Deeper than primes

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doronshadmi said:
If [3,5) is a collection of ifinitely many R members, and since no R member is an immediate successor of any other R member, then no one of the inifintely many R members of the interval [3,5) has 5 as its immediated successor exactly as shown at http://www.internationalskeptics.com/forums/showpost.php?p=4789115&postcount=3486 .
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Nobody asked you for the immediate successors of any of the intervals members. You were asked for the successor of the interval itself.
 
jsfisher said:
Nobody asked you for the immediate successors of any of the intervals members. You were asked for the successor of the interval itself.
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Your notion that [3,5) is something that is totally disconnected from its elements, is nothing but an illusion, exactly as if you claim that your body is totally disconnected from its atoms.
 
doronshadmi said:
Your notion that [3,5) is something that is totally disconnected from its elements, is nothing but an illusion, exactly as if you claim that your body is totally disconnected from its atoms.
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Focus, Doron, focus, don't get all airy-fairy now. You're almost there.

What is the next number after [3, 5)?
 
doronshadmi said:
Your notion that [3,5) is something that is totally disconnected from its elements, is nothing but an illusion, exactly as if you claim that your body is totally disconnected from its atoms.
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You seem to want to totally ignore the interval in favor of its elements. I, on the other hand, ignore neither the interval nor its membership. The object under consideration is the interval, but its properties are shaped by its elements.

Here's what I (and everyone else here) mean by "interval A precedes interval B". What do you mean?

[latex]$$ (A \prec B) \, \Leftrightarrow \, (\forall x \, \forall y \, (x \in A \wedge y \in B) \Rightarrow (x \prec y)) $$[/latex]​
 
jsfisher said:
Here's what I (and everyone else here) mean by "interval A precedes interval B". What do you mean?
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So each element of B is a successor of A.

But no element of B is an immediate successor of A.

For example:

Interval A = [3,5]

Interval B = any real number > 5

No number of interval B is an immediate successor of A.
 
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zooterkin said:
Right. So, the next number after the interval is 5.
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This next number is not an immediate successor of [3,5) since we deal here with infinitely many R members.

5 is an immediate successor of [3,5) if we deal only with the whole numbers of [3,5) interval.
 
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jsfisher said:
Since you claim the next number isn't 5, it would need to be less than 5.
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Look jsfisher, even if the next number of the infinitely many R elements of [3,5) is 5, 5 is not an immediate successor of any one of them.

Since [3,5) does not exist independently of these R elements, we can clearly say that 5 is not an immediate successor of [3,5).
 
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doronshadmi said:
This next number is not an immediate successor of [3,5) since we deal here with infinitely many R members.

5 is an immediate successor of [3,5) if we deal only with the whole numbers of [3,5) interval.
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Pretty much what we do with considering 2 as the imideate predecessor of 1 in the integers, in other words the reason we can claim 2 as the imideate successor of 1 in the integers is specifically because the real numbers in the interval (1,2) (of the reals) is excluded form consideration. In fact if we include intervals (specifically of the reals) in the definition of immediate predecessor or immediate successor we have for any open and entirely bounded interval the immediate predecessor for that interval is the lower bound and the immediate successor is the upper bound. This gives us specific predecessors and successors in relation to a given bounded open interval in the reals. Basically the opposite of what I think you are trying to claim in your notions Doron. You previously referred to your immediate successor or predecessor as one of your non-local elements in relation to some local element. However the definition I just gave would make them local elements (finite value) to what you might consider a non-local element or a range of values (interval). Don’t worry standard math is flexible enough to accommodate you. We can define the immediate predecessor as an upper open interval bound by an upper given value and the immediate successor as a lower open interval lower bound by that same value. Now we have non-local predecessor and successor to a local element in the reals.
 
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The Man said:
Pretty much what we do with considering 2 as the imideate predecessor of 1 in the integers, in other words the reason we can claim 2 as the imideate successor of 1 in the integers is specifically because the real numbers in the interval (1,2) (of the reals) is excluded form consideration. In fact if we include intervals (specifically of the reals) in the definition of immediate predecessor or immediate successor we have for any open and entirely bounded interval the immediate predecessor for that interval is the lower bound and the immediate successor is the upper bound. This gives us specific predecessors and successors in relation to a given bounded open interval in the reals. Basically the opposite of what I think you are trying to claim in your notions Doron. You previously referred to your immediate successor or predecessor as one of your non-local elements in relation to some local element. However the definition I just gave would make them local elements (finite value) to what you might consider a non-local element or a range of values (interval). Don’t worry standard math is flexible enough to accommodate you. We can define the immediate predecessor as an upper open interval bound by an upper given value and the immediate successor as a lower open interval lower bound by that same value. Now we have non-local predecessor and successor to a local element in the reals.
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1) Here we are talking only about Standard Math, and in Standard Math 5 is not an immediate successor of [3,5) if [3,5) is an open interval of infinitely many R elements.

2) As for non-local, and element is non-local iff it is in at least two different relations of the form =,< or > w.r.t to another element

(http://www.geocities.com/complementarytheory/OMPT.pdf pages 22-24).
 
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The Man said:
We can define the immediate predecessor as an upper open interval bound by an upper given value and the immediate successor as a lower open interval lower bound by that same value. Now we have non-local predecessor and successor to a local element in the reals.
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Please provide some example, based on infinitely many real numbers.
 
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doronshadmi said:
1) Here we are talking only about Standard Math, and in Standard Math 5 is not an immediate successor of [3,5) if [3,5) is an open interval of infinitely many R elements.
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The definitions given do not violate any tenants of standard math.


doronshadmi said:
2) As for non-local, and element is non-local iff it is in at least two different relations of the form =,< or > w.r.t to another element

(http://www.geocities.com/complementarytheory/OMPT.pdf pages 22-24).
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Any interval of the reals contains some element or elements that other portions of that interval are greater then and less then thus any interval is non-local by your ascriptions, since it does represent a line segment. Are you now claiming that a line segment is not your non-local element?

doronshadmi said:
Please provide some example, based on infinitely many real numbers.
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Done, that definition you cited was specifically about the reals, if you are insisting on an example no problem.

In the reals the interval (-∞,1) is the immediate and non-local predecessor, while the interval (1, ∞) is the immediate non-local successor of the local and finite value 1.
 
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