A collection of infinitely many distinct elements

[jsfisher]

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))

2 is "in" the domain of even numbers, and 2 is "in" the domain of prime numbers.

The latter does not take 2 "out" of the domain of even numbers. It is not both in and out of the domain of even numbers.

 

[doronshadmi]

2 is "in" the domain of even numbers, and 2 is "in" the domain of prime numbers.

The latter does not take 2 "out" of the domain of even numbers. It is not both in and out of the domain of even numbers.

Is the domain of even numbers is also the domain of prime numbers?

The answer is no.

In that case if 2 is considered only from the point of view of the domain of even numbers, then from this point of view, 2 as a member of the domain of prime numbers, is out of the domain of even numbers.

But from a point of view of 2, 2 is simultaneously a member (in) the domains of prime and even numbers.

If the domain is prime+even numbers, then 2 is local, form this point of view.

The idea is to examine a notion from different points of view.

If you do that, you will find that you use locality or non-locality.

If you disagree, than please show how locality or non-locality do not exist.

 

[jsfisher]

But from a point of view of 2, 2 is simultaneously a member (in) the domains of prime and even numbers.

Everything is simultaneously a member of multiple domains. For example,

[latex]$
\forall x : (x \in \{ x \}) \wedge (x \in \{ x, \{ x \} \})
$[/latex]

So, does that mean everything is non-local?

 

[ddt]

Again this is an informal dialog.

With the intention to mathematical precision. If you don't have that intention, we better stop now.

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)

Quoting your previous post:

Locality means: to be exactly in a one relation.

Non-locality means: to be in more than a one relation.

So now you obviously forget case 4:
4) both in and out of the domain.

Sigh.

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).

And these two relations are only 'in' and 'out', or can be any two relations? I asked this before and you didn't answer that. jsfisher has in the time between your post and this answer - and I think that's for the umpteenth time already - that the latter makes the notion of "local" or "non-local" quite senseless.

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.

And now you're applying the notion of "local" to an object and its membership in two - quite arbitrary - sets, instead of applying it to one object, one set and two predefined relations between objects and sets - as your previous examples seemed to indicate.

In general both options are fundamantals of the mathematical science.

They're not fundamental to mathematics. Math has done perfectly without them for 3000 years. It might be an interesting mathematical theory, but it has no impact whatsoever on logic.

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.

And that's a lousy notation - even without it yet being clear which of the three 'name's would be the thing that is (non)local and which would be the domain.

 

[ddt]

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

When you pretend to do something mathematical and fail at it abysmally again, then derision is your part.

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.

You're very good at not using formal thinking, yes.

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

What's the rush?

Not necessarily rush, but I'd like to have answers to the questions I ask. If you want to have a mathematical system, it has to be formalized sometime, so formal thinking is required. Why then shouldn't I employ formal thinking?

I note that as to now, you only reacted to one paragraph in my post #896, and that in a most confusing way that sets back analysis of what your idea of non-locality is again to point zero. You haven't reacted to the rest of it, neither in the affirmative, nor in the negative, although I explicitly wrote:

Right so far?

Since then, you've also written a bunch of other posts which make the whole thing even more confusing. This is one step forward, two steps back. You show with the various contradicting examples an utter inability at formalizing or abstracting things.

So, back to the roads-and-cities example and answer all questions that were raised by it.

 

[ddt]

Is the domain of even numbers is also the domain of prime numbers?

The answer is no.

And no-one said so.

In that case if 2 is considered only from the point of view of the domain of even numbers, then from this point of view, 2 as a member of the domain of prime numbers, is out of the domain of even numbers.

But from a point of view of 2, 2 is simultaneously a member (in) the domains of prime and even numbers.

If the domain is prime+even numbers, then 2 is local, form this point of view.

And now you're making one domain of "prime+even numbers"? I thought they were two domains.

Get your story straight.

The idea is to examine a notion from different points of view.

If you do that, you will find that you use locality or non-locality.

There are an infinite number of points of view. Never needed them. Why would I?

You're now using the idea of "locality" to an object and two arbitrary sets. Whenever I'm in the position that I have an object and happen to have two sets in my reasoning, should I think of locality?

And what about three sets? Or four? Etc.

If you disagree, than please show how locality or non-locality do not exist.

The onus is on you to show they do exist.

ETA: The story you presented in post #891 (and to which I reacted in post #896) seemed to be able to lead to something that might even have merit. However, your last posts lead to senseless definitions.

 

[nathan]

... sadly, this isn't clearer

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)

You're implicitly invoking something other than the domain, so it can be in one of the three states you've described. You've called these 'options' so I presume they are mutually exclusive.

#3 presumes the domain's border has a non-zero extent. If it had a zero extent, then *by definition* nothing can be exactly on the domain's border.

Also, you now seem to be saying nothing can be non-local, for nothing can be in multiple states. You're contradicting yourself again.

It's almost as if you're working towards flanges. These are where one object 'overlaps' another. One can describe X as flanging Y when the extent of X straddles the the boundary of Y. Fuzzy logic makes significant use of flanging.

 

[doronshadmi]

You all right,

I hardly try to communicate with you by using terms that you already understand, and then I fail to express my notions.

Let's start again:

There are 3 terms where no one of them is derived from the other:

1) In

2) Out

3) Middle (not In and not Out)

Locality: Simultaneously be one and only one term.

Non-locality: Simultaneously be more than a one term.

In that case the example of number 2 is like this (and it is different than what I said in the previous posts):

Number 2 cannot be simultaneously In and Out of a given domain.

For example:

Number 2 cannot be a member AND not a member of the set of even numbers.

In that case number 2 is a local thing (it is examined only by a one and only one term (In).

Number 2 is not a member of the empty set.

Number 2 cannot be a member AND not a member of the empty set.

In that case number 2 is a local thing (it is examined only by a one and only one term (Out).

Any relation of a one term with another term is non-local by nature, and it is not any of the related terms.

In that case we have at least Non-local\Local relation.


Let us stop here and examine the notions.

 

[ddt]

I hardly try to communicate with you by using terms that you already understand, and then I fail to express my notions.

Fixed it for you. You failed to answer the relevant questions raised; your use of "hardly" is therefore justified.

There are 3 terms where no one of them is derived from the other:

1) In

2) Out

3) Middle (not In and not Out)

They're relations between two things. Give examples of their use. Preferably with some example that is not from mathematics.

Why the third relation "Middle"??? You say yourself that Middle is "not In and not Out"; therefore, the equivalence

x Middle A <=> not (x In A) and not (x Out A)

holds and there is nothing fundamental about Middle.

Is this an affirmative answer to my earlier question whether this situation can exist?

Locality: Simultaneously be one and only one term.

Non-locality: Simultaneously be more than a one term.

In that case the example of number 2 is like this (and it is different than what I said in the previous posts):

Number 2 cannot be simultaneously In and Out of a given domain.

How do you use the word "local" or "non-local"? You've given mostly examples where a thing is local w.r.t. one other thing, but also where it's local w.r.t. two other things. You've managed to confuse to no end.

And your notion of "local" being simultaneously one term - does "term" here only apply to the three relations In, Out and Middle above? You failed to answer this question repeatedly. In general, you have refused to answer any question which tried to get precision into your statements.

As to the above: going with the above assumptions - "local" is used in the sense of "x is local w.r.t. A", and that the terms indeed are only Middle, In and Out - your system can easily be simplified. Note that Middle and In are mutually exclusive, and so are Middle and Out. Then it reduces to:

x is non-local w.r.t. A <=> x In A and x Out A

x is local w.r.t. A <=> not (x is local w.r.t. A)

What about examples - and please, no mathematical examples, as your primary quality in mathematics seems to be to torture and misrepresent it to no end.

Number 2 cannot be a member AND not a member of the set of even numbers.

In that case number 2 is a local thing (it is examined only by a one and only one term (In).

A doctor examines patients. Mathematical relations don't examine things.

ETA:

Any relation of a one term with another term is non-local by nature, and it is not any of the related terms.

In that case we have at least Non-local\Local relation.

What you write here is incomprehensible. Try again.

 

[nathan]

ddt, I suspect you mean:

x is non-local w.r.t. A <=> x In A and x Out A

x is local w.r.t. A <=> not (x is non-local w.r.t. A)

 

[doronshadmi]

Again, there are 3 fundamental terms:

1) In

2) Out

3) Middle


Locality: Simultaneously be in a relation by one and only one term, with a given domain.

Non-Locality: Simultaneously be in a relation by more than a one term, with a given domain.


Example:

A circle with r > 0 and < ∞ is examined by internal domain, external domain, border (which is not internal and not external).

Non-locality:

A line\circle relation can be: (in AND out), (in AND border), (out AND border).

In that case the line is non-local w.r.t the circle.



Locality:

A point\circle relation can be only: (in), (out), (border).

A point is local w.r.t the circle.

 

[nathan]

A line\circle relation can be: (in AND out), (in AND border), (out AND border).

Up to now your circle/line examples have all been planar. If so, how can a line be both in AND out, but not border? I don't see how it can not intersect the border.

 

[Little 10 Toes]

What if my line/circle relationship is that the line does not cross, does not intersect or does not become a tangent of the border of the circle? That will make my line local (since it only is in a relation with one term). My line is not in, it is not out, and it is not middle.

Perhaps it really is undefined since you only define in, out, middle with reguards to the circle.

 

[nathan]

A circle with r > 0 and < ∞ is examined by internal domain, external domain, border (which is not internal and not external).

Why are you giving the border a non-zero extent? Is it the fact that if you draw, with a pencil, a border, it has width? One can remove that from the discussion by simply saying the inside of the circle includes the border. So the plane is divided into exactly two regions -- inside and outside.

 

[jsfisher]

I note, too, we are back to the idea "in" and "out" and whatever meaning "at least partially in", etc. There is no requirement for "completely in".

 

[doronshadmi]

Up to now your circle/line examples have all been planar. If so, how can a line be both in AND out, but not border? I don't see how it can not intersect the border.

Being non-local means that it is undivided (an atom).

Without this atomic property the non-local cannot be both in AND out w.r.t a given donain.

In that case, the border is ignored if In AND out are considered.

 

[doronshadmi]

Why are you giving the border a non-zero extent? Is it the fact that if you draw, with a pencil, a border, it has width? One can remove that from the discussion by simply saying the inside of the circle includes the border. So the plane is divided into exactly two regions -- inside and outside.

The border is defined according to logic, which is: (not in AND not out), middle state.

The middle has exactly zero width.

 

[jsfisher]

The border is defined according to logic, which is: (not in AND not out), middle state.

The middle has exactly zero width.

Well, there is nothing about your example that requires it be a circle, its interior, and its exterior. Could just as easily be a red area, a blue area, and everything else. Also, there is no need for three relations of in, out, and middle. There is only one, let's use in; the other two can be derived from it.

In any case, there is a domain, A, with three distinct areas, R, B, and (A-R)-B. We also have three distinct "in" relations:

1. x in R
2. x in B
3. x in (A-R)-B


One last point: Since the in relation means at least partially in, it isn't going to translate well into set theory. Set membership is an all-or-nothing relationship

 

[nathan]

Being non-local means that it is undivided (an atom).

Without this atomic property the non-local cannot be both in AND out w.r.t a given donain.

In that case, the border is ignored if In AND out are considered.

This makes no sense at all. My understanding was that a line was non-local because it was both inside and outside the circle. but now you say that non-local means that it is undivided. Undivided by what? You only have two objects here -- the line and the circle, so the only thing that could be dividing the line would be the circle. And if the line is indeed both inside and outside the circle, any reasonable interpretation would conclude that the line is divided by the circle. It's divided into line segments, some segments reside entirely inside the circle and some line segments reside entirely outside. Those line segments make up the (original) extended line.

 

[nathan]

The border is defined according to logic, which is: (not in AND not out), middle state.

The middle has exactly zero width.

Isn't this just the law of excluded middle?