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skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-ბანახის-ტარსკის_პარადოქსი rootpage-ბანახის-ტარსკის_პარადოქსი skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">შინაარსზე გადასვლა</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="საიტი"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="მთავარი მენიუ" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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href="/wiki/%E1%83%95%E1%83%98%E1%83%99%E1%83%98%E1%83%9E%E1%83%94%E1%83%93%E1%83%98%E1%83%90:%E1%83%A0%E1%83%A9%E1%83%94%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%A1%E1%83%A2%E1%83%90%E1%83%A2%E1%83%98%E1%83%94%E1%83%91%E1%83%98"><span>რჩეული სტატიები</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%A8%E1%83%94%E1%83%9B%E1%83%97%E1%83%AE%E1%83%95%E1%83%94%E1%83%95%E1%83%98%E1%83%97" title="შემთხვევითი გვერდის გახსნა. [x]" accesskey="x"><span>შემთხვევითი გვერდი</span></a></li><li id="n-newpages" class="mw-list-item"><a href="/wiki/%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%90%E1%83%AE%E1%83%90%E1%83%9A%E1%83%98_%E1%83%92%E1%83%95%E1%83%94%E1%83%A0%E1%83%93%E1%83%94%E1%83%91%E1%83%98"><span>ახალი გვერდები</span></a></li> </ul> </div> </div> <div id="p-participation" class="vector-menu mw-portlet mw-portlet-participation" > <div class="vector-menu-heading"> მონაწილეობა </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-ინფორმაცია" class="mw-list-item"><a href="/wiki/%E1%83%95%E1%83%98%E1%83%99%E1%83%98%E1%83%9E%E1%83%94%E1%83%93%E1%83%98%E1%83%90:%E1%83%A1%E1%83%90%E1%83%96%E1%83%9D%E1%83%92%E1%83%90%E1%83%93%E1%83%9D%E1%83%94%E1%83%91%E1%83%98%E1%83%A1_%E1%83%92%E1%83%95%E1%83%94%E1%83%A0%E1%83%93%E1%83%94%E1%83%91%E1%83%98"><span>ინფორმაცია</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/%E1%83%95%E1%83%98%E1%83%99%E1%83%98%E1%83%9E%E1%83%94%E1%83%93%E1%83%98%E1%83%90:%E1%83%93%E1%83%90%E1%83%AE%E1%83%9B%E1%83%90%E1%83%A0%E1%83%94%E1%83%91%E1%83%90" title="სად გავიგოთ."><span>დახმარება</span></a></li><li id="n-ფორუმი" class="mw-list-item"><a href="/wiki/%E1%83%95%E1%83%98%E1%83%99%E1%83%98%E1%83%9E%E1%83%94%E1%83%93%E1%83%98%E1%83%90:%E1%83%A4%E1%83%9D%E1%83%A0%E1%83%A3%E1%83%9B%E1%83%98"><span>ფორუმი</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%91%E1%83%9D%E1%83%9A%E1%83%9D%E1%83%AA%E1%83%95%E1%83%9A%E1%83%98%E1%83%9A%E1%83%94%E1%83%91%E1%83%94%E1%83%91%E1%83%98" title="ვიკიპედიაში ბოლო ცვლილებების სია. 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type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="იერსახე" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">იერსახე</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu 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href="/w/index.php?title=%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%90%E1%83%9C%E1%83%92%E1%83%90%E1%83%A0%E1%83%98%E1%83%A8%E1%83%98%E1%83%A1_%E1%83%A8%E1%83%94%E1%83%A5%E1%83%9B%E1%83%9C%E1%83%90&returnto=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1+%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" title="ჩვენ გთავაზობთ, შექმნათ ანგარიში და შეხვიდეთ სისტემაში; თუმცა ეს არ არის აუცილებელი" class=""><span>ანგარიშის შექმნა</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%A8%E1%83%94%E1%83%A1%E1%83%95%E1%83%9A%E1%83%90&returnto=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1+%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" title="სასურველია დარეგისტრირდეთ, თუმცა აუცილებელი არ არის. [o]" accesskey="o" class=""><span>შესვლა</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="მეტი შესაძლებლობა" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="პირადი ხელსაწყოები" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">პირადი ხელსაწყოები</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="მომხმარებლის მენიუ" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ka.wikipedia.org&uselang=ka"><span>შემოწირულობები</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%90%E1%83%9C%E1%83%92%E1%83%90%E1%83%A0%E1%83%98%E1%83%A8%E1%83%98%E1%83%A1_%E1%83%A8%E1%83%94%E1%83%A5%E1%83%9B%E1%83%9C%E1%83%90&returnto=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1+%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" title="ჩვენ გთავაზობთ, შექმნათ ანგარიში და შეხვიდეთ სისტემაში; თუმცა ეს არ არის აუცილებელი"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>ანგარიშის შექმნა</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%A8%E1%83%94%E1%83%A1%E1%83%95%E1%83%9A%E1%83%90&returnto=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1+%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" title="სასურველია დარეგისტრირდეთ, თუმცა აუცილებელი არ არის. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>შესვლა</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> გვერდები სისტემიდან გამოსული რედაქტორებისათვის <a href="/wiki/%E1%83%95%E1%83%98%E1%83%99%E1%83%98%E1%83%9E%E1%83%94%E1%83%93%E1%83%98%E1%83%90:%E1%83%93%E1%83%90%E1%83%AE%E1%83%9B%E1%83%90%E1%83%A0%E1%83%94%E1%83%91%E1%83%90" aria-label="რედაქტირების შესახებ მეტის გაგება"><span>გაიგეთ მეტი</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%A9%E1%83%94%E1%83%9B%E1%83%98_%E1%83%AC%E1%83%95%E1%83%9A%E1%83%98%E1%83%9A%E1%83%98" title="რედაქტირებების სია შესრულებული ამ IP მისამართიდან [y]" accesskey="y"><span>წვლილი</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%A9%E1%83%94%E1%83%9C%E1%83%98_%E1%83%92%E1%83%90%E1%83%9C%E1%83%AE%E1%83%98%E1%83%9A%E1%83%95%E1%83%90" title="დისკუსია ამ ip მისამართის წაშლის შესახებ [n]" accesskey="n"><span>განხილვა</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eდამალვა\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- 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id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">შესავალი</div> </a> </li> <li id="toc-ლიტერატურა" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ლიტერატურა"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>ლიტერატურა</span> </div> </a> <ul id="toc-ლიტერატურა-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-რესურსები_ინტერნეტში" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#რესურსები_ინტერნეტში"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>რესურსები ინტერნეტში</span> </div> </a> <ul id="toc-რესურსები_ინტერნეტში-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-შენიშვნები" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#შენიშვნები"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>შენიშვნები</span> </div> </a> <ul id="toc-შენიშვნები-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="სარჩევი" 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="სარჩევის დამალვა/ჩვენება" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">სარჩევის დამალვა/ჩვენება</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">ბანახის-ტარსკის პარადოქსი</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="სხვაენოვან სტატიაზე გადასვლა. ხელმისაწვდომია 31 ენაზე" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-31" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">31 ენა</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%81%D8%A7%D8%B1%D9%82%D8%A9_%D8%A8%D8%A7%D9%86%D8%A7%D8%AE_%D8%AA%D8%A7%D8%B1%D8%B3%D9%83%D9%8A" title="مفارقة باناخ تارسكي — არაბული" lang="ar" hreflang="ar" data-title="مفارقة باناخ تارسكي" data-language-autonym="العربية" data-language-local-name="არაბული" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%BD%D0%B0_%D0%91%D0%B0%D0%BD%D0%B0%D1%85-%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D0%B8" title="Парадокс на Банах-Тарски — ბულგარული" lang="bg" hreflang="bg" data-title="Парадокс на Банах-Тарски" data-language-autonym="Български" data-language-local-name="ბულგარული" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Paradoxa_de_Banach-Tarski" title="Paradoxa de Banach-Tarski — კატალანური" lang="ca" hreflang="ca" data-title="Paradoxa de Banach-Tarski" data-language-autonym="Català" data-language-local-name="კატალანური" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Banach%C5%AFv%E2%80%93Tarsk%C3%A9ho_paradox" title="Banachův–Tarského paradox — ჩეხური" lang="cs" hreflang="cs" data-title="Banachův–Tarského paradox" data-language-autonym="Čeština" data-language-local-name="ჩეხური" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Banach-Tarski-Paradoxon" title="Banach-Tarski-Paradoxon — გერმანული" lang="de" hreflang="de" data-title="Banach-Tarski-Paradoxon" data-language-autonym="Deutsch" data-language-local-name="გერმანული" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox — ინგლისური" lang="en" hreflang="en" data-title="Banach–Tarski paradox" data-language-autonym="English" data-language-local-name="ინგლისური" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Paradoja_de_Banach-Tarski" title="Paradoja de Banach-Tarski — ესპანური" lang="es" hreflang="es" data-title="Paradoja de Banach-Tarski" data-language-autonym="Español" data-language-local-name="ესპანური" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Banachi-Tarski_paradoks" title="Banachi-Tarski paradoks — ესტონური" lang="et" hreflang="et" data-title="Banachi-Tarski paradoks" data-language-autonym="Eesti" data-language-local-name="ესტონური" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%A7%D8%B1%D8%A7%D8%AF%D9%88%DA%A9%D8%B3_%D8%A8%D8%A7%D9%86%D8%A7%D8%AE%E2%80%93%D8%AA%D8%A7%D8%B1%D8%B3%DA%A9%DB%8C" title="پارادوکس باناخ–تارسکی — სპარსული" lang="fa" hreflang="fa" data-title="پارادوکس باناخ–تارسکی" data-language-autonym="فارسی" data-language-local-name="სპარსული" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Banachin%E2%80%93Tarskin_paradoksi" title="Banachin–Tarskin paradoksi — ფინური" lang="fi" hreflang="fi" data-title="Banachin–Tarskin paradoksi" data-language-autonym="Suomi" data-language-local-name="ფინური" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Paradoxe_de_Banach-Tarski" title="Paradoxe de Banach-Tarski — ფრანგული" lang="fr" hreflang="fr" data-title="Paradoxe de Banach-Tarski" data-language-autonym="Français" data-language-local-name="ფრანგული" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1_%D7%A9%D7%9C_%D7%91%D7%A0%D7%9A-%D7%98%D7%A8%D7%A1%D7%A7%D7%99" title="הפרדוקס של בנך-טרסקי — ებრაული" lang="he" hreflang="he" data-title="הפרדוקס של בנך-טרסקי" data-language-autonym="עברית" data-language-local-name="ებრაული" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Paradoks_Banacha_i_Tarskoga" title="Paradoks Banacha i Tarskoga — ხორვატული" lang="hr" hreflang="hr" data-title="Paradoks Banacha i Tarskoga" data-language-autonym="Hrvatski" data-language-local-name="ხორვატული" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Banach%E2%80%93Tarski-paradoxon" title="Banach–Tarski-paradoxon — უნგრული" lang="hu" hreflang="hu" data-title="Banach–Tarski-paradoxon" data-language-autonym="Magyar" data-language-local-name="უნგრული" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Paradoks_Banach%E2%80%93Tarski" title="Paradoks Banach–Tarski — ინდონეზიური" lang="id" hreflang="id" data-title="Paradoks Banach–Tarski" data-language-autonym="Bahasa Indonesia" data-language-local-name="ინდონეზიური" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Banach%E2%80%93Tarski_%C3%BEvers%C3%B6gnin" title="Banach–Tarski þversögnin — ისლანდიური" lang="is" hreflang="is" data-title="Banach–Tarski þversögnin" data-language-autonym="Íslenska" data-language-local-name="ისლანდიური" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Paradosso_di_Banach-Tarski" title="Paradosso di Banach-Tarski — იტალიური" lang="it" hreflang="it" data-title="Paradosso di Banach-Tarski" data-language-autonym="Italiano" data-language-local-name="იტალიური" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%90%E3%83%8A%E3%83%83%E3%83%8F%EF%BC%9D%E3%82%BF%E3%83%AB%E3%82%B9%E3%82%AD%E3%83%BC%E3%81%AE%E3%83%91%E3%83%A9%E3%83%89%E3%83%83%E3%82%AF%E3%82%B9" title="バナッハ＝タルスキーのパラドックス — იაპონური" lang="ja" hreflang="ja" data-title="バナッハ＝タルスキーのパラドックス" data-language-autonym="日本語" data-language-local-name="იაპონური" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B2%BE%E0%B2%A8%E0%B2%BE%E0%B2%95%E0%B3%8D_%E0%B2%9F%E0%B2%BE%E0%B2%B0%E0%B3%8D%E0%B2%B8%E0%B3%8D%E0%B2%95%E0%B2%BF_%E0%B2%B5%E0%B2%BF%E0%B2%B0%E0%B3%8B%E0%B2%A7%E0%B2%BE%E0%B2%AD%E0%B2%BE%E0%B2%B8" title="ಬಾನಾಕ್ ಟಾರ್ಸ್ಕಿ ವಿರೋಧಾಭಾಸ — კანადა" lang="kn" hreflang="kn" data-title="ಬಾನಾಕ್ ಟಾರ್ಸ್ಕಿ ವಿರೋಧಾಭಾಸ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="კანადა" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%94%EB%82%98%ED%9D%90-%ED%83%80%EB%A5%B4%EC%8A%A4%ED%82%A4_%EC%97%AD%EC%84%A4" title="바나흐-타르스키 역설 — კორეული" lang="ko" hreflang="ko" data-title="바나흐-타르스키 역설" data-language-autonym="한국어" data-language-local-name="კორეული" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Paradoss_da_Banach-Tarski" title="Paradoss da Banach-Tarski — Lombard" lang="lmo" hreflang="lmo" data-title="Paradoss da Banach-Tarski" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Banach-tarskiparadox" title="Banach-tarskiparadox — ნიდერლანდური" lang="nl" hreflang="nl" data-title="Banach-tarskiparadox" data-language-autonym="Nederlands" data-language-local-name="ნიდერლანდური" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Paradoks_Banacha-Tarskiego" title="Paradoks Banacha-Tarskiego — პოლონური" lang="pl" hreflang="pl" data-title="Paradoks Banacha-Tarskiego" data-language-autonym="Polski" data-language-local-name="პოლონური" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Paradoxo_de_Banach%E2%80%93Tarski" title="Paradoxo de Banach–Tarski — პორტუგალიური" lang="pt" hreflang="pt" data-title="Paradoxo de Banach–Tarski" data-language-autonym="Português" data-language-local-name="პორტუგალიური" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%91%D0%B0%D0%BD%D0%B0%D1%85%D0%B0_%E2%80%94_%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D0%BE%D0%B3%D0%BE" title="Парадокс Банаха — Тарского — რუსული" lang="ru" hreflang="ru" data-title="Парадокс Банаха — Тарского" data-language-autonym="Русский" data-language-local-name="რუსული" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox — Simple English" lang="en-simple" hreflang="en-simple" data-title="Banach–Tarski paradox" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B0%D0%BD%D0%B0%D1%85-%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81" title="Банах-Тарски парадокс — სერბული" lang="sr" hreflang="sr" data-title="Банах-Тарски парадокс" data-language-autonym="Српски / srpski" data-language-local-name="სერბული" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Banach-Tarskis_paradox" title="Banach-Tarskis paradox — შვედური" lang="sv" hreflang="sv" data-title="Banach-Tarskis paradox" data-language-autonym="Svenska" data-language-local-name="შვედური" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%91%D0%B0%D0%BD%D0%B0%D1%85%D0%B0_%E2%80%94_%D0%A2%D0%B0%D1%80%D1%81%D1%8C%D0%BA%D0%BE%D0%B3%D0%BE" title="Парадокс Банаха — Тарського — უკრაინული" lang="uk" hreflang="uk" data-title="Парадокс Банаха — Тарського" data-language-autonym="Українська" data-language-local-name="უკრაინული" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ngh%E1%BB%8Bch_l%C3%BD_Banach%E2%80%93Tarski" title="Nghịch lý Banach–Tarski — ვიეტნამური" lang="vi" hreflang="vi" data-title="Nghịch lý Banach–Tarski" data-language-autonym="Tiếng Việt" data-language-local-name="ვიეტნამური" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B7%B4%E6%8B%BF%E8%B5%AB-%E5%A1%94%E6%96%AF%E5%9F%BA%E5%AE%9A%E7%90%86" title="巴拿赫-塔斯基定理 — ჩინური" lang="zh" hreflang="zh" data-title="巴拿赫-塔斯基定理" data-language-autonym="中文" data-language-local-name="ჩინური" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q737851#sitelinks-wikipedia" title="ენათშორისი ბმულების რედაქტირება" class="wbc-editpage">ბმულების რედაქტირება</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="სახელთა სივრცე"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" title="იხილეთ შიგთავსის გვერდი [c]" accesskey="c"><span>სტატია</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E1%83%92%E1%83%90%E1%83%9C%E1%83%AE%E1%83%98%E1%83%9A%E1%83%95%E1%83%90:%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=edit&redlink=1" rel="discussion" class="new" title="გვერდის შიგთავსის განხილვა (ჯერ არაა დაწერილი) [გ]" accesskey="გ"><span>განხილვა</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="ენის ვარიანტის შეცვლა" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">ქართული</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="გადახედვა"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98"><span>შემოწმებული სტატია</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&veaction=edit" title="ამ გვერდის რედაქტირება [v]" accesskey="v"><span>რედაქტირება</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=edit" title="ამ გვერდის წყაროს კოდის რედაქტირება [e]" accesskey="e"><span>წყაროს რედაქტირება</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=history" title="ამ გვერდის ბოლო ვერსიები [ი]" accesskey="ი"><span>ისტორია</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="ინსტრუმენტები გვერდებისთვის"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="ხელსაწყოები" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">ხელსაწყოები</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">ხელსაწყოები</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">გადატანა გვერდით ზოლზე</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">დამალვა</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> მოქმედებები </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98"><span>შემოწმებული სტატია</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&veaction=edit" title="ამ გვერდის რედაქტირება [v]" accesskey="v"><span>რედაქტირება</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=edit" title="ამ გვერდის წყაროს კოდის რედაქტირება [e]" accesskey="e"><span>წყაროს რედაქტირება</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=history"><span>ისტორია</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> ძირითადი </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:WhatLinksHere/%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" title="ყველა ვიკი გვერდის სია, რომლებშიც ეს გვერდია მითითებული [j]" accesskey="j"><span>ბმულები გვერდზე</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:RecentChangesLinked/%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" rel="nofollow" title="ბოლო ცვლილებები ამ გვერდთან დაკავშირებულ გვერდებზე [k]" accesskey="k"><span>დაკავშირებული ცვლილებები</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:%E1%83%92%E1%83%90%E1%83%9C%E1%83%A1%E1%83%90%E1%83%99%E1%83%A3%E1%83%97%E1%83%A0%E1%83%94%E1%83%91%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%92%E1%83%95%E1%83%94%E1%83%A0%E1%83%93%E1%83%94%E1%83%91%E1%83%98" title="ყველა სპეციალური გვერდის სია [q]" accesskey="q"><span>სპეციალური გვერდები</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&oldid=4301434" title="მუდმივი ბმული გვერდის ამ ვერსიაზე"><span>მუდმივი ბმული</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=info" title="მეტი ინფორმაცია ამ გვერდის შესახებ"><span>გვერდის ინფორმაცია</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98:CiteThisPage&page=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&id=4301434&wpFormIdentifier=titleform" title="ინფორმაცია ამ გვერდის ციტირების შესახებ"><span>გვერდის ციტირება</span></a></li><li id="t-urlshortener" class="mw-list-item"><a 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src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Banach-Tarski_Paradox.svg/350px-Banach-Tarski_Paradox.svg.png" decoding="async" width="350" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Banach-Tarski_Paradox.svg/525px-Banach-Tarski_Paradox.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Banach-Tarski_Paradox.svg/700px-Banach-Tarski_Paradox.svg.png 2x" data-file-width="445" data-file-height="100" /></a><figcaption>შესაძლებელია თუ არა ბურთი სივცრეში დაიშალოს სასრული რაოდენობის წერტილების სიმრავლედ და ხელახლა იქნას აწყობილი ორ, ორიგინალის იდენტურ ბურთად სივცრეში?</figcaption></figure> <p><b>ბანახის-ტარსკის პარადოქსი</b> — <a href="/w/index.php?title=%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%92%E1%83%94%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%90&action=edit&redlink=1" class="new" title="თეორიული გეომეტრია (ჯერ არაა დაწერილი)">თეორიული გეომეტრიის</a> ერთ-ერთი თეორემა, რომელიც მოიცავს შემდეგ არსს: როდესაც წრეს ენიჭება სამგანზომილებიანი სივრცე, შესაძლებელია იგი დაშლილ იქნას სასრული რაოდენობის არაგადამკვეთ სიმრავლეებად, რომელსაც შემდეგ შეუძლია წარმოქმნას ორიგინალის იდენტური ორი ასლი. სინამდვილეში, ეს პროცესი მხოლოდ მოიცავს ნაწილების გადაადგილებას და ბრუნვას, ფორმის ცვლილების გარეშე. თუმცაღა, მიღებული ბურთები არათუ მყარ ობიექტს, არამედ გაბნეულ ნაწილაკებს წარმოადგენენ. რეკონსტრუქცია შეიძლება მოიცავდეს მხოლოდ ხუთ ნაწილს. </p><p>თეორემის ძლიერი ვერსია გულისხმობს იმას, რომ მოცემული ორი „გონივრული“ მყარი ობიექტიდან შესაძლებელია (მაგ.: დიდი და პატარა ბურთები) ერთ-ერთმა გამოსახოს მეორე. ამ თეორემას ხშირად ადარებენ „დაჭრილ ბარდას, რომელსაც შეუძლია იქცეს მზედ“ და უწოდებენ „ბარდისა და მზის პარადოქსს“. </p><p>მიზეზი იმისა, თუ რატომ ეწოდება ბანახის-ტარსკის თეორემას პარადოქსი, არის ის, რომ იგი ეწინააღმდეგება ძირითად გეომეტრიულ ინტუიციას. „ბურთის გაორმაგება“ — მათი დაყოფა ნაწილებად და გადაადგილება, ყველანაირი დაჭიმვის, ან წერტილების დამატების გარეშე, როგორც ჩანს შეუძლებელია, რადგანაც შენარჩუნებულ უნდა იქნას მოცულობა. თეორიულად ეს შესაძლებელია და იგულისხმება მოცულობის ფორმალურ განმარტებებშიც. თუმცაღა, პრაქტიკულად არ შეიძლება განისაზღვროს განხილული მოცულობები, რადგანაც ისინი ხელახალი შეერთების შემდეგ აუცილებლად შეიცვლება. </p><p>გეომეტრიის სხვა თეორემებისგან განსხვავებით, ამ შედეგის მტკიცებულება დამოკიდებულია აქსიომების არჩევანზე მოცემული თეორიისათვის. ის შეიძლება დამტკიცებულ იქნას არჩევნის აქსიომით, რომელიც გულისხმობს უთვლად სიმრავლეებს, მაგ.: წერტილების სიმრავლე, რომელთაც არ გააჩნიათ მოცულობა პირდაპირი გაგებით, რომლის მნიშვნელობა მოითხოვს არჩევნის უთვლად რაოდენობას.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>2005 წელს დამტკიცდა — დაშლის დროს, ნაწილები არჩეულ უნდა იქნას იმ კრიტერიუმით, რომ მათ შეძლონ უწყვეტი გადაადგილება, ერთმანეთთან შეხვედრის გარეშე.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>  </p> <div class="mw-heading mw-heading2"><h2 id="ლიტერატურა"><span id=".E1.83.9A.E1.83.98.E1.83.A2.E1.83.94.E1.83.A0.E1.83.90.E1.83.A2.E1.83.A3.E1.83.A0.E1.83.90"></span>ლიტერატურა</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&veaction=edit&section=1" title="სექციის რედაქტირება:ლიტერატურა" class="mw-editsection-visualeditor"><span>რედაქტირება</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=edit&section=1" title="Edit section's source code: ლიტერატურა"><span>წყაროს რედაქტირება</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFBanachTarski,_Alfred1924" class="citation journal cs1">Banach, Stefan; Tarski, Alfred (1924). <a rel="nofollow" class="external text" href="http://www.emis.de/cgi-bin/JFM-item?50.0370.02">Review at JFM</a>. <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf">„Sur la décomposition des ensembles de points en parties respectivement congruentes“</a> <span class="cs1-format">(PDF)</span>. <i>Fundamenta Mathematicae</i>. <b>6</b>: 244–277.</cite><span 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a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style></li> <li><cite id="CITEREFChurkin2010" class="citation journal cs1">Churkin, V. A. (2010). „A continuous version of the Hausdorff–Banach–Tarski paradox“. <i>Algebra and Logic</i>. <b>49</b> (1): 91–98. <a href="/wiki/Doi" class="mw-redirect" title="Doi">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10469-010-9080-y">10.1007/s10469-010-9080-y</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algebra+and+Logic&rft.atitle=A+continuous+version+of+the+Hausdorff%E2%80%93Banach%E2%80%93Tarski+paradox&rft.volume=49&rft.issue=1&rft.pages=91-98&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2Fs10469-010-9080-y&rft.aulast=Churkin&rft.aufirst=V.+A.&rfr_id=info%3Asid%2Fka.wikipedia.org%3A%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1+%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" class="Z3988"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4042187"></li> <li><cite id="CITEREFStromberg1979" class="citation journal cs1">Stromberg, Karl (March 1979). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_1979-03_86_3/page/151">„The Banach–Tarski paradox“</a>. <i>The American Mathematical Monthly</i>. Mathematical Association of America. <b>86</b> (3): 151–161. <a href="/wiki/Doi" class="mw-redirect" title="Doi">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2321514">10.2307/2321514</a>. <a href="/w/index.php?title=JSTOR&action=edit&redlink=1" class="new" title="JSTOR (ჯერ არაა დაწერილი)">JSTOR</a> <a rel="nofollow" class="external text" href="//www.jstor.org/stable/2321514">2321514</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+Banach%E2%80%93Tarski+paradox&rft.volume=86&rft.issue=3&rft.pages=151-161&rft.date=1979-03&rft_id=info%3Adoi%2F10.2307%2F2321514&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2321514&rft.aulast=Stromberg&rft.aufirst=Karl&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_american-mathematical-monthly_1979-03_86_3%2Fpage%2F151&rfr_id=info%3Asid%2Fka.wikipedia.org%3A%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1+%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" class="Z3988"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4042187"></li> <li><cite id="CITEREFvon_Neumann1929" class="citation journal cs1">von Neumann, John (1929). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm13/fm1316.pdf">„Zur allgemeinen Theorie des Masses“</a> <span class="cs1-format">(<a href="/w/index.php?title=PDF&action=edit&redlink=1" class="new" title="PDF (ჯერ არაა დაწერილი)">PDF</a>)</span>. <i>Fundamenta Mathematicae</i>. <b>13</b>: 73–116.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fundamenta+Mathematicae&rft.atitle=Zur+allgemeinen+Theorie+des+Masses&rft.volume=13&rft.pages=73-116&rft.date=1929&rft.aulast=von+Neumann&rft.aufirst=John&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Ffm%2Ffm13%2Ffm1316.pdf&rfr_id=info%3Asid%2Fka.wikipedia.org%3A%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1+%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" class="Z3988"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4042187"></li></ul> <div class="mw-heading mw-heading2"><h2 id="რესურსები_ინტერნეტში"><span id=".E1.83.A0.E1.83.94.E1.83.A1.E1.83.A3.E1.83.A0.E1.83.A1.E1.83.94.E1.83.91.E1.83.98_.E1.83.98.E1.83.9C.E1.83.A2.E1.83.94.E1.83.A0.E1.83.9C.E1.83.94.E1.83.A2.E1.83.A8.E1.83.98"></span>რესურსები ინტერნეტში</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&veaction=edit&section=2" title="სექციის რედაქტირება:რესურსები ინტერნეტში" class="mw-editsection-visualeditor"><span>რედაქტირება</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=edit&section=2" title="Edit section's source code: რესურსები ინტერნეტში"><span>წყაროს რედაქტირება</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="infobox sisterproject"><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/%E1%83%95%E1%83%98%E1%83%99%E1%83%98%E1%83%A1%E1%83%90%E1%83%AC%E1%83%A7%E1%83%9D%E1%83%91%E1%83%98" title="ვიკისაწყობი"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/35px-Commons-logo.svg.png" decoding="async" width="35" height="47" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/53px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/70px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a><figcaption></figcaption></figure> <div style="margin-left: 50px;"><a href="/wiki/%E1%83%95%E1%83%98%E1%83%99%E1%83%98%E1%83%A1%E1%83%90%E1%83%AC%E1%83%A7%E1%83%9D%E1%83%91%E1%83%98" title="ვიკისაწყობი"><span id="fact">ვიკისაწყობში</span></a> არის გვერდი თემაზე: <div style="margin-left: 10px;"><i><b><a href="https://commons.wikimedia.org/wiki/Category:Banach%E2%80%93Tarski_paradox" class="extiw" title="commons:Category:Banach–Tarski paradox">ბანახის-ტარსკის პარადოქსი</a></b></i></div> </div> </div> <ul><li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/TheBanachTarskiParadox/">ბანახის-ტარსკის პარადოქსი</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="შენიშვნები"><span id=".E1.83.A8.E1.83.94.E1.83.9C.E1.83.98.E1.83.A8.E1.83.95.E1.83.9C.E1.83.94.E1.83.91.E1.83.98"></span>შენიშვნები</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&veaction=edit&section=3" title="სექციის რედაქტირება:შენიშვნები" class="mw-editsection-visualeditor"><span>რედაქტირება</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98&action=edit&section=3" title="Edit section's source code: შენიშვნები"><span>წყაროს რედაქტირება</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="font-size: 90%; list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> <span class="reference-text">Wagon, Corollary 13.3</span> </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <span class="reference-text"><cite id="CITEREFWilson2005" class="citation journal cs1">Wilson, Trevor M. (September 2005). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_journal-of-symbolic-logic_2005-09_70_3/page/946">„A continuous movement version of the Banach–Tarski paradox: A solution to De Groot's problem“</a>. <i>Journal of Symbolic Logic</i>. <b>70</b> (3): 946–952. <a href="/wiki/Doi" class="mw-redirect" title="Doi">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2178%2Fjsl%2F1122038921">10.2178/jsl/1122038921</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=A+continuous+movement+version+of+the+Banach%E2%80%93Tarski+paradox%3A+A+solution+to+De+Groot%27s+problem&rft.volume=70&rft.issue=3&rft.pages=946-952&rft.date=2005-09&rft_id=info%3Adoi%2F10.2178%2Fjsl%2F1122038921&rft.aulast=Wilson&rft.aufirst=Trevor+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_journal-of-symbolic-logic_2005-09_70_3%2Fpage%2F946&rfr_id=info%3Asid%2Fka.wikipedia.org%3A%E1%83%91%E1%83%90%E1%83%9C%E1%83%90%E1%83%AE%E1%83%98%E1%83%A1-%E1%83%A2%E1%83%90%E1%83%A0%E1%83%A1%E1%83%99%E1%83%98%E1%83%A1+%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%93%E1%83%9D%E1%83%A5%E1%83%A1%E1%83%98" class="Z3988"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4042187"></span> </li> </ol></div></div></div><script>(RLQ=window.RLQ||[]).push(function(){mw.log.warn("Gadget \"ReferenceTooltips\" was not loaded. 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