(The Principle of Mathematical Induction) For each positive integer , let be a statement. If
(1) is true and
(2) the implication
If , then
is true for ever positive integer , then is true for every positive integer .
Assume, to the contrary, that the theorem is false. Then conditions (1) and (2) are satisfied but there exist some positive integers for which is a false statement. Let
Since is a nonempty subset of , it follows by the Well-Ordering Principle that contains . Since is true, .
Here I am lost:
Thus and . Therefore, and so is a true statement. By condition (2), is also true
and so . This however, contradicts our assumption that .
Question: What is ?