A collection of infinitely many distinct elements

[doronshadmi]

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?

Definition 5:
If (x € A) xor (x ~€ A) then x is local.

Let x be the empty set (notated as {}) and let A be the domain (notated by "{" "}")

if ({} € {})=({{}}) xor ({} ~€ {})=({}{}) then {} is local.

{} is not the empty set, but it is the non-local property of the concept of Set.

Definition 6:
If (x € A) and (x ~€ A) then x is local.

Let x be non-local(notated as ___) and let A be the domain (notated by "{" "}")

if (___ € {})=({_}_) and (___ ~€ {})=({_}_) then ___ is non-local.

 

[doronshadmi]

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

My apology jsfisher, at has to be ddt, sorry again.

 

[Reality Check]

Definition 5:
If (x € A) xor (x ~€ A) then x is local.

Let x be the empty set (notated as {}) and let A be the domain (notated by "{" "}")

if ({} € {})=({{}}) xor ({} ~€ {})=({}{}) then {} is local.

{} is not the empty set, but it is the non-local property of the concept of Set.

Definition 6:
If (x € A) and (x ~€ A) then x is local.

Let x be non-local(notated as ___) and let A be the domain (notated by "{" "}")

(___ € {})=({_}_) and (___ ~€ {})=({_}_) then ___ is non-local.

But A is an element not a domain. This is in TOUM.pdf.

 

[jsfisher]

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

No, {} represents a very specific set, the null set.

2) If it refers to itself, then what is called set is its own member, and we get a non-empty set

Except that the null set does not refer to itself. {{}} is not the null set referring to itself. It is the set with the null set as its only member.

And therein lies a difficulty with the non-local/local view for sets you have expressed so far. You have told us that {} has a property (non-locality) that the sole member of {{}} does not have. That means {}, the member of {{}}, is distinct and different from {}, a set all by itself.

...
3) Please be aware that if non-locality refers to itself...

We have no case where non-locality refers to itself.

 

[jsfisher]

Let x be the empty set (notated as {}) and let A be the domain (notated by "{" "}")

Oh, geez, you really need to stop using notations contrary to any standard usage.

Do not introduce a notation for domain that looks like notation for a set.

Moreover, you don't need the new notation at all. You already said "let A be the domain" -- so use A as your notation.

 

[doronshadmi]

But A is an element not a domain. This is in TOUM.pdf.

A is element and element can be set or urelement, where a urelement can be local or non-local.

Non-locality is also a proprty of a set and it is notated by the first (outer) "{" "}" of any set (empty or not).

If you have an idea of how to notate the non-local property of a set (and a set or a urelement are both domains) I be glad to know it.

For example:

Locality = {{}}

Non-locality = {{}}

 

[doronshadmi]

We have no case where non-locality refers to itself.

If the outer (first) "{" "}" is for non-locality, then {{}} is non-locality that refers to itself.

The result of the self reference is local (notated by the inner "{" "}") but non-locality is still there (notated by the outer "{" "}"):

          ____
         / __ |
        / /  ||
       / /   ||
      / /    ||
      ||  [B]{[/B]  {}  [B]}[/B]
      ||  |      |
       \\_/     /
        \______/

Please do not look at this diagram as some physical or geometrical form.

 

[69dodge]

... 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.

...

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.

Ok, I like roads and cities. They're nice and concrete. (I understand that you're just using them as an analogy for something else, but I don't know what, so for now let's just talk some more about them.)

Can a single road connect any number of cities together, or can it only connect exactly two cities?

Can a road connect other roads, or can it only connect cities?

Are there other kinds of things besides roads and cities?

 

[doronshadmi]

Ok, I like roads and cities. They're nice and concrete. (I understand that you're just using them as an analogy for something else, but I don't know what, so for now let's just talk some more about them.)

Can a single road connect any number of cities together, or can it only connect exactly two cities?

Can a road connect other roads, or can it only connect cities?

Are there other kinds of things besides roads and cities?

We are talking in an informal way (by using analogies etc.) in order to first get the notions:

If a city is represented by | and a road is represented by ____ , then a road simultaneously connects as many cities (even infinitely many) as you wish, for example:


…_|__|___|_|_...


No city can be connected by another city, for example:

... | | | | ...

Not city can connect two roads, because | can be refer simultaneously to one and only one thing
for example:

___|___

In this example | can be refer to | xor ___ xor ___

___ connects as many ____ that you wish, for example:

_\___/__

 

[ddt]

Doron, you didn't yet react to my post #896.

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.

I'm fine with the roads and cities example. But lose the words about sets, because they're plain wrong.

With "roads" and "cities", there are two different kind of things. Is this what you want? Or do you want one kind of things? Or even more kinds of things?

And the "non-local" of the road is because it is associated with two (or more) cities. Does the notion of non-local only pertain to a road in connection with cities, or do you also speak of non-local when it would travel through two counties?

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).

And now you speak of a non-local aspect of two sets at the same time.

You should really get clear how your notion of non-local works. Is it a relation between two things? What kind of things? And let's stick to the "roads and cities" example - if that fits to express your ideas, that is.

(By the way, the difference between a set that it is its own member and a set that it is not its own member,

A set can't be its own member. I've shown that a couple of posts ago.

 

[ddt]

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.

Let's disregard set theory and your (wrong) ideas about what those braces mean. You wanted to build your own theory, didn't you? Then no set theory until you've followed a course on that, and use definite other notation than set notation for your ideas.

 

[ddt]

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.

My question about your diagram was to get an idea why you call something a "one of many" thing.

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.

How does this translate to the roads and cities case?

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.

Now you're saying something different. Above you said that if you had a horizontal point of view, you couldn't see vertical things, now you say you then can't see a vertical point of view. How can I see a point of view?

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.

First more examples, then we try to get the logic.

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:

But "locality" is some relation in your theory, not in logic.

 

[jsfisher]

If the outer (first) "{" "}" is for non-locality, then {{}} is non-locality that refers to itself.

The result of the self reference is local (notated by the inner "{" "}") but non-locality is still there (notated by the outer "{" "}")

You have given the notation a life of its own. All this does it confuse things. The braces are not a real part of the set in any sense, and drawing lines between one pair of braces and another doesn't give them any special meaning.

The set whose only member is the null set is not a self-reference to the null set. Expressing that as words, as I just did, or as {{}} is not a self-reference to {}, doesn't change reality in any way.

 

[ddt]

We are talking in an informal way (by using analogies etc.) in order to first get the notions:

If a city is represented by | and a road is represented by ____ , then a road simultaneously connects as many cities (even infinitely many) as you wish, for example:


…_|__|___|_|_...


No city can be connected by another city, for example:

... | | | | ...

OK, clear. Let's continue with the roads-and-cities example.

So you have two different kind of things - roads and cities?

And a city is always "local", whereas a road that connects two (or more) cities is "non-local"?

What about the road "Market Square", inside a city. Is it local?

Not city can connect two roads, because | can be refer simultaneously to one and only one thing
for example:

___|___

In this example | can be refer to | xor ___ xor ___

But you have here a (black) city A in the middle, where a red road on the left goes to city B (not shown in your diagram), and a blue road to the right goes to city C (again not shown). Doesn't city A connect those two roads? Or does the analogy between your system and the cities-and-roads break here?

 

[doronshadmi]

You have given the notation a life of its own. All this does it confuse things. The braces are not a real part of the set in any sense, and drawing lines between one pair of braces and another doesn't give them any special meaning.

The set whose only member is the null set is not a self-reference to the null set. Expressing that as words, as I just did, or as {{}} is not a self-reference to {}, doesn't change reality in any way.

Dear jsfisher, we are talking at this stage only by analogies, please ignore for a moment the formal point of view of the notations that I use, and look at them in an open mind, in order to get the notions that I try to show.

Please look at the dirageam at http://www.internationalskeptics.com/forums/showpost.php?p=3789826&postcount=927 in order to get the notion of non-locality that is refers to itself, and how the result is local (where the non-local is not ignored).

 

[doronshadmi]

OK, clear. Let's continue with the roads-and-cities example.

So you have two different kind of things - roads and cities?

And a city is always "local", whereas a road that connects two (or more) cities is "non-local"?

What about the road "Market Square", inside a city. Is it local?




But you have here a (black) city A in the middle, where a red road on the left goes to city B (not shown in your diagram), and a blue road to the right goes to city C (again not shown). Doesn't city A connect those two roads? Or does the analogy between your system and the cities-and-roads break here?

Being local is the ability to simultaneously connect one and only one thing.

Being non-local is the ability to simultaneously connect one or more things.

A road represents non-locality.

A city represents locality.

In the case of ___|___

| can be refer to ___ xor ___

What about the road "Market Square", inside a city. Is it local?

Anything that is completely in another thing, cannot be but local w.r.t this another thing.

And now you speak of a non-local aspect of two sets at the same time.

Yes, number 2 is the non-local connection between A,B sets, that are considered as locals form number 2 point of view, in this case.

But "locality" is some relation in your theory, not in logic.

In My theory locality and non-locality are also the fundamentals of logic.

 

[jsfisher]

Dear jsfisher, we are talking at this stage only by analogies, please ignore for a moment the formal point of view of the notations that I use, and look at them in an open mind, in order to get the notions that I try to show.

No, at this stage we are talking at the concept level. Analogies help convey the concept. However, that isn't what you are doing. Instead, you are letting the notation drive the analogy rather than the other way around.

 

[Reality Check]

A is element and element can be set or urelement, where a urelement can be local or non-local.

Non-locality is also a proprty of a set and it is notated by the first (outer) "{" "}" of any set (empty or not).

If you have an idea of how to notate the non-local property of a set (and a set or a urelement are both domains) I be glad to know it.

For example:

Locality = {{}}

Non-locality = {{}}

So a domain is an element? Why not call A an element?

 

[Little 10 Toes]

doronshadmi: please correctly reference your quotes of ddt in message #936. The first quote does come from post #934, but the second quote is from #930, and you do not show the third quote is post #932. If you cannot edit your post, please acknowledge what you have done.

 

[doronshadmi]

Let us examine the case of number 2 as non-local:


Set A = The set of prime numbers.

In the case of A 2 is considered as the result of the rule (n/n) AND (n/1)


Set B = The set of even numbers.

In the case of A 2 is considered as the result of the rule n/2.


A rule is not B rule, and in each case and 2 is a member of A or a member of B according to a partial aspect of number 2.

In that case number 2 (that has both aspects simultaneously) is non-local w.r.t A and B , by the formula:

A _2_ B