first-order theory

Definition. A theory $T$ is a deductively closed set of sentences in $L$; that is, a set $T$ such that for each sentence $\varphi$, $T\vdash\varphi$ only if $\varphi\in T$.

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 $T^{\vdash}:=\{\varphi\in L\mid T\vdash\varphi\}$. Furthermore, $T^{\vdash}$ is unique (it is the smallest deductively closed theory including $T$), and any structure  $M$ is a model of $T$ iff it is a model of $T^{\vdash}$.

Definition. A theory $T$ is consistent if and only if for some sentence $\varphi$, $T\not\vdash\varphi$. Otherwise, $T$ is inconsistent. A sentence $\varphi$ is consistent with $T$ if and only if the theory $T\cup\{\varphi\}$ is consistent.

Definition. A theory $T$ is complete     if and only if $T$ is consistent and for each sentence $\varphi$, either $\varphi\in T$ or $\neg\varphi\in T$.

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 $\varphi$, $\varphi\not\in T$ only if $T\cup\{\varphi\}$ is inconsistent. See this entry (http://planetmath.org/MaximallyConsistent) 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 (http://planetmath.org/DecidableSet) subset $\Delta$ such that $\Delta\vdash T$ (every sentence of $T$ is a logical consequence of $\Delta$), and finitely axiomatizable  if $\Delta$ can be made finite. Every complete axiomatizable theory $T$ is decidable; that is, there is an algorithm that given a sentence $\varphi$ as input yields $0$ if $\varphi\in T$, 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