Definition. A theory is a deductively
closed set of sentences in ; that is, a set such that for each
sentence , only if .
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 under this definition can be “extended” to a deductively closed theory . Furthermore, is unique (it is the smallest deductively closed theory including ), and any structure is a model of iff it is a model of .
Definition. A theory is consistent if and only
if for some sentence , .
Otherwise, is inconsistent. A sentence
is consistent with if and only if the
theory is consistent.
Definition. A theory is complete if and only
if is consistent and for each sentence , either
Lemma. A consistent theory is complete if and only if is
maximally consistent. That is, is complete if and only if for
each sentence , only if
is inconsistent. See this entry (http://planetmath.org/MaximallyConsistent) for a proof.
Theorem. (Tarski) Every consistent theory is included in a complete theory.
Proof : Use Zorn’s lemma on the set of consistent
theories that include .
Remark. A theory is axiomatizable if and only if includes a decidable (http://planetmath.org/DecidableSet) subset such that (every sentence of is a logical consequence of ), and finitely axiomatizable if can be made finite. Every complete axiomatizable theory is decidable; that is, there is an algorithm that given a sentence as input yields if , and otherwise.
|Date of creation||2013-03-22 12:43:04|
|Last modified on||2013-03-22 12:43:04|
|Last modified by||CWoo (3771)|
|Synonym||first order theory|
|Defines||finitely axiomatizable theory|