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?
 
jsfisher said:
Doron,

Very early on in this current tangent you said:



But just recently you said:




These two statements are not in universal harmony. Which one would you like to retract, or will you just let the contradiction stand?
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You mix up between OM and Standard Math.

My last dialog with you is only under Standard Math.
 
jsfisher said:
So, under a very reasonable assumption for partial ordering, the interval [3, 5) precedes 5. Equivalently, [3, 5) < 5.
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It does not mean that 5 is an immediate successor of anything that is related to [3,5).

5 is a successor of [3,5), as I claim

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

AGAIN: we are talking here only about Standard Math.
 
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Doron, what is the next number after [3, 5)?

If it is not 5, then what do you think [3, 5) means? What are the numbers which fall in that interval?
 
doronshadmi said:
You mix up between OM and Standard Math.

My last dialog with you is only under Standard Math.
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Well, then, you must be using some rather interesting partial ordering for your OM case. Tell us about it. What ordering were you implicitly assuming?
 
doronshadmi said:
It does not mean that 5 is an immediate successor of anything that is related to [3,5).

5 is a successor of [3,5), as I claim
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It took you quite a while to finally accept this claim (and yes, the whole time under an assumption of "standard math"). If fact, just a few posts back, you proclaimed, "Again, [3,5) < 5 is gibberish."

Still, glad to have you on board, finally.

5 is not an immediate successor of [3,5), as you claim.
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If 5 be not an immediate successor of [3, 5), then there'd need to be some number between [3, 5) and 5. What number could that be?
 
zooterkin said:
Doron, what is the next number after [3, 5)?

If it is not 5, then what do you think [3, 5) means? What are the numbers which fall in that interval?
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Any arbitrary value < 5, but it is not relevant to our case.

EDIT: the first row has a typo mistake, the right one is:

Any arbitrary value > 5, but it is not relevant to our case.

The relevant question here is: what is the next immediate number after [3, 5)?
 
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jsfisher said:
[3, 5) and 5. What number could that be?
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jsfisher, is 5 one of the elements of [3,5) interval ( if the answer is yes than 5 is the largest element, but then we are talking about [3,5] and not about [3,5) )?
 
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doronshadmi said:
jsfisher, is 5 one of the elements of [3,5) interval ( if the answer is yes than 5 is the largest element, but than we are talking about [3,5] and not about [3,5) ).
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Oh, dear, Doron, do keep up.

No, 5 is not included in the [3, 5) interval. By definition.
 
zooterkin said:
http://www.internationalskeptics.com/forums/showpost.php?p=4752300&postcount=3179

That's simply what the notation [X, Y) means. It refers to the finite interval starting with X, and including everything up to, but not including, Y.
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jsfisher said:
http://www.internationalskeptics.com/forums/showpost.php?p=4752496&postcount=3191

The phrase, up to, is neither a real number nor an interval, so it has no part in this discussion.
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doronshadmi said:
There is no "up to" here.
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jsfisher said:
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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So you contradict yourself once again, jsfisher.

Now we can see why you avoid this:
jsfisher said:
[3, 5) and 5. What number could that be?
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jsfisher, is 5 one of the elements of [3,5) interval ( if the answer is yes then 5 is the largest element, but than we are talking about [3,5] and not about [3,5) )?

and also you avoid this:

jsfisher said:
Yes, really. The proof showed that the set {X : X < Y} has no largest member. There was no half-open finite interval [X, Y) anywhere in the proof.
doronshadmi said:
jsfisher,

Is Y is a mamber of set {X : X < Y}?

Please answer by yes or no.
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doronshadmi said:
Any arbitrary value < 5, but it is not relevant to our case.
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Let's see. I'll take 2 as my arbitrary value less than 5. So, according to Doron, 2 is the next number after [3, 5).

The relevant question here is: what is the next immediate number after [3, 5)?
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Umm, that's the question zooterkin asked. (The word immediate is superfluous.)
 
doronshadmi said:
So you contradict yourself once again, jsfisher.
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What do you find contradictory?

On the one hand, I correctly point out that "up to" is not a number. You aren't claiming it is, are you? If so, is it bigger than 17?

On the other hand, I point out that you have difficulty understanding the phrase, "up to but not including Y".

Are you really saying those two statements are in conflict with each other?
 
jsfisher said:
What do you find contradictory?

On the one hand, I correctly point out that "up to" is not a number. You aren't claiming it is, are you? If so, is it bigger than 17?

On the other hand, I point out that you have difficulty understanding the phrase, "up to but not including Y".

Are you really saying those two statements are in conflict with each other?
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What is this up to that not including Y, please explicitly show its value.
 
jsfisher said:
Yes, really. The proof showed that the set {X : X < Y} has no largest member. There was no half-open finite interval [X, Y) anywhere in the proof.
doronshadmi said:
jsfisher,

Is Y is a mamber of set {X : X < Y}?

Please answer by yes or no.
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Still wating to your answer.
 
jsfisher said:
What do you find contradictory?

On the one hand, I correctly point out that "up to" is not a number. You aren't claiming it is, are you? If so, is it bigger than 17?

On the other hand, I point out that you have difficulty understanding the phrase, "up to but not including Y".

Are you really saying those two statements are in conflict with each other?
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Again you contradict yourself, because this "up to" case was given by zooterkin.

So you both accept and reject his "up to" phrase.
 
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