TITLE: Why do we do mathematical induction only for positive whole numbers?
QUESTION [21 upvotes]: 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 $0,1$ because it is usually easiest, but why do we only prove it works for $k+1$?
Why not prove it works for $k+\frac12$, assuming it works for $k=0$.
Applying some limits into this, why don't we prove that it works for $\lim_{n\to0}k+n$?
I want to do this because I realized that mathematical induction will only prove it works for $x=0,1,2,3,\dots$. assuming we start at $x=0$, meaning it is unknown if it will work for $0<x<1$ for all $x$.
And why not do $k-1$?  This way we can prove it for negative numbers as well, right?
What's so special about our positive whole numbers when doing mathematical induction?
And then this will only work for real numbers because we definitely can't do it for complex numbers, right?
And what about mapping values so that it becomes one of the above?  That is, change it so that we have $x\to\frac x2$?  Then proving for $x+1$ becomes a proof for all $x$ that is a multiple of $\frac12$!?

REPLY [2 votes]: 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 you define $X : PROOFS \to (PROOFS \to PROOFS) \to \mathbb{N} \to PROOFS$ such that $X(P,ind,0) = P$ and $X(P,ind,n) = ind(X(P,ind,n-1))$, then you can set $ind' = X(P,ind)$ and rewrite $PP$ as $\{ind'(0),ind'(1),...\}$.
Since you can always reduce any mathematical induction to an induction over the integers, you can assume the latter without loss of generality. Nothing is keeping you from defining induction over another countable set, however you must beware that, discounting proofs of infinite size, you can never inductively prove a property for all real $x$ such that $0 < x < 1$, since they do not form a countable set.
Edit: Sorry, I didn't realize the answers above already covered much of the ground, it's just a really good question and I got carried away ^^ If you want to learn more about what makes whole numbers special, I suggest you read about Gödel numberings, which establish interesting relations between the natural numbers and proof objects themselves. With such a numbering, you can freely replace $PROOFS$ by $\mathbb{N}$ in my earlier answer, perhaps making it much clearer.
Regarding the notion of countable set, it's just a fancy word for "a set whose elements can be listed", or equivalently "a set for which there is a bijection with the positive integers". Examples of such sets are $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and the set $PROOFS$ of all mathematical proofs used above. Examples of non-countable sets include $\mathbb{R}$ and $\mathbb{C}$ (see Cantor's diagonal argument for an intuitive proof of that last claim), and of course any nonempty interval $]a,b[$ in $\mathbb{R}$, since it can be bidirectionally mapped to $\mathbb{R}$ itself using the function $f(x) = tan(\frac{\pi (2x-a-b)}{2(b-a)})$, whose inverse is $f'(x) = \frac{a+b}{2} + \frac{2(b-a)arctan(x)}{2\pi}$. Hence my earlier claim that the real interval $]0,1[$ isn't countable.