mathematical induction

数学归纳法

Foundations & Logic · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For $P$ on $\mathbb N$ , the principle that $P(0)$ and every implication $P(n)\to P(n+1)$ together yield $\forall n\,P(n)$ .

自然数上的证明原理：零处成立且从任意 n 处成立能推出 n+1 处成立，则所有自然数处均成立。

$3\mid(4^n-1)$ follows from the base $n=0$ and $4^{n+1}-1=4(4^n-1)+3$ .

A successor implication without a true base case proves nothing about all indices.

Avigad, Hua, Lewis and van Doorn — Logic and Proof§17.1 The Principle of Induction; §17.2 Variants of Induction · Checked 2026-10-14

高等教育出版社 — 数学证明通往高等数学的阶梯（第4版）影印版，官方中文目录官方中文目录：第6章数学归纳法；§6.1 数学归纳法原理（名称证据，非正文定义证据） · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.