} the property of locality.


so either

• The {} in the second statement is not the {} in the second statement but there is only one empty set so this is not the case.
  Or
• The {} in the second statement differs from the {} in the first statement. But once again there is only one empty set (and the notation is the same).

Therefore {} cannot be the empty set. It must be 2 other things that you have defined as:
1. The empty set + non-locality when not a member of a set
2. The empty set + locality when a member of a set

You must use different symbols for these as they are not the empty set.


A Note on Notation
The standard interpretation of the notation "{" "}" used by every other mathematician in the world is:
The symbols { and } are the notation for "there exists a set that contains members that are the objects between the { and the }".

Thus the notation for a set that contains no members is: {}
And the notation for a set that contains the empty set is: {{}}

If you reinterpret standard notition for your own use then you must expect to be not understood.
Is this your intention?

 

[doronshadmi]

What you stated was:

The first statement assigns {} the property of non-locality.
The second statement assigns {} the property of locality.


so either

• The {} in the second statement is not the {} in the second statement but there is only one empty set so this is not the case.
  Or
• The {} in the second statement differs from the {} in the first statement. But once again there is only one empty set (and the notation is the same).

Therefore {} cannot be the empty set. It must be 2 other things that you have defined as:
1. The empty set + non-locality when not a member of a set
2. The empty set + locality when a member of a set

You must use different symbols for these as they are not the empty set.


A Note on Notation
The standard interpretation of the notation "{" "}" used by every other mathematician in the world is:
The symbols { and } are the notation for "there exists a set that contains members that are the objects between the { and the }".

Thus the notation for a set that contains no members is: {}
And the notation for a set that contains the empty set is: {{}}

If you reinterpret standard notition for your own use then you must expect to be not understood.
Is this your intention?

Let us examine the first "{" "}" that is used in any set's representation, whether it is empty or not.

I look at the first "{" "}" not as any special set, but as the representation of the non-local property of any given set, whether it is empty or not.

 

[Reality Check]

If you reinterpret standard notition for your own use then you must expect to be not understood.
Is this your intention?

 

[doronshadmi]

I'm not completely clear on that picture yet.
- Is the horizontal line one "not one of many" thing?
- What does it mean that the vertical lines touch/cross the horizontal line - is that some deliberate connection?

The horizontal (forget about the line) is non-local.

The vertical (forget about the line) is local.

We can look at the relation between the vertical and the horizontal from at least two different aspects:

If we look at this relation only form the vertical point of view, then the horizontal cannot be seen.

If we look at this relation only form the horizontal point of view, then the vertical cannot be seen.

In this case one point of view prevents the other point of view, and we are actually using a local point of view of both cases.

If we look at this relation from both horizontal and vertical points of view, then both vertical and horizontal points of view can be seen, and we are actually using a non-local point of view of both cases.

I have found that the non-local point of view can be represented by NXOR connective, where the local point of view can be represented as XOR connective.

You may say: But this is circular because you define things by using the things.

My answer is this:

We have this formula: <name> _<name>_ <name>

where _<name>_ notates non-locality, and <name> notates locality.

Now let us put names in this formula:


xor _xor_ nxor (in this case the name of the non-local is xor):

In this case one point of view prevents the other point of view, and we are actually using a local point of view of both cases.


xor _nxor_ nxor (in this case the name of the non-local is nxor):

In this case one point of view complements the other point of view and we are actually using a non-local point of view of both cases.


The name is not important, the formula is important.


In that case:

NXOR connective is _<name>_ of the formula: <name> _<name>_ <name>

XOR connective is <name> of the formula: <name> _<name>_ <name>

 

[doronshadmi]

If you reinterpret standard notition for your own use then you must expect to be not understood.
Is this your intention?

Yes, I am aware of it. But what can I do if I have another point of view of the same standard fundamental concepts?

If I use different notation, it will not change the fact that I am talking about the same fundamental concept, but from a new point of view.

 

[Reality Check]

If you use different notation then no one will be able to confuse your {}+locality and {}+non-locality with {}.

What fundamental concept are you talking about and what is the old point of view?

 

[doronshadmi]

If you use different notation then no one will be able to confuse your {}+locality and {}+non-locality with {}.

What fundamental concept are you talking about and what is the old point of view?

The cardinality of {}...is 0 (where {} is non-locality).
The cardinality of {{}}...is 1 (where the inner {} is local).

The first statement assigns {} the property of non-locality.
The second statement assigns {} the property of locality.

At both cases non-locality is assigned only to the first (outer) "{" "}".

 

[Reality Check]

Toum

The empty set itself must be local because of definition 5:
if (x € A) xor (x ~€ A) then x is local.

Which placeholder for an element (the definiton of A) is this true for? Or is there a hidden assumption of "for all A"?

The empty set in {{}} must be non-local through definition 6:
if (x € A) and (x ~€ A) then x is non-local.

I think that this means that the empty set is both local and non-local.
In that case what is the test that returns that it is non-local when it is included in a set?

 

[Reality Check]

At both cases non-locality is assigned only to the first (outer) "{" "}".

I should have said "the second statement assigns the inner {} (which is the empty set) the property of local.

Are all sets non-local or just ones containing the empty set?

 

[jsfisher]

I'm not completely clear on that picture yet.
- Is the horizontal line one "not one of many" thing?
- What does it mean that the vertical lines touch/cross the horizontal line - is that some deliberate connection?

Editing quotations, again, I see. Doron, I did not say that.