Mathematical logic and foundations Theorem

Principle of mathematical induction for positive integers

Statement

For each positive integer $n$ , let $S_n$ be a statement that is either true or false. Suppose that $S_1$ is true. Suppose also that for every positive integer $k$ , the truth of $S_k$ forces the truth of $S_{k+1}$ . Then $S_n$ is true for every positive integer $n$ .

Remarks

This is the form of induction that starts at $1$ . The proof of $S_1$ is the base case. The proof that $S_k$ implies $S_{k+1}$ is the induction step, and the assumption $S_k$ is the induction hypothesis.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

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Mathematical induction concept
Principle of mathematical induction for positive integers statement

Tag	`NZ273TPSG4`
URI	`m2://NZ273TPSG4`
Kind	statement, theorem
Writing	W1 stated
MSC	03F30
Status	This copy of the site has no build data. Run `m2 status NZ273TPSG4`.
Uses	Mathematical induction
Changed	4 November 2026

Principle of mathematical induction for positive integers

Statement

Remarks

Reading path

Further reading

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