Talk:Principle of Mathematical Induction

Alternative form
Given the Existence of Minimal Infinite Successor Set, the form of the Principle of Mathematical Induction that Halmos states (in quantified form) is

$\forall S\in\mathcal{P}(\omega)\,(\varnothing\in S \wedge \forall n\in S(n^+\in S) \Rightarrow S=\omega)$

is a consequence of $\omega$'s existence, its minimality property, and definition of set equality. I think its more immediate why the PMI applies to $\omega$ than to a Peano structure, as $\omega$'s existence has been demonstrated. Should we add this form? --Robertbiggs34 (talk) 21:16, 27 May 2013 (UTC)


 * What? The principle of mathematical induction is one of the Peano axioms. I don't see what you're getting at. --Dfeuer (talk) 21:25, 27 May 2013 (UTC)


 * Why are two variables (i.e n and k) needed? Why not just one? What is the advantage of utilizing k in step 2 below as opposed to n?


 * Suppose that:


 * $(1): \quad \map P {n_0}$ is true


 * $(2): \quad \forall n \in \Z: n \ge n_0 : \map P n \implies \map P {n + 1}$


 * Then:


 * $\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$.


 * Thanks in advance!
 * --Robkahn131 (talk) 13:46, 3 May 2021 (UTC)