Can you have more than one immediate successor?
I deliberately avoided the phrase immediate successor, because you seem to have an irrational reaction each time it is used, but yes, what I'm asking is, what is the immediate successor of [3, 5)?
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Can you have more than one immediate successor?
I deliberately avoided the phrase immediate successor, because you seem to have an irrational reaction each time it is used, but yes, what I'm asking is, what is the immediate successor of [3, 5)?
doronshadmi
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[3,5) is actually [3,... interval that followed by [5,... interval.
jfisher suggested to change [5,... by [5,5], and by doing that we can say that
[3,5) < [5,5] where [5,5] is an immediate successor of [3,5), but in order to say that, we must ignore the content (the elements) of both [3,5) and [5,5] intervals, as I wrote in http://www.internationalskeptics.com/forums/showpost.php?p=4791128&postcount=3537 .
But it is possible to say that [3,5) < [5,5] in the first place, exactly because we do not ignore the elements of [3,5) and [5,5] , so intervals of real numbers have no "<" between them without their contents.
In other words, [3,5) < [5,5] expression holds only if 5 is compared with the elements of [3,5), and as a result what is written in http://www.internationalskeptics.com/forums/showpost.php?p=4791128&postcount=3537 does not hold between intervals.
As for irrational reaction, each time you ask about an immediate successor of [3,5), this is indeed an irrational action, because [3,5) does not have an immediate successor, but you insist to repeat on this question again and again, which is an irrational re-action.
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Doron seems to be mixing up sets and set members. An interval is a set, in the case of [3,5) it is a set containing all the numbers on the real line which are greater or equal to 3 and are less than 5. This is obviously an infinite set. Within this set there is no largest member. However intervals belong to the real line, therefore the next number on the real line that succeeds this set is 5 and it is not a member of the interval. Doron thinks that because the set is infinite and contains real numbers it has no successor because he does not understand or refuses to accept that both intervals and numbers are part of the real line.
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.
doronshadmi
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sympathic ,
There is a successor to [3,5) elements, and it is called 5.
But 5 is not an immediate successor of any element of [3,5) interval, as clearly seen in [3,...[5,5] .
No, you refuse to accept that "<" relation between [3,5) and [5,5] has no meaning if we ignore the relations between the elements of [3,5) and the elements of [5,5], as jsfisher tries to force, in this case.
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That's a very odd way to phrase it. Why the superfluous word, "elements"?
Nobody is saying it is.
The Man
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jsfisher supporeted it because it is correct.
That is certainly not correct.
Um, this assertion Doron…
Is yours and yours alone, so it is up to you to show … “how [X,Y)< [Y,Y] by ignoring the elements of [X,Y) and the elements of [Y,Y].”
Please let us know when you’ve got that worked out, will ya.
What do you even mean by that?
doronshadmi
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because "<" is meaningless without these elements (thet exist both in [3,5) and [5,5]
[3,5) < [5,5] is gibberish if we ignore the elements of [3,5) and [5,5].
Since this is the case, then [3,5) does not have an immediate successor.
There aren't any elements that exist in both [3, 5) and [5, 5]
doronshadmi
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http://www.internationalskeptics.com/forums/showpost.php?p=4791399&postcount=3551
Here jsfisher tries to find [X,Y) < [Y,Y], and he claims that [Y,Y] is an immediate successor of [X,Y).jsfisher said:
But thtis is utter nonsense because "<" relation between [X,Y) and [Y,Y] does not exist unless Y element of [Y,Y] interval is compared with any element of [X,Y) interval.
In this case [X,Y) does not have an immediate successor.
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doronshadmi
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Ho no, you looked on the elements and not on the intervals, noty noty boy, did you forget what jsfisher told you?
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Name one element that is in [3, 5) and [5, 5].
(I'll give you a clue, there isn't one.)
doronshadmi
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So what?
Since "<" relation between [3,5) and [5,5] cannot be found without the elements of these intervals, then [3,5) < [5,5] has a meaning only because of the comparison between [5,5] elements and [3,5) elements.
According to this comparison (that without it "<" has no meaning) [3,5) does not have an immediate successor.
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well, you referred to:
Though I suspect that you didn't mean what you said.
If that's what you want to believe...
So close. I thought you were going to get it this time.
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Am I alone in finding it ironic that the person who has tremendous difficulty communicating in complete thoughts is quibbling over someones use of the colloquial "if...then"?
You asked for plain English, doron, and that's what you got. Whether you accept the colloquial "if...then" as equivalent to the mathematically precise "if and only if" isn't all that important. It is what he meant. Let's move on, please.
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Yes, and I also told you how ordering between intervals and numbers was to be defined. You must have skipped over that part.
I have already shown you how the ordering relation between intervals was to be defined. All the necessary conditions are satisfied, so [X, Y) < [Y, Y]. Or do you have a counter-example to contradict that?
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jsfisher
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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?
doronshadmi
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doronshadmi said:
By "these elements" I mean the elements of the intervals, and not particular elements that exist both in [3,5) and [5,5].
Next time try to get what you read according to the context (this is, by the way, the reason why you don't get that [3,5) does not have an immediate successor).
doronshadmi
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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.
We know that A<B only by their elements, so without the elements of A and B, A<B expression cannot be found.
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.
By doing that, it is clearly understood that A (which is a mathematical object that does not exist independently of its elements) does not have an immediate successor.
Anyone who claims that he can compare between A and B and conclude something about the relations between A and B (by ignoring the relations between the elements of A and the elements of B) uses an illusionary game with notations, without any reasoning and notion at the basis of this illusionary game.
Jsfisher, zooterkin and The Man, plays this illusionary game with notations, without any reasoning and notion at the basis of this game, in the case of A and B intervals.
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.
We know that A<B only by their elements, so without the elements of A and B, A<B expression cannot be found.
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.
By doing that, it is clearly understood that A (which is a mathematical object that does not exist independently of its elements) does not have an immediate successor.
Anyone who claims that he can compare between A and B and conclude something about the relations between A and B (by ignoring the relations between the elements of A and the elements of B) uses an illusionary game with notations, without any reasoning and notion at the basis of this illusionary game.
Jsfisher, zooterkin and The Man, plays this illusionary game with notations, without any reasoning and notion at the basis of this game, in the case of A and B intervals.
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doronshadmi
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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.
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jsfisher
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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?
Great. Now, would you be so kind as to answer my question?
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jsfisher
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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]
doronshadmi
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[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.
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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
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jsfisher, it does not work like this.
Please show us how interval A < interval B by totally ignore the contents of these intervals.
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doronshadmi
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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.
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doronshadmi
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jsfisher said:
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).
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The Man
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Nor in yours
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.
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 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.
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
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jsfisher said:
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.
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doronshadmi
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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.
I agree with you.
Please tell it to jsfisher.
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jsfisher
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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.
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.
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jsfisher
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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).
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.
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That makes you think that?
The Man
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By all means please show your work comparing each element in interval A to each element in interval B.
No you do not, stop lying, if not to everyone else at least to yourself.
He already knows you lie to yourself and do not agree with me.
doronshadmi
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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.
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.
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doronshadmi
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Is this a question or what?
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You, as is your custom, belabor the obvious.
Nobody except you is trying to use the less-than relation between intervals. The rest of us are applying an order relation.
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.
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
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jsfisher said:
jsfisher said:
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).
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The Man
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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.
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