nLab complete theory

Idea

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

Definition (in classical model theory)

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

The case of geometric logic

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

Proposition

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

This occurs as remark 2.5 in Caramello (2012).

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

Related entries

• Gödel's incompleteness theorem

• compactness theorem

• model complete theory

• substructure completeness

• two-valued topos

References

• Olivia Caramello, Atomic toposes and countable categoricity , Appl. Cat. Struc. 20 no. 4 (2012) pp.379-391. (arXiv:0811.3547)

• C. C. Chang, H. J. Keisler, Model theory , North-Holland Amsterdam 1973.