Succession of objects within a given infinite set, has nothing do to with order.
Fine, then by all means please define your “succession” without ordering.
This is the reason why, for example, {a,b,c,d,e,...} = {b,c,a,e,d,...}.
Much as it doesn’t surprise anyone, an example of a set with two different orderings is not a definition of “succession” without ordering.
Are you perhaps just confusing ordering with some specific ordering?
Please tell us what element of your set the first element of your set is a successor to?
Please tell us what element in your set is the successor to all the elements of your set.
As, by your own assertions, your “difference” requires your “interval” and the orderings of your set are different in your example, please tell us what is the “interval” that you require by that difference?
Once again you demonstrate your inability to understand simple notions, because your mind is tuned to think within agreed boxes.
As for generalization, it is up to you to get out of your boxes thinking style, in order to generalize what you read by using your own mind.
Unfortunately for you The Man your mind became lazy because of too many years of boxes thinking style, until you need others to organize everything in your agreed boxes, before you are able to deal with some subject.
The Man, with this kind of thinking style, you have no chance to think out of the box, where one on the fundamental abilities of thinking out of the box is called generalization, which is something that you simply can't do, and we see this inability over and over again all along this thread.
OH NO!!
Not the ‘get out of your box’ retort!!!
Again…
What a laugh.
This is your box, Doron. That you simply have as much distain (if not more) for your own box as you do for any other, that you would simply like to ascribe to others, is of consequence to no one but you. It is incumbent on no one but you to construct your box in, at the very least, a self consistent manor. Similarly filling your box with some particular meaning of your own choosing befalls no one but you. That you simply can’t be bothered to perform either of those tasks in no way gives anyone else the responsibility to do so for you. Why how deliberately lazy, of just you, Doron.
What Doron is attempting to advocate here, other than simply and deliberately trying to abdicate his own responsibility for his own notions and assertions, is a technique common to cold readers, con artists and other hucksters. By recommending, forcing or even just allowing the audience to provide their own meaning to such assertions the presenter frees themselves of the responsibility of any resulting discrepancies as well as gets the audience invested in such meaning that they choose to ascribe themselves. The problem is that we here on this forum are more than familiar with such antics. Additionally it may not even be intentional on Doron’s part but just a technique that has gotten him what he perceives as preferred results before. So he might have never found the need (nor found it beneficial) to take full responsibility for his own assertions.
The Man, A and B are different exactly because there is an interval between them, which is called difference.
If there is nothing between A and B, then there is one and only one thing, no matter how many names it has.
Again your boxes thinking style fails to generalize the notion of interval in order to get it also in terms of difference.
As already demonstrated the letters A and B are different (just as the orderings in your set example above are different), but with no interval as a consequence of that difference. Heck, technically an interval doesn’t even require a difference as in the case of [1,1], which results in the set {1}.
http://en.wikipedia.org/wiki/Interval_(mathematics)
The interval of numbers between a and b, including a and b, is often denoted [a,b]. The two numbers are called the endpoints of the interval.
[edit] Excluding the endpoints
To indicate that one of the endpoints is to be excluded from the set, many writers substitute a parenthesis for the corresponding square bracket. Thus, in set builder notation,
Note that (a,a), [a,a), and (a,a] denote the empty set, whereas [a,a] denotes the set {a}. When a > b, all four notations are usually assumed to represent the empty set.
http://en.wikipedia.org/wiki/Partially_ordered_set#Interval
Interval
For a ≤ b, the closed interval [a,b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains at least the elements a and b.
Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b. For example, the open interval (1,2) on the integers is empty since there are no integers i such that 1 < i < 2.
Sometimes the definitions are extended to allow a > b, in which case the interval is empty.
The half-open intervals [a,b) and (a,b] are defined similarly.
A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural ordering.
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders.
So by all means, please Doron, stop being so dang lazy and do try to at least learn something.
Certainly some differences do result in an interval just as some intervals can result from a difference, but neither specifically requires nor is required by the other.
The specific fallacy you are engaging in (at least in this particular case), Doron, is called the
fallacy of necessity.
So your assertion now is that you just want to call your “interval” a “difference” dispite the fact that there is no difference in some intervals? Talk about just being lazy, Doron.
