# 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 $\phi $, $T\u22a2\phi $ only if $\phi \in T$.

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 ${T}^{\u22a2}:=\{\phi \in L\mid T\u22a2\phi \}$. Furthermore, ${T}^{\u22a2}$ 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 ${T}^{\u22a2}$.

Definition. A theory $T$ is *consistent* if and only
if for some sentence $\phi $, $T\u22a2\u0338\phi $.
Otherwise, $T$ is *inconsistent*. A sentence
$\phi $ is *consistent with $T$* if and only if the
theory $T\cup \{\phi \}$ is consistent.

Definition. A theory $T$ is *complete ^{}* if and only
if $T$ is consistent and for each sentence $\phi $, either $\phi \in T$
or $\mathrm{\neg}\phi \in T$.

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 $\phi $, $\phi \notin T$ only if
$T\cup \{\phi \}$ 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 $\mathrm{\Delta}$ such that $\mathrm{\Delta}\u22a2T$ (every sentence of $T$ is a logical consequence of
$\mathrm{\Delta}$), and *finitely axiomatizable ^{}* if $\mathrm{\Delta}$ can be made finite. Every complete axiomatizable theory $T$ is decidable;
that is, there is an algorithm that given a sentence $\phi $ as
input yields $0$ if $\phi \in T$, 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 |