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I am studying analysis 1 by Tao. And here is the definition of strong induction in his book.

Strong induction: 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 $m≥m_0$, we have the following implication: if $P(m')$ is true for all natural numbers $m_0≤m'<m$, then $P(m)$ is also true.(In particular, this means that $P(m_0)$ is true, since in the case the hypothesis is vacuous.) Then we can conclude that $P(m)$ is true for all natural numbers $m≥m_0$.

I saw in few posts on MSE including Strong Induction Requires No Base Case?
I saw that write the strong induction as $$ ∀m ≥m_0[ ∀m'(m_0≤m'<m \implies P(m'))] \implies P(m)$$ But according to me the logical form would look like this $$ ∀m[ ∀m_0≤m'<m, P(m')] \implies P(m)$$

My Question: why people are introducing extra implication? that is $$ ∀m'(m_0≤m'<m \implies P(m')).$$

Mauro ALLEGRANZA
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Afzal
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  • Your "logical form" is not correct. – Anne Bauval Jul 14 '23 at 15:29
  • @AnneBauval Oh I see. Can you please tell me what is correct and why? And why it isn't correct. – Afzal Jul 14 '23 at 15:30
  • What is correct is the formula you reported just before "But according to me", and it is correct because it is precisely the formal transcription of Tao's definition of "Strong induction". – Anne Bauval Jul 14 '23 at 15:35
  • @AnneBauval oh. But I can't see this implication in the definition of strong induction! It simply says "If $P(m')$ is true for all $m_0≤m'<m$". This doesn't looks me an implication that "If for all $m_0≤m'<m$, then $P(m')$ is true" – Afzal Jul 14 '23 at 15:36
  • The full formal transcription of Tao's definition is: $$\left[(\forall m\ge m_0)(Q(m)\implies P(m))\right]\implies(\forall m\ge m_0)P(m).$$ where $Q(m)$ is $$(\forall m')(m_0\le m'<m\implies P(m').$$ – Anne Bauval Jul 14 '23 at 15:49
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    Your writing "forall $m_0\le m'<m$" is formally incorrect. the "forall" must immediately be followed by $m'$. Then, the formalization of "$P(m')$ is true for all $m'$ such that $R(m')$", or "forall $m'$ such that $R(m')$, we have $P(m')$", requires an implication sign:$$(\forall m')(R(m')\implies P(m')).$$ – Anne Bauval Jul 14 '23 at 15:51
  • @AnneBauval Aah i see. So correct wording is "for all $m'$ s.t $m_0≤m'<m$ we have $P(m')$" this can be written in implication form as you wrote. I understood thank you. – Afzal Jul 14 '23 at 17:19

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