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Axiom
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<p>An axiom, postulate, or assumption is a <a href="/facts/Statement_(logic)/CjpWAyxk">statement</a> that is taken to be <a href="/facts/Truth/NyQmBtUC">true</a>, to serve as a <a href="/facts/Premise/HluL8lcd">premise</a> or starting point for further reasoning and arguments. The word comes from the <a href="/facts/Ancient_Greek/LIK7NlUh">Ancient Greek</a> word ἀξίωμα (<i>axíōma</i>), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. </p><p>The precise <a href="/facts/Definition/5SlGKKDJ">definition</a> varies across fields of study. In <a href="/facts/Classic_philosophy/73cd664F">classic philosophy</a>, an axiom is a statement that is so <a href="/facts/Self-evidence/I9WhGuT5">evident</a> or well-established, that it is accepted without controversy or question. In modern <a href="/facts/Logic/8JnpJLtt">logic</a>, an axiom is a premise or starting point for reasoning. </p><p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an <i>axiom</i> may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (<i>A</i> and <i>B</i>) implies <i>A</i>), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example <i>a</i> + 0 = <i>a</i> in integer arithmetic. </p><p>Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the <a href="/facts/Parallel_postulate/xhSxUX4u">parallel postulate</a> in <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain. </p><p>Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the <a href="/facts/Philosophy_of_mathematics/z7xBcdWA">philosophy of mathematics</a>. </p>