nLab first-order theory

Idea

A first-order theory is a theory written in the language of first-order logic i.e it is a set of formulas or sequents (or generally, axioms over a signature) whose quantifiers and variables range over individuals of the underlying domain, but not over subsets of individuals nor over functions or relations of individuals etc.

A first-order theory is called infinitary when the expressions contain infinite disjunctions or conjunctions, else it is called finitary.

Another name for a first-order theory is elementary theory which is found in the older literature but by now has gone extinct, though it has left its traces in names like elementary topos, elementary embedding, ETCC etc.

The characterization of set-theoretic models of (finitary) first-order theories is the topic of traditional model theory.

Examples

Formulations of set theory are usually first-order theories (see first-order set theory), such as

• material set theory: ZFC

• structural set theory: fully formal ETCS

Remark

In particular, the precise formulation of the former in first-order predicate logic by Skolem in 1922 provided the impetus for the hold that first-order predicate logic gained over mathematical logic in the following.

It is somewhat ironic that classical first-order logic owes its promotion to prominence to intuitionistic mathematicians like Weyl (1910,1918) and Skolem.

Properties

Proposition

Let $\mathbb{T}$ be a finitary first-order theory. There exists a Grothendieck topos $\mathcal{E}$ and a $\mathbb{T}$-model $N_{\mathbb{T}}$ in $\mathcal{E}$ such that the first-order sequents satisfied in $N_{\mathbb{T}}$ are precisely those provable in $\mathbb{T}$.

This result due to P. Freyd is discussed in Johnstone (2002, p.900) or Freyd-Scedrov (1990, p.130).

If $\mathbb{T}$ is a theory over a countable signature, then $\mathcal{E}$ can be taken as the topos $Sh(2^{\mathbb{N}})$ of sheaves on the Cantor space $2^{\mathbb{N}}$. This completeness result shows that $Sh(2^{\mathbb{N}})$ plays in constructive first-order logic a role similar to the role of $Set$ in classical first-order logic.

Related entries

• dependent type theory

• second-order logic

• geometric theory

• coherent logic

• Skolem paradox

References

• Carsten Butz, Peter Johnstone, Classifying toposes for first-order theories , APAL 91 (1998) pp.33-58.

• P. J. Freyd, A. Scedrov, Categories, Allegories , North-Holland Amsterdam 1990. (pp.129ff)

• Peter Johnstone, Sketches of an Elephant II , Oxford UP 2002. (D1.3, D3.1, pp.899f)