2.3. The Principle of Induction

Example 2.3.5(b):

Suppose the induction principle defined above does not contain the assumption that every element except for the smallest has an immediate predecessor. Then show that it could be proved that every natural number must be even (which is, of course, not true so the additional assumption on the induction principle is necessary).
In other words, we assume that the induction principles was stated as follows: If this principle was true, we could prove that every natural number must be even as follows:

Consider the natural numbers with the ordering << defined as follows:

We have already proved that this set is well-ordered. We want to show that every number is even. Therefore:
Q is the property that every element is 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.

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