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# first-order theory

In what follows, references to sentences 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 $\varphi$, $T\vdash\varphi$ 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 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 $\Delta$ 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.

## Mathematics Subject Classification

03C07*no label found*03B10

*no label found*

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