Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can one prove the principle of mathematical induction using the well ordering principle?

share|cite|improve this question

Suppose that both

$$P(1)\,\,\,\wedge\,\,\,\left(\,P(n)\longrightarrow P(n+1)\,\right)$$

are true. Let $\,K\subset\Bbb N\,$ the set of all natural numbers for which $\,P(k)\,$ is true.

Let $\,C=\Bbb N-K\,$ . If $\,C\neq \emptyset\,$ then the WOP tells us it has a first element (in the natural, usual order), say $\,c\,$ . By assumption, $\,c>1\,$ (why?), so we can say that

$$c-1\in\Bbb N-C=K\Longrightarrow P(c-1)\,\,\,\text{is true, so also }\,\,\,P(c-1+1)=P(c)\,\,\,\text{is true}$$

But this contradicts $\,c\notin K\,$ !

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.