2.3. The Principle of Induction
- Let S be a well-ordered set Then: if Q is a property such that:
- the smallest element of S has the property Q
- if s S has property Q then the successor of s also has property Q
- Then the property Q holds for every element in S
Consider the natural numbers with the ordering << defined 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
Q is the property that every element is even.
- The smallest element of our set in the << ordering is 2, which is even.
- Also, if s has property Q then so does the successor of s. That is because in our ordering, the successor of an even number is always the next even number, never an odd number, and if s has property Q, then s must be even.
Therefore, by the incorrect induction principle, every natural number is even - which is, of course, not true.
The actual induction principle as we have defined it does, however, not apply to this example, since 1 does not have an immediate predecessor.
This example was suggested by Karl Hahn who pointed out that there is another principle, called Transfinite Induction which - suitably stated - does apply to every well-ordered set. He also suggested the book Set Theory and Logic by Stoll, published by Dover, for further reference on this and other set theoretical topics.