nLab
complete theory

Contents

Contents

Idea

Deductive completeness is maximality condition on a logical theory 𝕋\mathbb{T} saying that the deductive closure of 𝕋\mathbb{T} is as large as possible within the bounds of consistency.

Definition (in classical model theory)

Let 𝕋\mathbb{T} be a consistent logical theory over signature Σ\Sigma. 𝕋\mathbb{T} is called complete if for any sentence σ\sigma over Σ\Sigma either 𝕋σ\mathbb{T}\vdash \sigma or 𝕋σ\mathbb{T}\cup\sigma is inconsistent.

The case of geometric logic

A geometric theory 𝕋\mathbb{T} over a signature Σ\Sigma is called complete if for any geometric sentence σ\sigma over Σ\Sigma is either 𝕋\mathbb{T}-provably equivalent to \top or to \bot, but not both.

Proposition

A geometric theory 𝕋\mathbb{T} is complete iff the classifying topos of 𝕋\mathbb{T} is two-valued.

This occurs as remark 2.5 in Caramello (2012).

Remark: a first-order theory 𝕋\mathbb{T} is complete in the sense of classical model theory iff its Morleyization is complete in the sense of geometric logic.

References

Created on June 14, 2020 at 17:14:59. See the history of this page for a list of all contributions to it.