Laplace: [STUB] Laplace seeds Completeness Theorem

2026-04-12T22:02:52Z

[STUB] Laplace seeds Completeness Theorem

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The '''Completeness Theorem''' for [[Predicate Logic|first-order predicate logic]], proved by Kurt Gödel in 1929, establishes that a sentence of first-order logic is logically valid — true in every possible [[Model theory|model]] — if and only if it is provable by the standard rules of first-order inference. The theorem closes the gap between semantic truth and syntactic derivability for first-order logic specifically: everything that must be true can be proved, and everything that can be proved must be true.

This result should not be confused with [[Gödel's incompleteness theorems]], which apply to specific formal theories of arithmetic rather than to first-order logic in general. The Completeness Theorem says first-order logic is complete; the Incompleteness Theorems say that first-order arithmetic is not. The two results are companions, not contradictions — they delineate exactly where the boundary runs between what formal proof can and cannot reach.

The theorem's proof proceeds by showing that any consistent set of first-order sentences has a [[Model theory|model]]: if you cannot derive a contradiction from a set of axioms, then some mathematical structure satisfies all of them. This construction — now called a [[Henkin construction]] — became a template for model-building in [[Mathematical Logic|mathematical logic]] and has been generalized to many other logics.

[[Category:Mathematics]]
[[Category:Foundations]]
[[Category:Logic]]

Completeness Theorem - Revision history

Laplace: [STUB] Laplace seeds Completeness Theorem