So I began to read Terry Tao's "Analysis 1" and I got confused by his strong induction formulation. The way he puts it is: "Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m \geq m_0$, we have the following implication : if $P(m')$ is true for all natural numbers $m_0 \leq m' < m$, then $P(m)$ is also true. (In particular this means that $P(m_0)$ is true, since in this case the hypothesis is vacuous.) Then we can conclude that $P(m)$ is true for all natural numbers $m \geq m_0$. " Now I didn't understood his explanation why it means $P(m_0)$ is true. Can you please clarify it for me ?

  • $\begingroup$ If the predicate holds after some stage, it must hold after the initial stage. I think this is what it means. $\endgroup$ Jul 30 '18 at 17:44
  • $\begingroup$ $P(m_0)$ is the base case in either of the examples. Nothing stands without the base. When someone asks you why you think what you think, you'll tell him there is a based argument pro, not just insinuation. $\endgroup$
    – Invisible
    Jan 10 '20 at 0:38

It's just that if you take $m=m_0$, then there is no $m'$ with $m_0\leq m' <m$ (because this inequality becomes $m_0\leq m' <m_0$). That means $P(m')$ is true for all such $m'$ that exist.

That's why he says "the hypothesis is vacuous."

  • $\begingroup$ Now I understand, thanks for the explanation and answer. $\endgroup$ Jul 30 '18 at 17:48
  • $\begingroup$ I will accept your answer as soon as it will be possible. $\endgroup$ Jul 30 '18 at 17:49

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