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id="toc-Axiom_of_choice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Axiom_of_choice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Axiom of choice</span> </div> </a> <button aria-controls="toc-Axiom_of_choice-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Axiom of choice subsection</span> </button> <ul id="toc-Axiom_of_choice-sublist" class="vector-toc-list"> <li id="toc-Equivalents_of_AC" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalents_of_AC"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Equivalents of AC</span> </div> </a> <ul id="toc-Equivalents_of_AC-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stronger_than_AC" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stronger_than_AC"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Stronger than AC</span> </div> </a> <ul id="toc-Stronger_than_AC-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weaker_than_AC" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weaker_than_AC"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Weaker than AC</span> </div> </a> <ul id="toc-Weaker_than_AC-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternates_incompatible_with_AC" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternates_incompatible_with_AC"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Alternates incompatible with AC</span> </div> </a> <ul id="toc-Alternates_incompatible_with_AC-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_axioms_of_mathematical_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_axioms_of_mathematical_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Other axioms of mathematical logic</span> </div> </a> <ul id="toc-Other_axioms_of_mathematical_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Geometry</span> </div> </a> <ul id="toc-Geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_axioms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Other axioms</span> </div> </a> <ul id="toc-Other_axioms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" 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In <a href="/wiki/Epistemology" title="Epistemology">epistemology</a>, the word <i>axiom</i> is understood differently; see <a href="/wiki/Axiom" title="Axiom">axiom</a> and <a href="/wiki/Self-evidence" title="Self-evidence">self-evidence</a>. Individual axioms are almost always part of a larger <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic system</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="ZF_(the_Zermelo–Fraenkel_axioms_without_the_axiom_of_choice)"><span id="ZF_.28the_Zermelo.E2.80.93Fraenkel_axioms_without_the_axiom_of_choice.29"></span>ZF (the <a href="/wiki/Zermelo%E2%80%93Fraenkel_axioms" class="mw-redirect" title="Zermelo–Fraenkel axioms">Zermelo–Fraenkel axioms</a> without the axiom of choice)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=1" title="Edit section: ZF (the Zermelo–Fraenkel axioms without the axiom of choice)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Together with the axiom of choice (see below), these are the</i> <a href="/wiki/De_facto" title="De facto">de facto</a> <i>standard axioms for contemporary <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> or <a href="/wiki/Set_theory" title="Set theory">set theory</a>. They can be easily adapted to analogous theories, such as <a href="/wiki/Mereology" title="Mereology">mereology</a>.</i> </p> <ul><li><a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">Axiom of extensionality</a></li> <li><a href="/wiki/Axiom_of_empty_set" title="Axiom of empty set">Axiom of empty set</a></li> <li><a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">Axiom of pairing</a></li> <li><a href="/wiki/Axiom_of_union" title="Axiom of union">Axiom of union</a></li> <li><a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">Axiom of infinity</a></li> <li><a href="/wiki/Axiom_schema_of_replacement" title="Axiom schema of replacement">Axiom schema of replacement</a></li> <li><a href="/wiki/Axiom_of_power_set" title="Axiom of power set">Axiom of power set</a></li> <li><a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">Axiom of regularity</a></li> <li><a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">Axiom schema of specification</a></li></ul> <p>See also <a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo set theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Axiom_of_choice"><a href="/wiki/Axiom_of_choice" title="Axiom of choice">Axiom of choice</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=2" title="Edit section: Axiom of choice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>With the Zermelo–Fraenkel axioms above, this makes up the system <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZFC</a> in which most mathematics is potentially formalisable.</i> </p> <div class="mw-heading mw-heading3"><h3 id="Equivalents_of_AC">Equivalents of AC</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=3" title="Edit section: Equivalents of AC"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Hausdorff_maximality_theorem" class="mw-redirect" title="Hausdorff maximality theorem">Hausdorff maximality theorem</a></li> <li><a href="/wiki/Well-ordering_theorem" title="Well-ordering theorem">Well-ordering theorem</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Stronger_than_AC">Stronger than AC</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=4" title="Edit section: Stronger than AC"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Axiom_of_global_choice" title="Axiom of global choice">Axiom of global choice</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Weaker_than_AC">Weaker than AC</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=5" title="Edit section: Weaker than AC"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">Axiom of countable choice</a></li> <li><a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">Axiom of dependent choice</a></li> <li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Axiom_of_uniformization" class="mw-redirect" title="Axiom of uniformization">Axiom of uniformization</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Alternates_incompatible_with_AC">Alternates incompatible with AC</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=6" title="Edit section: Alternates incompatible with AC"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Axiom_of_real_determinacy" title="Axiom of real determinacy">Axiom of real determinacy</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Other_axioms_of_mathematical_logic">Other axioms of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=7" title="Edit section: Other axioms of mathematical logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_axioms" class="mw-redirect" title="Von Neumann–Bernays–Gödel axioms">Von Neumann–Bernays–Gödel axioms</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a> and <a href="/wiki/Continuum_hypothesis#The_generalized_continuum_hypothesis" title="Continuum hypothesis">its generalization</a></li> <li><a href="/wiki/Freiling%27s_axiom_of_symmetry" title="Freiling's axiom of symmetry">Freiling's axiom of symmetry</a></li> <li><a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">Axiom of determinacy</a></li> <li><a href="/wiki/Axiom_of_projective_determinacy" title="Axiom of projective determinacy">Axiom of projective determinacy</a></li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a></li> <li><a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">Axiom of constructibility</a></li> <li><a href="/wiki/Rank-into-rank" title="Rank-into-rank">Rank-into-rank</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_axioms" class="mw-redirect" title="Kripke–Platek axioms">Kripke–Platek axioms</a></li> <li><a href="/wiki/Diamond_principle" title="Diamond principle">Diamond principle</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Geometry"><a href="/wiki/Geometry" title="Geometry">Geometry</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=8" title="Edit section: Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Parallel_postulate" title="Parallel postulate">Parallel postulate</a></li> <li><a href="/wiki/Birkhoff%27s_axioms" title="Birkhoff's axioms">Birkhoff's axioms</a> (4 axioms)</li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a> (20 axioms)</li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's axioms</a> (10 axioms and 1 schema)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Other_axioms">Other axioms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_axioms&action=edit&section=9" title="Edit section: Other axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Axiom_of_Archimedes" class="mw-redirect" title="Axiom of Archimedes">Axiom of Archimedes</a> (<a href="/wiki/Real_number" title="Real number">real number</a>)</li> <li><a href="/wiki/Axiom_of_countability" title="Axiom of countability">Axiom of countability</a> (<a href="/wiki/Topology" title="Topology">topology</a>)</li> <li><a href="/wiki/Dirac%E2%80%93von_Neumann_axioms" title="Dirac–von Neumann axioms">Dirac–von Neumann axioms</a></li> <li><a href="/wiki/Fundamental_axiom_of_analysis" class="mw-redirect" title="Fundamental axiom of analysis">Fundamental axiom of analysis</a> (<a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>)</li> <li><a href="/wiki/Gluing_axiom" title="Gluing axiom">Gluing axiom</a> (<a href="/wiki/Sheaf_theory" class="mw-redirect" title="Sheaf theory">sheaf theory</a>)</li> <li><a href="/wiki/Haag%E2%80%93Kastler_axioms" class="mw-redirect" title="Haag–Kastler axioms">Haag–Kastler axioms</a> (<a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>)</li> <li><a href="/wiki/Huzita%27s_axioms" class="mw-redirect" title="Huzita's axioms">Huzita's axioms</a> (<a href="/wiki/Origami" title="Origami">origami</a>)</li> <li><a href="/wiki/Kuratowski_closure_axioms" title="Kuratowski closure axioms">Kuratowski closure axioms</a> (<a href="/wiki/Topology" title="Topology">topology</a>)</li> <li><a href="/wiki/Peano%27s_axioms" class="mw-redirect" title="Peano's axioms">Peano's axioms</a> (<a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>)</li> <li><a href="/wiki/Probability_axioms" title="Probability axioms">Probability axioms</a></li> <li><a href="/wiki/Separation_axiom" title="Separation axiom">Separation axiom</a> (<a href="/wiki/Topology" title="Topology">topology</a>)</li> <li><a href="/wiki/Wightman_axioms" title="Wightman axioms">Wightman axioms</a> (<a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>)</li> <li><a href="/wiki/Action_axiom" title="Action axiom">Action axiom</a> (<a href="/wiki/Praxeology" title="Praxeology">praxeology</a>)</li></ul> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 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