# Talk:Principle of Mathematical Induction

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## 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)