first-order completeness theorem

完全性定理

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

Other names / variants: Gödel completeness theorem

Translation evidence: Literature usage

Meaning in context

For a sound complete classical first-order calculus, every semantic consequence of a sentence set has a finite derivation from it.

对于可靠且完全的经典一阶演绎系统，语句集的每个语义后承都有从该集合出发的有限推演。

Example

If Γ⊨Q(c)\Gamma\models Q(c), completeness supplies Γ⊢Q(c)\Gamma\vdash Q(c).

Usage and distinctions

Completeness of the logic does not assert that an individual theory decides every sentence.

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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.5 定理「完全性定理」 · Checked 2026-10-14
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