first-order theory

In what follows, references to sentencesMathworldPlanetmath and sets of sentences are all relative to some fixed first-order language L.

Definition. A theory T is a deductively closed set of sentences in L; that is, a set T such that for each sentence φ, Tφ only if φT.

Remark. Some authors do not require that a theory be deductively closed. Therefore, a theory is simply a set of sentences. This is not a cause for alarm, since every theory T under this definition can be “extended” to a deductively closed theory T:={φLTφ}. Furthermore, T is unique (it is the smallest deductively closed theory including T), and any structureMathworldPlanetmath M is a model of T iff it is a model of T.

Definition. A theory T is consistent if and only if for some sentence φ, T⊬φ. Otherwise, T is inconsistent. A sentence φ is consistent with T if and only if the theory T{φ} is consistent.

Definition. A theory T is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if and only if T is consistent and for each sentence φ, either φT or ¬φT.

Lemma. A consistent theory T is complete if and only if T is maximally consistent. That is, T is complete if and only if for each sentence φ, φT only if T{φ} is inconsistent. See this entry ( for a proof.

Theorem. (Tarski) Every consistent theory T is included in a complete theory.

Proof : Use Zorn’s lemma on the set of consistent theories that include T.

Remark. A theory T is axiomatizable if and only if T includes a decidable ( subset Δ such that ΔT (every sentence of T is a logical consequence of Δ), and finitely axiomatizablePlanetmathPlanetmath if Δ can be made finite. Every complete axiomatizable theory T is decidable; that is, there is an algorithm that given a sentence φ as input yields 0 if φT, and 1 otherwise.

Title first-order theory
Canonical name FirstorderTheory
Date of creation 2013-03-22 12:43:04
Last modified on 2013-03-22 12:43:04
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 19
Author CWoo (3771)
Entry type Definition
Classification msc 03C07
Classification msc 03B10
Synonym first order theory
Related topic PropertiesOfConsistency
Related topic MaximallyConsistent
Defines theory
Defines complete theory
Defines axiomatizable theory
Defines deductively closed
Defines finitely axiomatizable theory