first-order compactness theorem

紧致性定理

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

Translation evidence: Literature usage

Meaning in context

A set of classical first-order sentences has a model if and only if every finite subset has a model.

经典一阶语句集有模型，当且仅当其每个有限子集有模型。

Example

The axioms $c>\bar n$ for all standard $n$, together with arithmetic, have a nonstandard model.

Usage and distinctions

Finite fragments may use different models; the theorem does not force a model on a fixed domain.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Jeremy Avigad — Logic and Computation (January 2002)§6.3, Theorems 6.3.1–6.3.2; §7.2, Theorem 7.2.1 · Checked 2026-10-14

Translation sources

1. 杨睿之（复旦大学）— 数理逻辑，第17讲PDF p.22 定理「紧致性定理」 · Checked 2026-10-14

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