nLab complete theory

Contents

model theory

Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

Contents

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.

References

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