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Metatheorem
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<h2 id="examples">Examples</h2> <p>Examples of metatheorems include: </p> <ul><li>The <a href="/facts/Deduction_theorem/wy4veysT">deduction theorem</a> for <a href="/facts/First-order_logic/lcISqWIg">first-order logic</a> says that a sentence of the form φ→ψ is provable from a set of axioms <i>A</i> if and only if the sentence ψ is provable from the system whose axioms consist of φ and all the axioms of <i>A</i>.</li> <li>The <a href="/facts/Von_Neumann%25E2%2580%2593Bernays%25E2%2580%2593G%25C3%25B6del_set_theory/5wLhAIMz">class existence theorem</a> of <a href="/facts/Von_Neumann%25E2%2580%2593Bernays%25E2%2580%2593G%25C3%25B6del_set_theory/5wLhAIMz">von Neumann–Bernays–Gödel set theory</a> states that for every formula whose <a href="/facts/Quantifier_(logic)/bgmvwxoo">quantifiers</a> range only over sets, there is a <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> consisting of the <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> satisfying the formula.</li> <li><a href="/facts/Consistency_proof/IHkdnA9M">Consistency proofs</a> of systems such as <a href="/facts/Peano_arithmetic/DhhDnNug">Peano arithmetic</a>.</li> <li><a href="/facts/G%25C3%25B6del%2527s_completeness_theorem/yq02MUyw">Gödel's completeness theorem</a> states that <a href="/facts/First-order_logic/lcISqWIg">first-order logic</a> is <a href="/facts/Completeness_(logic)/GzK9zLnu">complete</a>.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Metamathematics/yeIm8vqC">Metamathematics</a></li> <li><a href="/facts/Use%25E2%2580%2593mention_distinction/h9hlBElb">Use–mention distinction</a></li></ul> <ul><li><a href="/facts/Geoffrey_Hunter_(logician)/SBTYATog">Geoffrey Hunter</a> (1969), <i>Metalogic</i>.</li> <li>Alasdair Urquhart (2002), "Metatheory", <i>A companion to philosophical logic</i>, Dale Jacquette (ed.), p. 307</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.encyclopediaofmath.org/index.php/Meta-theorem"><i>Meta-theorem</i> at Encyclopaedia of Mathematics</a></li> <li>Barile, Margherita. <a href="https://mathworld.wolfram.com/Metatheorem.html">"Metatheorem"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>