Deeper than primes

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The Man said:
The definitions given do not violate any tenants of standard math.




Any interval of the reals contains some element or elements that other portions of that interval are greater then and less then thus and interval is non-local by your ascriptions, since it dose represent a line segment. Are you now claiming that a line segment is not your non-local element?



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 no-local predecessor, while the interval (1, ∞) is the immediate non-local successor of the local and finite value 1.
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Nonsense.

-∞ or ∞ are not real numbers (do not forget that we are talking here only about standard real analysis).
 
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zooterkin said:
What do you think [3, 5) means?
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Good question.

The answer is (according to Standard Math):

[3,5) is an ordered collection of infinitely many R members, where this collection does not have the largest element.

As a result 5 (that is greater than any element of this collection) is not an immediate successor of this collection exactly because
(5 is not an element of this collection) AND (this collection does not have the largest element).
 
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doronshadmi said:
Nonsense.

-∞ or ∞ are not real numbers (do not forget that we are talking here only about standard real analysis).
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Do not forget that (-∞, ∞) is an interval of all real numbers. Don't like them then replace them with some numbers more to your preferance.
 
doronshadmi said:
Good question.

The answer is (according to Standard Math):

[3,5) is an ordered collection of infinitely many R members, where this collection does not have the largest element.

As a result 5 (that is greater than any element of this collection) is not an immediate successor of this collection exactly because
(5 is not an element of this collection) AND (this collection does not have the largest element).
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5 is the immediate successor of this collection exactly becouse it is the upper boundary of that caollection AND "is not an element of this collection".
 
doronshadmi said:
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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There is no requirement any number in the interval B be an immediate successor of any number in interval A. There's only two conditions that need to be satisfied:

1. A < B, and
2. There is no C such that A < C < B.

For A = [3, 5] and B = (5, +inf), both conditions are met, so B is an immediate successor to A.
 
Doron,

Since we have told you what we mean by successor with respect to intervals and what we mean by immediate successor, and since you seem to be using those terms completely differently, I think is is well past time you tell us your usage.

What do you mean by successor with respect to real intervals?

What do you mean by immediate successor?

It would be great if you could provide this information in a precise, comprehensive way, for example, using first-order predicates.
 
doronshadmi said:
Nonsense.

-∞ or ∞ are not real numbers (do not forget that we are talking here only about standard real analysis).
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Oh, geez. You aren't even acquainted with standard interval notation. [K, ∞) is a perfectly fine interval of all the numbers <= K.
 
jsfisher said:
There is no requirement any number in the interval B be an immediate successor of any number in interval A. There's only two conditions that need to be satisfied:

1. A < B, and
2. There is no C such that A < C < B.

For A = [3, 5] and B = (5, +inf), both conditions are met, so B is an immediate successor to A.
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Pure nonsense.

< has a meaning only if it is used between things that can be compared with each other.

[3,5) cannot be compared with 5, unless the comparison is between 5 and some element of [3,5).

Do you want to play, so let us play this game:

[3,5) = [3,…[5,…

As we see [3,… is an open interval of real numbers that has no largest element.

Following [3,… there is [5,… that is another interval of real numbers that has no largest element.

No number of [5,… is an immediate successor of [3,… because [3,… does not have the largest element.

End of game.


EDIT:


jsfisher Please avoid your twisted games (now you enter to the game A and B intervals).

In http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 you clearly say:

"Y is in fact an immediate successor to [X,Y)".

Well, Y (where Y is a real number) is not an immediate successor to [X,Y) interval , as clearly shown in the example above.

(Also you can't use "<" relation between A and B intervals, because any use of "<" relation has a meaning only if it used between the elements of A and B intervals).
 
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jsfisher said:
Doron,

Since we have told you what we mean by successor with respect to intervals and what we mean by immediate successor, and since you seem to be using those terms completely differently, I think is is well past time you tell us your usage.

What do you mean by successor with respect to real intervals?

What do you mean by immediate successor?

It would be great if you could provide this information in a precise, comprehensive way, for example, using first-order predicates.
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Nothing is different here, immediate successor has a one and only one meaning in Standard Math which is:

If a and b are two whole numbers and b immediately follows a such that a<b, then b is called the immediate successor of a.

(you can't take a and b as two intervals, because any use of "<" relation has a meaning only if it used between the elements of a and b intervals).

Any other use of the term 'immediate successor' under Standard Math is a load of crap.

Please see http://www.internationalskeptics.com/forums/showpost.php?p=4790953&postcount=3530 for more details, about this subject.
 
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The Man said:
Do not forget that (-∞, ∞) is an interval of all real numbers. Don't like them then replace them with some numbers more to your preferance.
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So what if (-∞, ∞) is the interval of all real numbers?

We are taliking about the relation "<" that has a meaning only if it used between the numbers of this interval.
 
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doronshadmi said:
Prove it.
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Did you read the following?
jsfisher said:
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]​
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zooterkin said:
Did you read the following?
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You know what ?

If we wish to extend the use of "<" relation beyond numbers, then this relation must be used between things of the same type.

For example: if A and B are intervals, then B is indeed the immediate successor of A, as long as we ignore A and B elements.

But jsfisher claims that Y (which is a real number) is an immediate successor of [X,Y) interval.

By doing this, jsfisher mixes between [X,… (which is an interval that has no largest element) and Y, where Y is not another interval but it is the first element of [Y,… interval, where [Y,… interval is not Y element.

A = [X,…

B = [Y,…

B immediately follows A, and B is an immediate successor of A exactly because A and B are of the same type (both of them are intervals, in this case).

This is defiantly not the case in [3,5) < 5, where two different types are mixed with each other, and as a result "<" relation has no meaning in "[3,5) < 5" gibberish expression.


EDIT:


In http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 jsfisher clearly says:

"Y is in fact an immediate successor to [X,Y)".

By doing that he mixes between an interval and an element of an interval, and the result is an utter gibberish.
 
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doronshadmi said:
You know what ?

If we wish to extend the use "<" relation beyond numbers, then this relation must be used between things of the same type.

For example: if A and B are intervals, then B is indeed the immediate successor of A, as long as we ignore A and B elements.

But jsfisher claims that Y (which is a real number) is an immediate successor of [X,Y) interval.

By doing this, jsfisher mixes between [X,… (which is an interval that has no largest element) and Y, where Y is not another interval but it is the first element of [Y,… interval, where [Y,… interval is not Y element.

A = [X,…

B = [Y,…

B immediately follows A, and B is an immediate successor of A exactly because A and B are of the same type (both of them are intervals, in this case).

This is defiantly not the case in [3,5) < 5, where two different types are mixed with each other, and as a result "<" relation has no meaning in "[3,5) < 5" gibberish expression.


EDIT:


In http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 jsfisher clearly says:

"Y is in fact an immediate successor to [X,Y)".

By doing that he mixes between an interval and an element of an interval, and the result is an utter gibberish.
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So, did you read it but not understand it, or not read it?

ETA:
Do you think jsfisher is making this up as he goes along? There is only one person in this thread doing that.
 
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zooterkin said:
So, did you read it but not understand it, or not read it?

ETA:
Do you think jsfisher is making this up as he goes along? There is only one person in this thread doing that.
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No, you quated it without undertand it.

Now let us see how you explain it in details, since you are the one who used it in order to support your claims.

Please do it, the stage is yours, I am wating ...
 
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