(related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
The indicator function provides new insight into the proving principle by complete induction. If p is a property which is valid for some natural numbers m\in\mathbb N, then it defines a unique indicator function \chi(m) defined by \chi(m):=\begin{cases}1&\text{if }p(m)\text{ is true}\\0&\text{else, i.e if }p(m)\text{ is false.}\end{cases} With this interpretation, the proving principle goes as follows: * Base case: Prove that \chi(m)=1 for some m. * Induction step: Show that if n\ge m and \chi(m)=1, then \chi(n+1)=\chi(n). * Conclusion: Conclude that \chi(n)=1 for all n\ge m.
In other words, the proving principle uses the fact that the indicator function \chi is constant locally, i.e. for all the two natural numbers (n,n+1), if n\ge m and \chi(m)=\chi(n)=1. This allows the conclusion, that \chi is constant globally, i.e. for all consecutive pairs (n,n+1), (n+1,n+2), etc, we have \chi(m)=\chi(n)=\chi(n+1)=\chi(n+2)=\ldots. This, in turn, is equivalent to proving that the property p is true for all natural numbers n\ge m.
