Deeper than primes

[doronshadmi]

Ok, I must have missed this. Has doron invented yet another bit of notation?

Doron, what do you mean by this "[3,..." and "[5,..." stuff?

Let us notate it in a general way:

[X,Y) = [X,...[Y,Y]

The term successor does not hold without "<" relation, and "<" relation holds only if we compare Y value with any value of [X,Y) that is less than Y.

Only by comparing Y value with any value of [X,Y) that is less than Y, we can use "<" relation and concude something about [X,Y), and in this case we conclude that [X,Y) does not have an immediate successor.

 

[jsfisher]

Darn, I accidentally sent this as a private message. I hate it when that happens. My apologies, Doron. I didn't mean to PM you.

Ok, I must have missed this. Has doron invented yet another bit of notation?

Doron, what do you mean by this "[3,..." and "[5,..." stuff?

Let us notate it in a general way:

[X,Y) = [X,...[Y,Y]

Great. Now, would you be so kind as to answer my question?

 

[jsfisher]

Interval A = [X,Y)

Interval B = [Y,Y]

We know that A and B are different and A and B have no common locations along the real-line, only if we look at their elements.

Yes.

We know that A<B only by their elements, so without the elements of A and B, A<B expression cannot be found.

Not quite, but close enough.

Since this is the case, then 'immediate successor' has a meaning here only in terms of the comparison between the elements of B interval and the elements of A interval.

No. This completely misses the definition for immediate successor.

Since this seems to be such a great source of confusion for Doron, I will be explicit about all four possibilities for order relations among intervals and real numbers, and the immediate qualifier. (In the following, the curvy greater-than is the order relation while the straight greater-than is the ordinary numeric comparator. In plain text, when the meaning is clear, the ">" will be used for both.)

"A succeeds B" means:

(1) A, a real number, B, a real number:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (A > B)) $$[/latex]​

(2) A, a real number, B, an interval:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (\forall y \, (y \in B) \Rightarrow (A \succ y)) $$[/latex]​

(3) A, an interval, B, a real number:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (\forall x \, (x \in A) \Rightarrow (x \succ B)) $$[/latex]​

(4) A, an interval, B, an interval):

[latex]$$ (A \succ B) \, \Leftrightarrow \, (\forall x \, \forall y \, (x \in A \wedge y \in B) \Rightarrow (x \succ y)) $$[/latex]​

"A immediately succeeds B" means:

[latex]$$(A \, immediately \succ B) \Leftrightarrow ((A \succ B) \wedge \nexists C \, (A \succ C \succ B))$$[/latex]​

 

[doronshadmi]

Darn, I accidentally sent this as a private message. I hate it when that happens. My apologies, Doron. I didn't mean to PM you.



Great. Now, would you be so kind as to answer my question?

[X,... is a collection of real numbers that has no largest element and [Y,Y] is a single real number that is greater than any number of [X,... collection.

 

[integral]

Because Doron did not go past his bizarre interpretations of simple mathematical concepts onto more advanced topics like calculus that rely on intervals for concepts as domain and others he looks at things from a very naive and uneducated standpoint. The sad thing is that he can not bring himself to admit his errors and accept other peoples authority in Math.

Ditto. After following the thread to this point, I'm beginning to think that the problem has as much to do with a language barrier as it does a lack of mathematical reasoning skills.

 

[doronshadmi]

Not quite, but close enough.

jsfisher, it does not work like this.

Please show us how interval A < interval B by totally ignore the contents of these intervals.

 

[doronshadmi]

Yes.



Not quite, but close enough.



No. This completely misses the definition for immediate successor.

Since this seems to be such a great source of confusion for Doron, I will be explicit about all four possibilities for order relations among intervals and real numbers, and the immediate qualifier. (In the following, the curvy greater-than is the order relation while the straight greater-than is the ordinary numeric comparator. In plain text, when the meaning is clear, the ">" will be used for both.)

"A succeeds B" means:

(1) A, a real number, B, a real number:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (A > B)) $$[/latex]​

(2) A, a real number, B, an interval:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (\forall y \, (y \in B) \Rightarrow (A \succ y)) $$[/latex]​

(3) A, an interval, B, a real number:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (\forall x \, (x \in A) \Rightarrow (x \succ B)) $$[/latex]​

(4) A, an interval, B, an interval):

[latex]$$ (A \succ B) \, \Leftrightarrow \, (\forall x \, \forall y \, (x \in A \wedge y \in B) \Rightarrow (x \succ y)) $$[/latex]​

"A immediately succeeds B" means:

[latex]$$(A \, immediately \succ B) \Leftrightarrow ((A \succ B) \wedge \nexists C \, (A \succ C \succ B))$$[/latex]​

Nothing is new in your post, jsfisher.

Again, let us write

[latex]$$ (A \prec B) \, \Leftrightarrow \, (\forall x \, \forall y \, (x \in A \wedge y \in B) \Rightarrow (x \prec y)) $$[/latex]​

in simple English and show how you are using the elements of A and B in order to show that "interval A precedes interval B".

(A < B) iff (for all x and for all y (such that x is a member of B AND y is a member of B) implies (x < y))

It is clearly seen that A<B because we compare between each element of A (called x) and each element of B (called y).

So A<B cannot be found in this expression, independently of the elements of A and B.

Again:

Since you show that A<B expression depends on the elements of A (called x) and the elements of B (called y), and since no y is an immediate successor of any x, then (by following this reasoning) you cannot use

[latex]$$ (A \prec B) \, \Leftrightarrow \, (\forall x \, \forall y \, (x \in A \wedge y \in B) \Rightarrow (x \prec y)) $$[/latex]​

in order to claim that B is an immediate successor of A, because by your expression above A<B has a meaning only according the relations of A and B elements (notated as x and y, where no y element of interval B is an immediate successor of any x element of interval A).

Stings with nice notations will not help here.

 

[doronshadmi]

[latex]$$(A \, immediately \succ B) \Leftrightarrow ((A \succ B) \wedge \nexists C \, (A \succ C \succ B))$$[/latex]​

C has no elements so A<C<B expression = A<B expression.

A<B expression totally depends on the comparison between the elements of A (called x) and the elements of B (called y).

 

[The Man]

Nothing is new in your post, jsfisher.

Nor in yours

Again, let us write

[latex]$$ (A \prec B) \, \Leftrightarrow \, (\forall x \, \forall y \, (x \in A \wedge y \in B) \Rightarrow (x \prec y)) $$[/latex]​

in simple English and show how you are using the elements of A and B in order to show that "interval A precedes interval B".

(A < B) iff (for all x and for all y (such that x is a member of B AND y is a member of B) implies (x < y))

It is clearly seen that A<B because we compare between each element of A (called x) and each element of B (called y).

If that is the case then be my guest, try A=[1,2) and B=[2,3] in the reals. Since both intervals contain an infinite number of elements, start comparing each element of A to each element of B and let us know when you are done.

So A<B cannot be found in this expression, independently of the elements of A and B.

The interval A is its elements just as the interval B is its elements, to claim anything is done with or by them “independently” of those elements is simply ludicrous, almost as ludicrous as claiming one needs to “compare between each element”. How’s that element by element comparison coming by the way?

Again:

Since you show that A<B expression depends on the elements of A (called x) and the elements of B (called y), and since no y is an immediate successor of any x, then (by following this reasoning) you cannot use

[latex]$$ (A \prec B) \, \Leftrightarrow \, (\forall x \, \forall y \, (x \in A \wedge y \in B) \Rightarrow (x \prec y)) $$[/latex]​

in order to claim that B is an immediate successor of A, because by your expression above A<B has a meaning only according the relations of A and B elements (notated as x and y, where no y element of interval B is an immediate successor of any x element of interval A).

Again any relationship between interval A and interval B depends on their ‘elements’ just not in the ludicrous way you are requiring. By the way, you done with that element by element comparison you were insisting on yet? You know we can’t wait forever.

Stings with nice nitations will not help here.

Ludicrous requirements to prop up your straw man arguments resulting in your erroneous conclusions certainly do not help you. Done with your element by element comparison yet?

 

[doronshadmi]

(1) A, a real number, B, a real number:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (A > B)) $$[/latex]​

(2) A, a real number, B, an interval:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (\forall y \, (y \in B) \Rightarrow (A \succ y)) $$[/latex]​

(3) A, an interval, B, a real number:

[latex]$$ (A \succ B) \, \Leftrightarrow \, (\forall x \, (x \in A) \Rightarrow (x \succ B)) $$[/latex]​

Here you are using comparison between real numbers (1) and a mix comparison between intervals and real numbers (2),(3).

But actually there is no mixing here because A>B expression holds only if the comparison is done between real numbers.

 

[doronshadmi]

If that is the case then be my guest, try A=[1,2) and B=[2,3] in the reals. Since both intervals contain an infinite number of elements, start comparing each element of A to each element of B and let us know when you are done.

Done, and still A<B expression has no meaning independently of the elements of A and B.

Also [1,2) has no immediate successor by that comparison.

The interval A is its elements just as the interval B is its elements,...

I agree with you.

Please tell it to jsfisher.

 

[jsfisher]

Here you are using comparison between real numbers (1) and a mix comparison between intervals and real numbers (2),(3).

Technically, there's only one comparison in the whole list of 4 (not just the 3 you showed)^*. The first has a comparison (the numeric greater-than). The rest only use the successor order relation.

But actually there is no mixing here because A>B expression holds only if the comparison is done between real numbers.

Huh? What are you trying to say here, Doron? No where in the list of four predicates is the numeric comparison operator used for anything but comparing numbers.

If you take your time and actually understand all four, you may notice that (2) through (4) reduce any case involving an interval into cases involving only real numbers.




ETA:
^*At least for comparisons in the normal sense of less than, greater than, and equal. Order relations are comparisons, too, but not in the sense that I think Doron meant.

 

[jsfisher]

Done, and still A<B expression has no meaning independently of the elements of A and B.

Ok, good. So, we all agree for the example [1, 2) and [2, 3] that [1, 2) precedes [2, 3] and [2, 3] succeeds [1, 2).

Also [1,2) has no immediate successor by that comparison.

Except "that comparison" doesn't figure into it. Perhaps if you used the actual definition for immediate successor rather than one you just made up, you might get the correct answer.

 

[jsfisher]

C has no elements...

That makes you think that?

 

[The Man]

Done, and still A<B expression has no meaning independently of the elements of A and B.

By all means please show your work comparing each element in interval A to each element in interval B.

Also [1,2) has no immediate successor by that comparison.


I agree with you.

No you do not, stop lying, if not to everyone else at least to yourself.

Please tell it to jsfisher.

He already knows you lie to yourself and do not agree with me.

 

[doronshadmi]

If you take your time and actually understand all four, you may notice that (2) through (4) reduce any case involving an interval into cases involving only real numbers.

If you take your time and actually understand all four, then you get that "<" relation can be used only between real numbers.

By following that fact, no real number is an immediate successor of any other real number, and any attempt to use "<" relation between intervals (instead of between real numbers) does not hold water.

jsfisher, you try to claim that B interval is an immediate successor of A interval, but as I clearly explained "A interval < B interval" expression holds exactly because we reduce intervals into cases involving only real numbers, and since this is the case (and there is no other case here) A (that is determined only by its real numbers) has no immediate successor exactly because B also determined only by its real numbers.

So our game is only between real numbers, and no real number has an immediate successor (by Standard Math).

End of game.

Except "that comparison" doesn't figure into it. Perhaps if you used the actual definition for immediate successor rather than one you just made up, you might get the correct answer.

You can use any fancy words that you like, but it does not change the facts I wrote above.

Perhaps if you used the actual definition for immediate successor (the one that is based on real numbers) rather than one you just made up (by using twisted maneuvers with fancy notations) you might get the correct answer.

 

[doronshadmi]

That makes you think that?

Is this a question or what?

 

[jsfisher]

If you take your time and actually understand all four, then you get that "<" relation can be used only between real numbers.

You, as is your custom, belabor the obvious.

By following that fact, no real number is an immediate successor of any other real number, and any attempt to use "<" relation between intervals (instead of between real numbers) does not hold water.

Nobody except you is trying to use the less-than relation between intervals. The rest of us are applying an order relation.

jsfisher, you try to claim that B interval is an immediate successor of A interval, but as I clearly explained "A interval < B interval" expression holds exactly because we reduce intervals into cases involving only real numbers, and since this is the case (and there is no other case here) A (that is determined only by its real numbers) has no immediate successor exactly because B also determined only by its real numbers.

The order relation is defined in terms of real number comparisons. The immediate successor is not. In fact, the definition of immediate successor is independent of how successor is defined.

...
Perhaps if you used the actual definition for immediate successor (the one that is based on real numbers) rather than one you just made up (by using twisted maneuvers with fancy notations) you might get the correct answer.

The actual definition isn't based on real numbers. The definition of immediate successor is based on the meaning of successor, and how successor is determined is immaterial to the definition.

And as for that notation, gee, I'm sorry. Trying to discuss Mathematics using Mathematics was so inconsiderate of me, wasn't it.

 

[doronshadmi]

In fact, the definition of immediate successor is independent of how successor is defined.

The definition of immediate successor is based on the meaning of successor, and how successor is determined is immaterial to the definition.

There is a big difference between successor and immediate successor.

Exactly because of this difference Y number is a successor of the numbers of [X,Y) and not an immediate successor of the numbers of [X,Y), as you claim (and again, to claim that an interval has an immediate successor or a successor is an utter gibberish, because "<" relation (that without it there is no meaning to order, in this case) holds only between the real numbers of the given intervals).

 

[The Man]

There is a big difference between successor and immediate successor.

Exactly because of this difference Y number is a successor of the numbers of [X,Y) and not an immediate successor of the numbers of [X,Y), as you claim (and again, to claim that an interval has an immediate successor or a successor is an utter gibberish, because "<" relation (that without it there is no meaning to order, in this case) holds only between the real numbers of the given intervals).

Please identify the real number between the real number interval [1,2) and the real number 2 or any real number in that interval that is not less then 2? As there is order in an interval the meaning of "<" (an ordering relation) is inherently part of that interval.