A collection of infinitely many distinct elements

[jsfisher]

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

[Edited slightly from the original for format.]

So, then any member of any set would be non-local, since it would also be a member of another, different set because of some other comprehension rule.

 

[ddt]

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

Bump.

 

[ddt]

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

[...]

Why are you still defending the TOUM paper? I thought the conclusion a couple of days ago was that it was passé and you'd start anew getting your ideas on - this time - solid footing?

 

[nathan]

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

Aside from jsfisher's point, this is a different situation to what you've previously been talking about. You defined two relationships between x and A -- in and out. And went on to define non-local in terms of whether x had both the in and out properties wrt A. At no time did you introduce a third object that affected the locality of x wrt A.

Here you're talking about relationships between x and A and between x and B. See the difference?

In your example, 2 is in A (the set of primes), because it is prime. It is not out of A -- to be outside of A it would have to be a non-prime.

 

[ddt]

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.

You didn't answer my question. Does your theory have two different kinds of things, like roads and cities?

In the case of ___|___

| can be refer to ___ xor ___

What do you mean? The sentence is grammatically wrong. I do not understand it.

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

OK. So we have (at least) two relations between things:
1) "in"
2) "local to"
Both are relations between two things. And for things x and y, we have the rule that
x in y => x local to y

Right?

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

I haven't seen yet any indication that this notion should be at the logic level.

 

[doronshadmi]

[Edited slightly from the original for format.]

So, then any member of any set would be non-local, since it would also be a member of another, different set because of some other comprehension rule.

No.

A member is an element of a single set, where the single set is based on some rule.

For example, if we are talking about the set of both prime and even numbers, then 2 is local w.r.t to this set.

In my previous example we have two deffient sets where 2 is a member of each set according to some partial property of 2.

In each single set 2 is local acording to this partial property that defines the set, but if we take 2 not according to any partial property of it, than 2 is non-local w.r.t A,B sets.

 

[doronshadmi]

In your example, 2 is in A (the set of primes), because it is prime. It is not out of A -- to be outside of A it would have to be a non-prime.

2 also has the property of rule n/2 (the rule of even numbers), which is not the rule (n/n) AND (n/1) (the rule of prime numbers).

From this point of view 2 is not exclusively in A or B, or in other words, it is non-local w.r.t A,B sets.

 

[ddt]

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)

That is a very roundabout - and wrong - way of saying that "2 is a member of A because it is only divisible by 1 and itself".

Stop that circumlocutious way of writing. It is abhorrent, and it only serves to hide what you mean. It may be en vogue on some managers' circles - but then, when the revolution comes, they'll be the first ones to be put against the wall (those of the Sirius Cybernetics Corporation the very first of course).

 

[jsfisher]

No.

A member is an element of a single set, where the single set is based on some rule.

For example, if we are talking about the set of both prime and even numbers, then 2 is local w.r.t to this set.

In my previous example we have two deffient sets where 2 is a member of each set according to some partial property of 2.

In each single set 2 is local acording to this partial property that defines the set, but if we take 2 not according to any partial property of it, than 2 is non-local w.r.t A,B sets.

(As an aside, I notice with some amusement that "w.r.t." has now crept into your lexicon.)

What I tried to express before was this: Consider any case where Q is a member of some set M and Q is also a member of some set N. (In your example Q is 2 and M and N are the sets of primes and even numbers, respectively.) By your latest definition, Q is non-local with respect to the sets M and N.

For any Q, any Q at all, I can always identify two different sets for which Q is a member, so everything is non-local with respect to some pair of sets.

Also, please note you have migrated from a definition of the non-local that was absolute to one where non-local is relative.

 

[nathan]

2 also has the property of rule n/2 (the rule of even numbers), which is not the rule (n/n) AND (n/1) (the rule of prime numbers).

From this point of view 2 is not exclusively in A or B, or in other words, it is non-local w.r.t A,B sets.

You're missing the point. You are now using a different definition of non-local -- one you have not previously used, nor do your axioms and definitions say anything about. Please provide the axiom of non-locality that mentions an object x and *two* sets (domains or whatever you're now calling the container objects).

 

[doronshadmi]

That is a very roundabout - and wrong - way of saying that "2 is a member of A because it is only divisible by 1 and itself".

Stop that circumlocutious way of writing. It is abhorrent, and it only serves to hide what you mean. It may be en vogue on some managers' circles - but then, when the revolution comes, they'll be the first ones to be put against the wall (those of the Sirius Cybernetics Corporation the very first of course).

You are right about the rule (n/n) AND (n/1).

In the case of A it has to be "only (n/n) AND (n/1)".

 

[ddt]

That is a very roundabout - and wrong - way of saying that "2 is a member of A because it is only divisible by 1 and itself".

Stop that circumlocutious way of writing. It is abhorrent, and it only serves to hide what you mean. It may be en vogue on some managers' circles - but then, when the revolution comes, they'll be the first ones to be put against the wall (those of the Sirius Cybernetics Corporation the very first of course).

You are right about the rule (n/n) AND (n/1).

In the case of A it has to be "only (n/n) AND (n/1)".

And you still have it wrong. n/n is a numerical expression, not a logical expression.

But why do you react to only two words in this post? There's a lot of other posts waiting for your replies. You seem not to be very fast in responding to questions about your ideas what non-locality means, or other concepts in your theories-to-be-formed.

 

[doronshadmi]

For any Q, any Q at all, I can always identify two different sets for which Q is a member, so everything is non-local with respect to some pair of sets.

Also, please note you have migrated from a definition of the non-local that was absolute to one where non-local is relative.

Relation is a general word for both "a member of" or "not a member of".

Here is how I thought about it according to the last part of our dialog:

1) There are two fundamental states, whether they are relative (w.r.t) to other thing(s) or not.

2) Locality is defined whenever a thing is simultaneously in exactly one and only one relation with a one thing.

3) Non-locality is defined whenever a thing is simultaneously in more than one relation with one or more things.

4) The universal formula of this notion is:

<name> _<name>_ <name>

where name can be any thinkable thing, <name> notates locality and _<name>_ notates non-locality.

5) The empty set is defined according things that are not its members. So the empty set's relation is based on simultaneously not be the domain of infinitely many things, which fits to (3).

 

[doronshadmi]

And you still have it wrong. n/n is a numerical expression, not a logical expression.

ddt,

At this stage do not try to understand my replies by using any formal thinking.

By not using any formal thinking I can write X AND Y in order to say that a prime number is any whole number that can be defined only by (n/n) AND (n/1) ratios.

I am doing it for purpose because I want to be sure that you do not use any formal thinking at this stage.

But why do you react to only two words in this post? There's a lot of other posts waiting for your replies. You seem not to be very fast in responding to questions about your ideas what non-locality means, or other concepts in your theories-to-be-formed.

What's the rush?

Let us do it step by step.

 

[jsfisher]

Relation is a general word for both "a member of" or "not a member of".

Here is how I thought about it according to the last part of our dialog:

1) There are two fundamental states, whether they are relative (w.r.t) to other thing(s) or not.

"W.r.t." doesn't mean what you think it means.

2) Locality is defined whenever a thing is simultaneously in exactly one and only one relation with a one thing.

3) Non-locality is defined whenever a thing is simultaneously in more than one relation with one or more things.

There is always a relation that can be defined between two things. Always. There is always a second relation that can be defined between two things. Always.

So, the only conclusion I can draw is that #2 never applies and #3 always applies, to everything is non-local with respect to everything else.

4) The universal formula of this notion is:

<name> _<name>_ <name>

where name can be any thinkable thing, <name> notates locality and _<name>_ notates non-locality.

Don't rush ahead to a notation when you don't have the concept ironed out.

 

[doronshadmi]

So, the only conclusion I can draw is that #2 never applies and #3 always applies, to everything is non-local with respect to everything else.

But in the case of 2 the general rule n/2 does not hold in the case of the set of prime numbers.

You cannot use a any property of a thing which is based on the fact that it is a member of some domain, for example:

2 is a member of the set of even numbers according to (the rule n/2) AND (2 is the smallest number of the set of even numbers).

The second one does not hold because it is based on the fact that 2 is a mamber to the set of even numbers.


2 is a member of the set of even numbers according to one and only one rule, which is n/2, and this relation is local.

If you can show that 2 is also a member of another set, then and only then 2 is considered as non-local.

 

[jsfisher]

But in the case of 2 the general rule n/2 does not hold in the case of the set of prime numbers.

You cannot use a any property of a thing which is based on the fact that it is a member of some domain, for example:

2 is a member of the set of even numbers according to (the rule n/2) AND (2 is the smallest number of the set of even numbers).

The second one does not hold because it is based on the fact that 2 is a mamber to the set of even numbers.


2 is a member of the set of even numbers according to one and only one rule, which is n/2, and this relation is local.

If you can show that 2 is also a member of another set, then and only then 2 is considered as non-local.

I have no idea what point you are trying to make here.

My point was simply this:

(1) You defined non-local as there being more than one relation between something and one or more somethings else.
(2) I claim everything has multiple relations with everything else.

Therefore, everything is non-local. Nothing is local...at least according to your definitions.

 

[nathan]

But in the case of 2 the general rule n/2 does not hold in the case of the set of prime numbers.

You cannot use a any property of a thing which is based on the fact that it is a member of some domain, for example:

But that's exactly what you are doing. You're saying 2 is non-local wrt primes because it's a member of the set of even numbers.

2 is a member of the set of even numbers according to (the rule n/2) AND (2 is the smallest number of the set of even numbers).

Incorrect. 2 is a member of the set of even numbers because it is exactly divisible by 2. The fact that it is the smallest even number is irrelevant here.

The second one does not hold because it is based on the fact that 2 is a mamber to the set of even numbers.

Second what?

2 is a member of the set of even numbers according to one and only one rule, which is n/2, and this relation is local.

Here you're contradicting your earlier assertion about its property of being the smallest even number as relevant. Now we're back to a single set, you're saying it's local. So why isn't 2 local wrt the set of primes? There's only one rule that determines primality.

If you can show that 2 is also a member of another set, then and only then 2 is considered as non-local.

sure. It's a member of {2}. So is it now non-local again?

 

[doronshadmi]

Bump.

You wrote at http://www.internationalskeptics.com/forums/showpost.php?p=3785550&postcount=896 :

However, the last two examples of the list above are cases where none of them is true. That's contradictory. So you have to decide whether you allow such a case or not.

Again this is an informal dialog.

My notion is this.

There is a domain.

There can be 3 options:

1) in the domain
2) out of the domain
3) not in and not out of the domain (exactly on the domain's border)


Local thing can be in a one and only one relation with some domain (where this relation is based on only one of the options).

Non-local thing can be in at least two relations with some domain (where these relations are based on at least two of the options above).

For example:

2 is both (in the domaim of the even numbers) AND (out of the domaim of the even numbers (it is also a member of the domain of prime numbers))

If 2 is considered only as the member of, for example, the domaim of the even numbers, then 2 is local.

In general both options are fundamantals of the mathematical science.

The general formula of this notion is:

<name> _<name>_ <name>

where name can be any thinkable thing, <name> notates locality and _<name>_ notates non-locality.

 

[doronshadmi]

But that's exactly what you are doing. You're saying 2 is non-local wrt primes because it's a member of the set of even numbers.


Incorrect. 2 is a member of the set of even numbers because it is exactly divisible by 2. The fact that it is the smallest even number is irrelevant here.



Second what?



Here you're contradicting your earlier assertion about its property of being the smallest even number as relevant. Now we're back to a single set, you're saying it's local. So why isn't 2 local wrt the set of primes? There's only one rule that determines primality.


sure. It's a member of {2}. So is it now non-local again?

You are right Nathan, I am still not clear. Maybe http://www.internationalskeptics.com/forums/showpost.php?p=3792924&postcount=959 is better.