Answer by Some Guy for Why do we do mathematical induction only for positive...
If you let $P$ be your initial proof, and $ind : PROOFS \to PROOFS$ be your induction step, then the set $PP$ of all the proofs you can inductively prove from them is $\{P,ind(P),ind(ind(P)),...\}$. If...
View ArticleAnswer by Milo Brandt for Why do we do mathematical induction only for...
Induction really only requires that we can always make some sort of path between our base case and the cases we want to prove. The normal form of induction makes a path that looks like:$$\rightarrow...
View ArticleAnswer by Jack M for Why do we do mathematical induction only for positive...
All of the contexts in which you propose to apply induction are implicitly using the positive integers anyway.For instance, suppose we have a property $P$, and we prove that $P(x)\implies P(x+\frac 1...
View ArticleAnswer by Mohsen Shahriari for Why do we do mathematical induction only for...
$ \def \t {\quad \rightarrow \quad} $Let's forget about natural numbers for a while and take a look at what mathematical induction is proving. At the base step, it's proven that a property, say $ P $,...
View ArticleAnswer by drhab for Why do we do mathematical induction only for positive...
Induction works on a set $A$ that is equipped with a relation $R\subseteq A\times A$ such that every non-empty subset $B$ of $A$ contains an element $b\in B$ with $\{x\in A\mid x \mathrel{R} b\}\cap...
View ArticleAnswer by Mauro ALLEGRANZA for Why do we do mathematical induction only for...
Proof by induction"works" because the inductive step is :for any $n$, if $P(n)$ holds, then $P(n+1)$ holds.The key feature here is that for any $n$ we have a successor : $n+1$.If we move to e.g....
View ArticleWhy do we do mathematical induction only for positive whole numbers?
After reading a question made here, I wanted to ask "Why do we do mathematical induction only for positive whole numbers?"I know we usually start our mathematical induction by proving it works for...
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