strong induction

强数学归纳法

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

Translation evidence: Literature usage

Meaning in context

The principle that proving $P(n)$ from all $P(k)$ with $k<n$, for each $n\in\mathbb N$, proves $P$ everywhere.

对每个自然数 n，在假设所有 k<n 的情形成立时证明 n 的情形，即可得到全部情形成立的归纳原理。

Example

Factorization existence for $n\geq2$: a composite $n=ab$ uses the earlier cases $2\leq a,b<n$.

Usage and distinctions

The n=0 obligation is not omitted; its earlier-case hypothesis is empty.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

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

Translation sources

1. 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：§6.3 强数学归纳法原理（名称证据，非正文定义证据） · Checked 2026-10-14

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