Deeper than primes

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doronshadmi said:
I agree with you (it can't be), so you have to ask jsfisher about it (he is the one that claims that there is no connection between set {X:X<Y} and [X,Y) ).
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To avoid further ambiguity, where does he do that?

ETA: What connection do you think there is? Can you remember what point you were trying to make when you started this particular digression?

It means that Y is not a member to the interval [X,Y).
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Yes, but not only that. It means that Y is the immediate successor to [X, Y).
 
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doronshadmi said:
I agree with you (it can't be), so you have to ask jsfisher about it (he is the one that claims that there is no connection between set {X:X<Y} and [X,Y) ).
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No, I claimed I didn't reference any [X, Y) interval in the proof. I also claimed that X never appeared as a free variable in the proof.

I also also claimed that you keep misrepresenting {X : X < Y} as [X, Y).

It means that Y is not a member to the interval [X,Y).
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It means considerably more than that.
 
doronshadmi said:
No, you are unclear of how that would works, so please direct us to some professional source that clearly talks about Y as am immediate successor of [X,Y) (as you claim).
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Well, how about we start with your claim, since it did come first, after all. The current discussion tangent arose from your insistence the interval [X, Y) guaranteed Y must have an immediate predecessor.

So, please direct us to some professional source that clearly talks about Y having an immediate predecessor in [X,Y) (as you claim).
 
jsfisher said:
Well, how about we start with your claim, since it did come first, after all. The current discussion tangent arose from your insistence the interval [X, Y) guaranteed Y must have an immediate predecessor.

So, please direct us to some professional source that clearly talks about Y having an immediate predecessor in [X,Y) (as you claim).
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You will not find any professional source (based on Standard Math) that clearly talks about Y having an immediate predecessor in [X,Y).

Since you are using only Standard Math and claim that Y is an immediate successor of [X,Y) (and now we are talking only on Standard Math framework), then please provide this Standard Math source.

Also this time please do not avoid the question in http://www.internationalskeptics.com/forums/showpost.php?p=4786190&postcount=3400, as you did in your http://www.internationalskeptics.com/forums/showpost.php?p=4786549&postcount=3406 reply.
 
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doronshadmi said:
You will not find any professional source (based on Standard Math) that clearly talks about Y having an immediate predecessor in [X,Y).
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Is anyone claiming you will?

Since you are using only Standard Math and claim that Y is an immediate of [X,Y), then please provide this Standard Math source.
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If you mean 'immediate successor', then that's what the notation means. Why don't you get this? http://en.wikipedia.org/wiki/Interval_(mathematics)
 
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Doroshani appears to be disputing the meaning of 'immediate successor'. Perhaps he might explain what he means by it if not Y of [X, Y).
 
jsfisher said:
I separately alleged the set { X : X < Y } for some Y has no largest element. You seem hung up on this one, too, so I will support the claim.
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Here jsfisher defines set A such that any member of set A < some Y, and he wants to show that A has no largest member.

By this construction it is clear that Y is not a member of A, otherwise Y is the largest member of A, and we cannot prove that A does not have the largest member.

Since Y must not be a member of A and A's members are non-finite R members that each one of them < Y, then A members are some non-finite R elements that exist in the clopen interval [W,Y), where W is the smallest member of A set.

jsfisher said:
I will use a simple proof by contradiction. As with all such proofs, it begins with an assumption then proceeds to construct a contradiction, thereby showing the assumption to be false.

Assume the set {X : X<Y} does have a largest element, Z.

For Z to be an element of the set, Z < Y.
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Now jsfisher provides the assumption, which claims that the non-finite set A that is based on the clopen interval [W,Y), has the largest member called Z, which is a member of A set (and an element of [W,Y) interval).
jsfisher said:
Let h be any element of the interval (Z,Y).
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Here we clearly see the open side ",Y)" of the clopen interval [W,Y), that jsfisher claims that this clopen interval is not a part of his proof. But the fact is that Z is a member of A and < Y, exactly because we deal here with the clopen interval [W,Y).
jsfisher said:
By the construction of h, Z < h < Y.
Since h < Y, h is an element of the set {X : X<Y}.
Since Z < h, the assumption Z was the largest element of the set has been contradicted.

Therefore, the set {X : X<Y} does not have a largest element.
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We can clearly see that since A has no largest member, then jsfisher claim ( see http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 ) that Y is the immediate successor of A set ( that its members are the elements of the clopen interval [W,Y) ), is a false claim.
 
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doronshadmi said:
We can clearly see that since A has no largest member, then jsfisher claim ( see http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 ) that Y is the immediate successor of A set ( that its members are the elements of the clopen interval [W,Y) ), is a false claim.
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Obviously some previously unknown use of the word 'clearly'.

Let's try a concrete example. Take X=3, Y=5, so [3, 5).

Any number >=3 and <5 is in the interval. Pick any number you like, and it's possible to say exactly whether it is in the set or not. 5 is the immediate successor of the interval.
 
zooterkin said:
Obviously some previously unknown use of the word 'clearly'.

Let's try a concrete example. Take X=3, Y=5, so [3, 5).

Any number >=3 and <5 is in the interval. Pick any number you like, and it's possible to say exactly whether it is in the set or not. 5 is the immediate successor of the interval.
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No, 5 is a successor of any arbitrary element of [3,5) (which is trivial) and it is not an immediate successor of any arbitrary element of [3,5) , as jsfisher claims, because given any arbitrary element Z in [3,5) there is h in [3,5) such that Z<h<5, so 5 is not an immediate successor of Z.
 
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doronshadmi said:
No, 5 is a successor of any arbitrary element of [3,5) (which is trivial) and it is not an immediate successor of any arbitrary element of [3,5) , as jsfisher claims, because given any arbitrary element Z in [3,5) there is h in [3,5) such that Z<h<5, so 5 is not an immediate successor of Z.
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What?

5 is the immediate successor of the interval, not of any arbitrary element in the interval.
 
zooterkin said:
What?

5 is the immediate successor of the interval, not of any arbitrary element in the interval.
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Again:

If jsfisher claims (in this case) that 5 is an immediate successor of the open interval [3,5) (by avoiding any particular element of it) he actually takes anything that starts with 3 and < 5 as a one mathematical object, exactly as the non-finite sequence 0.9+0.09+0.009+ … is considered as a one mathematical object (=1).

But in the case of a one mathematical object that is based on 0.9+0.09+0.009+ …=1 we really deal (according to Standard Math) with the non-finite (0.9+0.09+0.009+ … =1 and <1 does not hold), where in the case of [3,5) we do not deal with the non-finite, and this is exactly the reason of why "<5" expression holds (in the first place) in 3<5, Z<5 or Z<h<5 cases.

zooterkin said:
5 is the immediate successor of the interval ...
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Immediate successor means that there is no object between 5 and any given member of [3,5) interval, if "<5" is used.

Since "<5" is used and h is between 5 and any given member of [3,5) interval, then 5 is not the immediate successor of [3,5) interval.
 
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doronshadmi said:
If jsfisher claims (in this case) that 5 is an immediate successor of the open interval [3,5)
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It's not a claim, that's what [3,5) means.
(by avoiding any particular element of it) he actually takes anything that starts with 3 and < 5 as a one mathematical object, exactly as the non-finite sequence 0.9+0.09+0.009+ … is considered as a one mathematical object (=1).
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1) Why do you keep saying 'a one mathematical object'? It's not English.
2) Why are you trying to draw some equivalence between an interval and a single number?
3) Why do you keep using 'non-finite'?
Immediate successor means that there is no object between 5 and any given member of [3,5) interval, if "<5" is used.
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Not so. The interval starts at 3 and includes all the numbers up to, but not including, 5. There will always be at least one value between any given number in the interval and 5.
Since "<5" is used and h is between 5 and any given member of [3,5) interval, then 5 is not the immediate successor of [3,5) interval.
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Quite right, 5 is not the immediate successor of 'h'. However, it is the immediate successor of [3,5) because that's what [3,5) means.
 
zooterkin said:
Quite right, 5 is not the immediate successor of 'h'. However, it is the immediate successor of [3,5) because that's what [3,5) means.
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No, immediate successor has an exect meaning in Standard Math, which is:

Given x and y, y is an immediate successor of y iff x<y and there is no other element between x and y.

zooterkin said:
However, it is the immediate successor of [3,5) because that's what [3,5) means.
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It is like saying: "True is actually false because that what true means."
 
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doronshadmi said:
No, immediate successor has an exect meaning in Standard Math, which is:

Given x and y, y is an immediate successor of y iff x<y and there is no other element between x and y.
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Please show where ‘immediate successor’ is defined that or any other way by any standard math reference.
 
zooterkin said:
Not so. The interval starts at 3 and includes all the numbers up to, but not including, 5. There will always be at least one value between any given number in the interval and 5.
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"up to" is not involved here. You can ask jsfisher if you wish.
 
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doronshadmi said:
Here jsfisher defines set A such that any member of set A < some Y, and he wants to show that A has no largest member.
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You belabor the obvious.

By this construction it is clear that Y is not a member of A
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More obvious belaborment.

...otherwise Y is the largest member of A and we cannot prove that A does not have the largest member.
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(intermixed with gibberish)

Since Y must not be a member of A
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(Note the use of bold. Clearly this is important.)

and A's members are non-finite R members
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(infinitely many, actually.)

that each one of them < Y, then A members are some non-finite R elements that exist in the clopen interval [W,Y), where W is the smallest member of A set.
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No. And I'll bet just about everyone except Doron knows why his statement is just wrong.

Now jsfisher provides the assumption, which claims that the non-finite set A that is based on the clopen interval [W,Y)
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Nope.

has the largest member called Z, which is a member of A set (and an element of [W,Y) interval).
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Nope.

Here we clearly see the open side ",Y)" of the clopen interval [W,Y), that jsfisher claims that this clopen interval is not a part of his proof.
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Yes, indeed. It is not part of the proof. It is just some bit of irrelevance Doron keeps trying to introduce.

But the fact is that Z is a member of A and < Y, exactly because we deal here with the clopen interval [W,Y).
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Nope.

We can clearly see that since A has no largest member
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Excellent!! You have accepted the proof as valid. Ok, then, we are done. Doron accepts that {X : X < Y} has no largest element.

Why'd it take so long?

then jsfisher claim ( see http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 ) that Y is the immediate successor of A set ( that its members are the elements of the clopen interval [W,Y) ), is a false claim.
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Whoa! Talk about a leap of mush into unrelated nonsense.... There is no logical connection between this and anything that preceded it. (And even if there were, since the premises are false, the conclusion would be irrelevant.)

By the way, I never claimed what Doron claims I have claimed, nor does the set have its elements entirely in the half-open interval [W, Y).
 
doronshadmi said:
http://209.85.229.132/search?q=cach...ate+successor"+mathematics&cd=9&hl=en&ct=clnk


http://books.google.com/books?id=qL...-OX3Cg&sa=X&oi=book_result&ct=result&resnum=4


http://books.google.com/books?id=GU...OX3Cg&sa=X&oi=book_result&ct=result&resnum=10
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Very good but one minor correction on your previous post.

doronshadmi said:
No, immediate successor has an exect meaning in Standard Math, which is:

Given x and y, y is an immediate successor of y iff x<y and there is no other element between x and y.
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* bolding added

The bolded Y should actually be X since the assertion is that Y in the immediate successor of X.

Now with that out of the way. please show an immediate successor of X in the real numbers as you have claimed based on the given requirement.
 
doronshadmi said:
No, immediate successor has an exect meaning in Standard Math, which is:

Given x and y, y is an immediate successor of y iff x<y and there is no other element between x and y.
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In this case, x is the interval [3, 5), and there is no other element between it and y, which is 5.

It is like saying: "True is actually false because that what true means."
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No.
 
I made a typo mistake (as you have alreadt noticed) so here is the right one:

No, immediate successor has an exect meaning in Standard Math, which is:

Given x and y, y is an immediate successor of x iff x<y and there is no z such that x<z and z<y.
 
zooterkin said:
In this case, x is the interval [3, 5), and there is no other element between it and y, which is 5.
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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 ).
The Man said:
Now with that out of the way. please show an immediate successor of X in the real numbers as you have claimed based on the given requirement.
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I claim the opposite. Y is not an immediate successor of [X,Y), as jsfisher claims in http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 .
 
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doronshadmi said:
I made a typo mistake (as you have alreadt noticed) so here is the right one:

No, immediate successor has an exect meaning in Standard Math, which is:

Given x and y, y is an immediate successor of x iff x<y and there is no z such that x<z and z<y.
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In the case of [3, 5), please tell me what lies between [3, 5) and 5.
 
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.
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jsfisher,

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

Please answer by yes or no.
 
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)
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You're making things up again. The immediate successor of [X, Y) is Y; because that's what [X,Y) means. There is no requirement for there to be a single value which is the immediate predecessor of Y.


I claim the opposite. Y is not an immediate successor of [X,Y), as jsfisher claims in http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 .
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What is the immediate successor of [X, Y), then?
 
jsfisher said:
By the way, I never claimed what Doron claims I have claimed, nor does the set have its elements entirely in the half-open interval [W, Y).
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Please avoid your twisted games.

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)".

This last long dialog is based on this claim.

If you claim now that your proof by contradiction has nothing to do with [X,Y), then it is not relevent to this dialog, that is based on your "Y is in fact an immediate successor to [X,Y)".
 
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