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.