To get some idea as to how complicated your question really is.
Have a look at this: Dedekind_cuts
How is the sequential order of counting numbers (1, 2, 3, ...) actually defined? Why is it not (1, 7, 3, 6, etc.)?
I assume it would be defined simply as a series where each number A(n) follows a number A(n-1) such that A(n) = A(n-1) + 1.
Fine. But then what about the sequential order or all reals? Here the difference between A(n) and A(n-1) is not determinable. So the definition must be more general.
Q1: Would it be correct to define the sequential order of all reals as: A series of real numbers where each number A(n) follows a number A(n-1) such that A(n) > A(n-1) and A(n) – A(n-1) is the smallest difference between A(n-1) and any number not A(n-1)?
Q2 (Main Question): If sequential order of reals is correctly defined above (in Q1), then is it also correct to state the converse (as follows)?
If a set of randomly ordered real numbers is to be re-ordered such that the difference between each number and the number preceding it has the smallest value of all possible difference values, then the re-ordered set can only be a sequentially ordered series of real numbers.
Any clarity on this would be much appreciated.
In formal contexts, a successor function like the one you defined is in fact used, although my experience with this is limited and I should hit the books sometime. See: Natural numbers, subheading: Formal definitions on Wikipedia.
As for the reals, I again have limited experience but here is a reference: Construction of the real numbers - Wikipedia, the free encyclopedia.
Your definition for reals will not work, because this smallest difference you mention must be a number right? But what happens if you take half of that number?
In fact your definition implies that there exists a bijection between the naturals and the reals, which is false.
Thank you undefined.
My understanding of your point is that the problem with defining sequential order of reals (as stated in my Q1) is that a "smallest difference" is not determinable.
OK. But what if, instead of attempting to include all reals in the definition, we limit it to a set of N discrete elements, where each element is a real number > 0?
Would my definitions under Q1 and (most importantly) Q2 be correct for such a set?
Ordered sets are usually referred to as tuples. (Ordered pair is another name for 2-tuple, an ordered triple is a 3-tuple, etc.)
Note that since you're given the subset of the reals rather than trying to construct the set, you could use a simpler definition. These following definitions are equivalent:
Let . Let the N-tuple be an ordering of the elements of .
Definition 1: is in sequential order .
Definition 2: is in sequential order .
Definition 3: is in sequential order .