1. ## Induction

The principle of mathematical induction can be extended as follows. A list $\displaystyle P_m, P_{m+1}, \cdot \cdot \cdot$ of propositions is true provided (i) $\displaystyle P_m$ is ture, (ii) $\displaystyle P_{n+1}$is true whenever $\displaystyle P_n$ is true and $\displaystyle n\geq m$.

Question: Is the above what we called the Strong Induction?

2. Originally Posted by novice
The principle of mathematical induction can be extended as follows. A list $\displaystyle P_m, P_{m+1}, \cdot \cdot \cdot$ of propositions is true provided (i) $\displaystyle P_m$ is ture, (ii) $\displaystyle P_{n+1}$is true whenever $\displaystyle P_n$ is true and $\displaystyle n\geq m$.

Question: Is the above what we called the Strong Induction?
It looks like it, but I must say the formulation seems a bit lacking in precision.
The principle of mathematical induction could be stated, I guess from your question, like this:
If (i) $\displaystyle P_m$ is true and (ii) for all $\displaystyle n\geq m$: $\displaystyle P_{n+1}$ is true, provided $\displaystyle P_n$ is true
then $\displaystyle P_n$ is true for all $\displaystyle n\geq m$
Now a somewhat unhappy formulation of the principle of strong induction would be
If (i) $\displaystyle P_m$ is true and (ii) for all $\displaystyle n\geq m$: $\displaystyle P_{n+1}$ is true provided $\displaystyle P_k$ is true for all $\displaystyle k$ with $\displaystyle m\leq k \leq n$
then $\displaystyle P_n$ is true for all $\displaystyle n\geq m$
I wrote "unhappy formulation" because the principle of strong induction should, in my opinion, really be formulated like this:
If for all $\displaystyle n\geq m$: $\displaystyle P_n$ is true provided that $\displaystyle P_k$ is true for all $\displaystyle k$ with $\displaystyle m\leq k<n$
then $\displaystyle P_n$ is true for all $\displaystyle n\geq m$
Note that in this formulation we do not need a separate clause to require that $\displaystyle P_m$ holds, because if you set $\displaystyle n := m$ in the above you can see, that this requires you to be able to prove $\displaystyle P_m$ without presupposing the truth of any other $\displaystyle P_k$, with $\displaystyle k\geq m$, since for this choice of n there is no k that satisfies $\displaystyle m\leq k<n$.

This would become much clearer if we were allowed to write the above with connectives of implication, conjunction, and quantification from first-order logic, I think.

3. The "strong" principle of induction usually is stated as

If (1) P(1) is true and (2) if P(k) is true for all [tex]k\le i[/itex] then P(i+ 1) is true
then P(n) is true for all positive integers n.

Changing "P(1)" to "P(m)" and asserting that "then P(n) is true for all $\displaystyle n\ge m$ is not precisely the "strong" principle but follows easily from it.

4. Originally Posted by HallsofIvy
The "strong" principle of induction usually is stated as

If (1) P(1) is true and (2) if P(k) is true for all [tex]k\le i[/itex] then P(i+ 1) is true
then P(n) is true for all positive integers n.

Changing "P(1)" to "P(m)" and asserting that "then P(n) is true for all $\displaystyle n\ge m$ is not precisely the "strong" principle but follows easily from it.
Is there a name for it?
You mean using $\displaystyle m$ as basis and prove $\displaystyle p(n), n\geq m$?
If $\displaystyle P(m)$ is true, when $\displaystyle m\leq n$, won't $\displaystyle P(n)$ automatically become true?