Axiomatic system

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Logic/8JnpJLtt">logic</a>, an axiomatic system is a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of <a href="/facts/Formal_language/crDTyP8q">formal statements</a> (i.e. <a href="/facts/Axiom/mLjP14Mc">axioms</a>) used to logically derive other statements such as <a href="/facts/Lemma_(mathematics)/zchvB1Ax">lemmas</a> or <a href="/facts/Theorem/f2Pe1FMD">theorems</a>. A <a href="/facts/Formal_proof/6nEhV4cw">proof</a> within an axiom system is a sequence of <a href="/facts/Deductive_reasoning/HadA1zWS">deductive steps</a> that establishes a new statement as a consequence of the axioms. An axiom system is called <a href="/facts/Completeness_(logic)/GzK9zLnu">complete</a> with respect to a property if every formula with the property can be derived using the axioms. The more general term <a href="/facts/Theory_(mathematical_logic)/mYe5gQpI">theory</a> is at times used to refer to an axiomatic system and all its derived theorems.
</p><p>In its pure form, an axiom system is effectively a syntactic construct and does not by itself refer to (or depend on) a <a href="/facts/Structure_(mathematical_logic)/GsMx1fEb">formal structure</a>, although axioms are often defined for that purpose. The more modern field of <a href="/facts/Model_theory/vLWVV8sQ">model theory</a> refers to mathematical structures. The relationship between an axiom systems and the models that correspond to it is often a major issue of interest.
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Axiomatic system open-in-new

Axiomatic system