In 370 BC, Plato's Parmenides may have contained traces of an early example of an implicit inductive proof, however, the earliest implicit proof by mathematical induction was written by al-Karaji around 1000 AD, who … See more

The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps:
1. See more

In practice, proofs by induction are often structured differently, depending on the exact nature of the property to be proven. All variants of … See more

One variation of the principle of complete induction can be generalized for statements about elements of any well-founded set, … See more

Sum of consecutive natural numbers
Mathematical induction can be used to prove the following statement P(n) for all natural numbers n.
This states a … See more

In second-order logic, one can write down the "axiom of induction" as follows:
In words, the base case P(0) and the induction step … See more

The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms. It is strictly stronger than the well-ordering principle in the context of the … See more