Interactive Real Analysis
- part of Example 2.3.5(a):
Impose a new ordering labeled << on the natural
numbers as follows:
Back
The natural numbers, ordered by the ordering <<,
could be listed in order as follows:
- if n and m are both even, then define n << m if n < m
- if n and m are both odd, then define n << m if n < m
- if n is even and m is odd, we always define n << m
Back2, 4, 6, 8, ....., 1, 3, 5, 7, 9, ..... ,To show it is well-ordered, take any subset A of natural numbers.
- If it contains only odd numbers, then the smallest number in the usual ordering is the smallest element of A
- If it contains only even numbers, then the smallest number in the usual ordering is the smallest element of A
- If it contains both even and odd numbers, then the smallest of the even numbers in the usual ordering is the smallest element of A
But, not every element has an immediate predecessor. For example, the set:
A = {1, 3, 5, 7, ...}has a smallest element (namely 1), but 1 does not have an immediate predecessor, since every even number is smaller than 1 by definition.