For questions about mathematical induction, a method of mathematical proof. Mathematical induction generally proceeds by proving a statement for some integer, called the *base case*, and then proving that if it holds for one integer then it holds for the next integer. This tag is primarily meant for questions about induction over natural numbers but is also appropriate for other kinds of induction such as transfinite, structural, double, backwards, etc.
Mathematical induction is a form of deductive reasoning. Its most common use is induction over well ordered sets, such as natural numbers, or ordinals. While induction can be expanded to class relations which are well-founded, this tag is aimed mostly at questions about induction over natural numbers.
In general use, induction means inference from the particular to the general. This is used in terms such as inductive reasoning, which involves making an inference about the unknown based on some known sample. Mathematical induction is not true induction in this sense, but is rather a form of concrete, rigorous proof.
Induction over the natural numbers generally proceeds with a base case and an inductive step:
- (Base Case) First prove the statement for the base case, which is usually $n=0$ or $n=1$.
- (Inductive Step) Next, assume that the statement is true for an input $n$, and prove that it is true for the input $n+1$.
The following variant goes without a base case: Assuming the statement is true for all $n\in\mathbb N$ with $n < N$, prove that is true for $N$, too. This has to be done for all $N\in\mathbb N$.