Deeper than primes

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doronshadmi said:
http://www.internationalskeptics.com/forums/showpost.php?p=4787704&postcount=3424


Care to share where Standard Math uses an expression like "[X,Y) < Y" ?
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You're the one who keeps using that expression.

Now, I'm going to type this slowly, so you can concentrate on the meaning:

[X, Y) is defined as the interval that starts with X and includes all the numbers up to, but not including, Y. The next number after [X, Y) is, by definition, Y. If you don't want to call that the immediate successor, then fine, but that's what it is, for all practical purposes.

In the concrete example, [3, 5), if you don't think the next number after [3, 5) is 5, then please state what it is.
 
zooterkin said:
You're the one who keeps using that expression.

Now, I'm going to type this slowly, so you can concentrate on the meaning:

[X, Y) is defined as the interval that starts with X and includes all the numbers up to, but not including, Y. The next number after [X, Y) is, by definition, Y. If you don't want to call that the immediate successor, then fine, but that's what it is, for all practical purposes.

In the concrete example, [3, 5), if you don't think the next number after [3, 5) is 5, then please state what it is.
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There is no "up to" here.
 
doronshadmi said:
No, you do not get what is your "up to" element.

Your "up to" element is the largest value of [3,5) that is < 5.

Please show us this "up to" element.
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'Up to' does not refer to an element. Every value between 3 and 5, but not including 5, is in the interval.

Are you having trouble accepting that such an interval can exist?

The interval is well defined, we can tell for any number whether it is in the interval or not. There is no need to specify the largest value in the interval.
 
zooterkin said:
'Up to' does not refer to an element. Every value between 3 and 5, but not including 5, is in the interval.

Are you having trouble accepting that such an interval can exist?

The interval is well defined, we can tell for any number whether it is in the interval or not. There is no need to specify the largest value in the interval.
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5 is a successor of [3,5).

5 is not an immediate successor of [3,5).

You do not distinguish between 'successor' and 'immediate successor', and again "up to" is not invoved here (as jsfisher already told you).
 
doronshadmi said:
5 is a successor of [3,5).

5 is not an immediate successor of [3,5).

You do not distinguish between 'successor' and 'immediate successor', and again "up to" is not invoved here (as jsfisher already told you).
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Ok, what is the distinction you are making between 'successor' and 'immediate successor'?

Is 7 also a successor of [3, 5)?
 
doronshadmi said:
x and y must be of the same type. since y represent a single value, then x also represent (an arbitrary) single value in [3,5)
( [3,5) < 5 is gibberish ).
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Same type? You made that up, didn't you?

All you need is a partial ordering. And there is a simple and obvious way to extend the normal ordering for the reals to real intervals and then to a combination of the two.

For the real numbers, the common ordering we use is the based on the less-than relation. There is a common notion for ordering among real intervals as well. [1, 2] precedes [8, 20], for example.

We can express this more formally as the interval A precedes interval B if and only if every value along the interval A precedes every value along interval B. When there is any overlap between the intervals, then there is no relationship, so this is a partial ordering.

Now, suppose you wanted to extend these concepts to include both the reals and real intervals together? The most natural way to do this is to consider any real, R, to be equivalent to the interval [R, R] for the purpose of determining order relations. With this approach, the reals maintain their normal complete ordering among each other, and the intervals maintain their normal partial ordering.

So, under a very reasonable assumption for partial ordering, the interval [3, 5) precedes 5. Equivalently, [3, 5) < 5.
 
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doronshadmi said:
There is no "up to" here.
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Doron, you have such difficulty with so many simple expressions. By the way, the full expression, as was used, here, was up to but not including. To you have similar problems with between? If we describe [3, 5) as all the real numbers between 3 and 5 and including 3, will that cause you similar consternation?

Really, what's your hang-up with zooterkin's original wording?
 
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