Editing Axiom (section)

===<span id="role">Role in mathematical logic</span>===

====Deductive systems and completeness====
A '''[[deductive system]]''' consists of a set <math>\Lambda</math> of logical axioms, a set <math>\Sigma</math> of non-logical axioms, and a set <math>\{(\Gamma, \phi)\}</math> of ''rules of inference''.  A desirable property of a deductive system is that it be '''complete'''.  A system is said to be complete if, for all formulas <math>\phi</math>,
<div class="center">
<math>\text{if }\Sigma \models \phi\text{ then }\Sigma \vdash \phi</math>
</div>

that is, for any statement that is a ''logical consequence'' of <math>\Sigma</math> there actually exists a ''deduction'' of the statement from <math>\Sigma</math>.  This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". [[Gödel's completeness theorem]] establishes the completeness of a certain commonly used type of deductive system.

Note that "completeness" has a different meaning here than it does in the context of [[Gödel's first incompleteness theorem]], which states that no ''recursive'', ''consistent'' set of non-logical axioms <math>\Sigma</math> of the Theory of Arithmetic is ''complete'', in the sense that there will always exist an arithmetic statement <math>\phi</math> such that neither <math>\phi</math> nor <math>\lnot\phi</math> can be proved from the given set of axioms.

There is thus, on the one hand, the notion of ''completeness of a deductive system'' and on the other hand that of ''completeness of a set of non-logical axioms''.  The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.