Answer by Peter Webb for Can mathematical inductions work for other sets?
These are very good questions, in that they directly address the question of why induction works.For your first question concerning p(k+n). If the based case is n=1, then induction can be used to prove...
View ArticleAnswer by celtschk for Can mathematical inductions work for other sets?
The most general case where induction works for a claim $P$ on elements of a set $X$ is:You have a set of base cases where you can prove the claim directly, that is, a set $S_0\subset X$ so that you...
View ArticleAnswer by Thomas for Can mathematical inductions work for other sets?
if you are interested in $p(k+n)$ with always the same $n$ a renumbering will do, of course -- this way you can, of course, prove statements for all $m\in \{ k_0+ rn: r\in \mathbb{N}\}$, say.For the...
View ArticleCan mathematical inductions work for other sets?
I know that induction works only for the natural numbers $\mathbb{N}$.We first have to prove the base case. And we then prove that if the statement $p(k)$ holds then $\color{blue}{\textbf{p(k+1)}}$...
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