{"id":9266,"date":"2025-10-17T20:14:23","date_gmt":"2025-10-17T11:14:23","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9266"},"modified":"2026-09-27T12:08:17","modified_gmt":"2026-09-27T03:08:17","slug":"ch10-axiom-of-choice","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch10-axiom-of-choice\/","title":{"rendered":"\uc120\ud0dd \uacf5\ub9ac"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>10. \uc120\ud0dd \uacf5\ub9ac<\/h2>\n\n --><\/p>\n<p><a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">9\uc7a5<\/a>\uc5d0\uc11c ZF\uc5d0 \uc120\ud0dd\uacf5\ub9ac\ub97c \ub354\ud55c \uccb4\uacc4\ub97c ZFC\ub77c\uace0 \uc815\uc758\ud558\uc600\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub294 \uc720\ud55c\ud55c \uc120\ud0dd\uc5d0\uc11c\ub294 \ub4dc\ub7ec\ub098\uc9c0 \uc54a\uc9c0\ub9cc, \uc11c\ub85c \uc544\ubb34 \uad00\ub828\uc774 \uc5c6\uace0 \ubb34\ud55c\ud788 \ub9ce\uc73c\uba70 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc5d0\uc11c \uc6d0\uc18c\ub97c \ud558\ub098\uc529 \ub3d9\uc2dc\uc5d0 \uace8\ub77c\uc57c \ud560 \ub54c \ud575\uc2ec\uc801\uc778 \uc5ed\ud560\uc744 \ud55c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uacf5\ub9ac 9.12\uc758 \uc5ec\ub7ec \ub3d9\uce58\ud615\ud0dc\uc640 \ub300\ud45c\uc801\uc778 \uc751\uc6a9\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uc120\ud0dd\uacf5\ub9ac\ubcf4\ub2e4 \uc57d\ud55c \uc120\ud0dd\uc6d0\ub9ac\ub3c4 \uad6c\ubd84\ud55c\ub2e4.<\/p>\n<p>\uad34\ub378\uc740 ZF\uac00 \ubb34\ubaa8\uc21c\uc774\ub77c\uba74 ZFC\ub3c4 \ubb34\ubaa8\uc21c\uc784\uc744 \ubcf4\uc600\uace0, \ucf54\uc5b8\uc740 ZF\uac00 \ubb34\ubaa8\uc21c\uc774\ub77c\uba74 \\(\\mathrm{ZF}+\\neg\\mathrm{AC}\\)\ub3c4 \ubb34\ubaa8\uc21c\uc784\uc744 \ubcf4\uc600\ub2e4. \ub530\ub77c\uc11c ZF\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \uc804\uc81c\ub85c \ud558\uba74 \uc120\ud0dd\uacf5\ub9ac\ub294 ZF\ub85c\ubd80\ud130 \uc99d\uba85\ud560 \uc218\ub3c4 \ubc18\uc99d\ud560 \uc218\ub3c4 \uc5c6\ub2e4.<\/p>\n<h3>1. \uc120\ud0dd \uacf5\ub9ac\uc758 \uc758\ubbf8<\/h3>\n<p><a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">\uacf5\ub9ac 9.12<\/a>\uc5d0\uc11c \uc815\uc758\ud55c \uc120\ud0dd\ud568\uc218\ub97c \ub2e4\uc2dc \uc0b4\ud3b4\ubcf4\uc790. \uacf5\uc9d1\ud569\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc871 \\(\\mathcal F\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218<br \/>\n\\[f\\colon\\mathcal F\\to\\bigcup\\mathcal F\\]<br \/>\n\uac00 \ubaa8\ub4e0 \\(A\\in\\mathcal F\\)\uc5d0 \ub300\ud574 \\(f(A)\\in A\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(f\\)\ub97c \\(\\mathcal F\\)\uc758 \uc120\ud0dd\ud568\uc218\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<p>\uc9c1\uad00\uc801\uc73c\ub85c \uc120\ud0dd\ud568\uc218\ub294 \uac01 \uc9d1\ud569\uc5d0\uc11c \uc6d0\uc18c\ub97c \ud558\ub098\uc529 \uc120\ud0dd\ud558\ub294 \ud568\uc218\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[\\mathcal F=\\{\\{1,2\\},\\{3,4,5\\},\\{6\\}\\}\\]<br \/>\n\uc5d0 \ub300\ud574\uc11c\ub294 \uc120\ud0dd\ud568\uc218\uac00 \\(2\\cdot3\\cdot1=6\\)\uac1c \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc774 \uc720\ud55c \uac1c\ubfd0\uc774\uba74 \uc120\ud0dd\ud568\uc218\uc758 \uc874\uc7ac\ub294 ZF\uc5d0\uc11c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \uc120\ud0dd\uacf5\ub9ac\uac00 \ud544\uc694\ud55c \uc9c0\uc810\uc740 \uc784\uc758\uc758 \ud06c\uae30\ub97c \uac16\ub294 \uc9d1\ud569\uc871\uc5d0 \ub300\ud574 \uc774\ub7ec\ud55c \uc120\ud0dd\uc744 \ud55c\uaebc\ubc88\uc5d0 \uc218\ud589\ud560 \ub54c\uc774\ub2e4.<\/p>\n<p>\uc120\ud0dd\uacf5\ub9ac\ub294 \ub2e4\uc74c \ub450 \ud615\ud0dc\ub85c\ub3c4 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li>\uacf5\uc9d1\ud569\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uc784\uc758\uc758 \uc9d1\ud569\uc871 \\(\\mathcal F\\)\uc5d0\ub294 \uc120\ud0dd\ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\uac01 \\(i\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(A_i\\ne\\varnothing\\)\uc778 \uc9d1\ud569\uc871 \\(\\{A_i\\}_{i\\in I}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\prod_{i\\in I}A_i\\ne\\varnothing\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ub458\uc9f8 \ud45c\ud604\uc5d0\uc11c \uacf1\uc9d1\ud569\uc758 \uc6d0\uc18c\ub294 \uac01 \\(i\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(i)\\in A_i\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\ubbc0\ub85c \ub450 \ud45c\ud604\uc740 \uac19\uc740 \ub0b4\uc6a9\uc744 \ub9d0\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.1.<\/span><br \/>\n\\(I=\\{0,1,2\\}\\)\uc774\uace0<br \/>\n\\[<br \/>\nA_0=\\{0,1\\},\\quad A_1=\\{a,b,c\\},\\quad A_2=\\{\\ast\\}<br \/>\n\\]<br \/>\n\ub77c \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\displaystyle\\prod_{i\\in I}A_i\\)\uc758 \uc6d0\uc18c\ub97c \ubaa8\ub450 \ub098\uc5f4\ud558\uc2dc\uc624.<\/li>\n<li>\uc774 \uc9d1\ud569\uc871\uc758 \uc120\ud0dd\ud568\uc218\uc758 \uac1c\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\uacf1\uc9d1\ud569\uc758 \uc6d0\uc18c\uc640 \uc120\ud0dd\ud568\uc218\uac00 \uc5b4\ub5bb\uac8c \ub300\uc751\ud558\ub294\uc9c0 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.2.<\/span><br \/>\n\uac01 \\(n\\in\\mathbb N\\)\uc5d0 \ub300\ud558\uc5ec \\(A_n=\\{n,n+1\\}\\)\uc774\ub77c \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f(A_n)=n\\)\uc774\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(\\{A_n\\}_{n\\in\\mathbb N}\\)\uc758 \uc120\ud0dd\ud568\uc218\uac00 \ub428\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\uc774 \uc608\uc5d0\uc11c \uc120\ud0dd\ud568\uc218\uc758 \uc874\uc7ac\ub97c \ubcf4\uc774\uae30 \uc704\ud558\uc5ec \uac00\uc0b0 \uc120\ud0dd \uacf5\ub9ac\ub97c \ub530\ub85c \uc0ac\uc6a9\ud560 \ud544\uc694\uac00 \uc5c6\ub294 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>2. \uc120\ud0dd \uacf5\ub9ac\uc640 \ub3d9\uce58\uc778 \uba85\uc81c\ub4e4<\/h3>\n<p>ZF \ud558\uc5d0\uc11c \uc120\ud0dd\uacf5\ub9ac\uc640 \ub3d9\uce58\uc778 \uba85\uc81c\ub294 \ub9e4\uc6b0 \ub9ce\ub2e4. \uc5ec\uae30\uc11c\ub294 \uc815\ub82c \uc815\ub9ac, \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac, \ud558\uc6b0\uc2a4\ub3c4\ub974\ud504\uc758 \uadf9\ub300 \uc6d0\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<p>\uc9d1\ud569 \\(A\\) \uc704\uc5d0 \uc5b4\ub5a4 \uc815\ub82c\uc21c\uc11c\ub97c \uc904 \uc218 \uc788\uc73c\uba74 \\(A\\)\ub97c <span class=\"defined\">\uc815\ub82c \uac00\ub2a5<\/span>(well-orderable)\ud558\ub2e4\uace0 \ud55c\ub2e4. \uc815\ub82c\uc21c\uc11c\uc758 \uc815\uc758\ub294 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch08-ordinal-numbers\">\uc815\uc758 8.1<\/a>\uc744 \ub530\ub978\ub2e4.<\/p>\n<p>\ubd80\ubd84\uc21c\uc11c\uc9d1\ud569 \\((P,\\leq)\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(C\\subseteq P\\)\uac00 \uc784\uc758\uc758 \ub450 \uc6d0\uc18c\ub07c\ub9ac \ube44\uad50 \uac00\ub2a5\ud558\uba74 \\(C\\)\ub97c <span class=\"defined\">\uc0ac\uc2ac<\/span>(chain)\uc774\ub77c\uace0 \ud55c\ub2e4. \uc6d0\uc18c \\(m\\in P\\)\uc5d0 \ub300\ud558\uc5ec \\(m&lt;x\\)\uc778 \\(x\\in P\\)\uac00 \uc5c6\uc73c\uba74 \\(m\\)\uc744 <span class=\"defined\">\uadf9\ub300\uc6d0\uc18c<\/span>(maximal element)\ub77c\uace0 \ud55c\ub2e4. \uadf9\ub300\uc6d0\uc18c\ub294 \ubaa8\ub4e0 \uc6d0\uc18c\ubcf4\ub2e4 \ud070 \ucd5c\ub300\uc6d0\uc18c\uc640 \uad6c\ubcc4\ud574\uc57c \ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.3.<\/span><br \/>\n\\(P=\\{\\varnothing,\\{1\\},\\{2\\}\\}\\)\uc5d0 \ud3ec\ud568\uad00\uacc4\ub85c \ubd80\ubd84\uc21c\uc11c\ub97c \uc8fc\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(P\\)\uc758 \uc6d0\uc18c \uac00\uc6b4\ub370 \uc11c\ub85c \ube44\uad50 \uac00\ub2a5\ud55c \ub450 \uc6d0\uc18c\ub97c \ubaa8\ub450 \ucc3e\uc73c\uc2dc\uc624.<\/li>\n<li>\\(P\\)\uc758 \ud3ec\ud568\uad00\uacc4\uc5d0 \ub300\ud558\uc5ec \uadf9\ub300\uc778 \uc0ac\uc2ac\uc744 \ubaa8\ub450 \ucc3e\uc73c\uc2dc\uc624.<\/li>\n<li>\\(P\\)\uc758 \uadf9\ub300\uc6d0\uc18c\ub97c \ubaa8\ub450 \ucc3e\uc73c\uc2dc\uc624. \ucd5c\ub300\uc6d0\uc18c\ub3c4 \uc874\uc7ac\ud558\ub294\uac00?<\/li>\n<li>\\(P\\)\uc758 \ubaa8\ub4e0 \uc0ac\uc2ac\uc774 \\(P\\) \uc548\uc5d0\uc11c \uc0c1\uacc4\ub97c \uac00\uc9d0\uc744 \ud655\uc778\ud558\uace0, \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\uc758 \uacb0\ub860\uacfc \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 10.1. (\uc120\ud0dd\uacf5\ub9ac\uc758 \ub300\ud45c\uc801\uc778 \ub3d9\uce58\ud615\ud0dc)<\/span><\/p>\n<p>ZF \uc704\uc5d0\uc11c \ub2e4\uc74c \uba85\uc81c\ub4e4\uc740 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc120\ud0dd\uacf5\ub9ac.<\/li>\n<li><span class=\"defined\">\uc815\ub82c \uc815\ub9ac<\/span>(Well-Ordering Theorem): \ubaa8\ub4e0 \uc9d1\ud569\uc740 \uc815\ub82c \uac00\ub2a5\ud558\ub2e4.<\/li>\n<li><span class=\"defined\">\ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac<\/span>(Zorn&#8217;s Lemma): \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubd80\ubd84\uc21c\uc11c\uc9d1\ud569 \\(P\\)\uc5d0\uc11c \ubaa8\ub4e0 \uc0ac\uc2ac\uc774 \\(P\\) \uc548\uc758 \uc0c1\uacc4\ub97c \uac00\uc9c0\uba74 \\(P\\)\uc5d0\ub294 \uadf9\ub300\uc6d0\uc18c\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li><span class=\"defined\">\ud558\uc6b0\uc2a4\ub3c4\ub974\ud504\uc758 \uadf9\ub300 \uc6d0\ub9ac<\/span>(Hausdorff maximal principle): \ubaa8\ub4e0 \ubd80\ubd84\uc21c\uc11c\uc9d1\ud569\uc740 \ud3ec\ud568\uad00\uacc4\uc5d0 \ub300\ud558\uc5ec \uadf9\ub300\uc778 \uc0ac\uc2ac\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc644\uc804\ud55c \ub3d9\uce58 \uc99d\uba85\uc740 \ucd08\ud55c\uc7ac\uadc0\uc640 \ud558\ub974\ud1a1\uc2a4 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\ub294 \ud45c\uc900\uc801\uc778 \uc9d1\ud569\ub860 \ub17c\uc99d\uc744 \ud544\uc694\ub85c \ud558\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4. \ub2e4\ub9cc \uba87 \ubc29\ud5a5\uc740 \uc27d\uac8c \ud655\uc778\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc815\ub82c \uc815\ub9ac\ub97c \uac00\uc815\ud558\uba74 \\(\\bigcup\\mathcal F\\)\uc5d0 \uc815\ub82c\uc21c\uc11c\ub97c \ud558\ub098 \uc904 \uc218 \uc788\ub2e4. \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uac01 \\(A\\in\\mathcal F\\)\uc5d0\uc11c \uc774 \uc815\ub82c\uc21c\uc11c\uc5d0 \ub300\ud55c \ucd5c\uc18c\uc6d0\uc18c\ub97c \ud0dd\ud558\uba74 \ub418\ubbc0\ub85c \uc120\ud0dd\uacf5\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.4.<\/span><br \/>\n\uc815\ub82c \uc815\ub9ac\ub97c \uac00\uc815\ud558\uace0, \uacf5\uc9d1\ud569\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc871 \\(\\mathcal F\\)\ub97c \uc0dd\uac01\ud558\uc790. \\(\\bigcup\\mathcal F\\)\uc5d0 \uc815\ub82c\uc21c\uc11c \\(\\preceq\\)\ub97c \ud558\ub098 \uc8fc\uace0<br \/>\n\\[<br \/>\nf(A)=\\text{\\(A\\)\uc758 \\(\\preceq\\)-\ucd5c\uc18c\uc6d0\uc18c}\\quad(A\\in\\mathcal F)<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f\\)\uac00 \uc798 \uc815\uc758\ub41c \ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(f\\)\uac00 \\(\\mathcal F\\)\uc758 \uc120\ud0dd\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\uac01 \\(A\\in\\mathcal F\\)\ub97c \ub530\ub85c \uc815\ub82c\ud560 \ud544\uc694 \uc5c6\uc774 \\(\\bigcup\\mathcal F\\) \ud558\ub098\ub9cc \uc815\ub82c\ud574\ub3c4 \ucda9\ubd84\ud55c \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \uac00\uc815\ud558\uc790. \uc9d1\ud569\uc871 \\(\\mathcal F\\)\uc758 \ubd80\ubd84 \uc120\ud0dd\ud568\uc218\ub4e4\uc744 \uc815\uc758\uc5ed\uc758 \ud3ec\ud568\uacfc \ud568\uc218\uc758 \ud655\uc7a5\uad00\uacc4\ub85c \uc21c\uc11c\ud654\ud55c\ub2e4. \uc0ac\uc2ac\uc5d0 \uc18d\ud55c \ubd80\ubd84 \uc120\ud0dd\ud568\uc218\ub4e4\uc758 \ud569\uc9d1\ud569\uc740 \ub2e4\uc2dc \ubd80\ubd84 \uc120\ud0dd\ud568\uc218\uc774\ubbc0\ub85c \uc0ac\uc2ac\uc758 \uc0c1\uacc4\uac00 \ub41c\ub2e4. \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub85c \uadf9\ub300 \ubd80\ubd84 \uc120\ud0dd\ud568\uc218 \\(g\\)\ub97c \uc5bb\ub294\ub2e4. \ub9cc\uc57d \\(\\operatorname{dom}(g)\\ne\\mathcal F\\)\uc774\uba74 \\(A\\in\\mathcal F\\setminus\\operatorname{dom}(g)\\)\uc640 \\(a\\in A\\)\ub97c \ud558\ub098 \ud0dd\ud558\uc5ec \\(g\\)\ub97c \\(A\\)\uae4c\uc9c0 \ud655\uc7a5\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \uadf9\ub300\uc131\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(g\\)\ub294 \\(\\mathcal F\\) \uc804\uccb4\uc758 \uc120\ud0dd\ud568\uc218\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.5.<\/span><br \/>\n\uacf5\uc9d1\ud569\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc871 \\(\\mathcal F\\)\uc5d0 \ub300\ud558\uc5ec \\(P\\)\ub97c \\(\\mathcal F\\)\uc758 \ubaa8\ub4e0 \ubd80\ubd84 \uc120\ud0dd\ud568\uc218\uc758 \uc9d1\ud569\uc774\ub77c \ud558\uc790. \uc989 \\(g\\in P\\)\uc774\uba74 \\(\\operatorname{dom}(g)\\subseteq\\mathcal F\\)\uc774\uace0 \ubaa8\ub4e0 \\(A\\in\\operatorname{dom}(g)\\)\uc5d0 \ub300\ud558\uc5ec \\(g(A)\\in A\\)\uc774\ub2e4. \ud568\uc218\uc758 \ud655\uc7a5\uad00\uacc4\ub85c \\(P\\)\ub97c \uc21c\uc11c\ud654\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\uacf5\ud568\uc218\uac00 \\(P\\)\uc5d0 \uc18d\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(P\\)\uc758 \uc0ac\uc2ac \\(\\mathcal C\\)\uc5d0 \ub300\ud558\uc5ec \\(h=\\bigcup\\mathcal C\\)\uac00 \ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(h\\)\uac00 \ub2e4\uc2dc \ubd80\ubd84 \uc120\ud0dd\ud568\uc218\uc774\uba70 \\(\\mathcal C\\)\uc758 \uc0c1\uacc4\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub85c \uc5bb\uc740 \uadf9\ub300 \ubd80\ubd84 \uc120\ud0dd\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc774 \ubc18\ub4dc\uc2dc \\(\\mathcal F\\) \uc804\uccb4\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\ud558\uc6b0\uc2a4\ub3c4\ub974\ud504\uc758 \uadf9\ub300 \uc6d0\ub9ac\ub97c \uac00\uc815\ud558\uace0 \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\uc758 \uac00\uc815\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubd80\ubd84\uc21c\uc11c\uc9d1\ud569 \\(P\\)\ub97c \uc0dd\uac01\ud558\uc790. \uadf9\ub300 \uc0ac\uc2ac \\(C\\)\ub97c \ud0dd\ud558\uace0 \uadf8 \uc0c1\uacc4\ub97c \\(u\\in P\\)\ub77c \ud558\uc790. \\(C\\cup\\{u\\}\\)\ub3c4 \uc0ac\uc2ac\uc774\ubbc0\ub85c \uadf9\ub300\uc131\uc5d0\uc11c \\(u\\in C\\)\uc774\ub2e4. \ub9cc\uc57d \\(u&lt;v\\)\uc778 \\(v\\in P\\)\uac00 \uc874\uc7ac\ud558\uba74 \\(C\\cup\\{v\\}\\)\ub3c4 \uc0ac\uc2ac\uc774 \ub418\uc5b4 \ub2e4\uc2dc \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(u\\)\ub294 \\(P\\)\uc758 \uadf9\ub300\uc6d0\uc18c\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 10.6.<\/span><br \/>\n\ud558\uc6b0\uc2a4\ub3c4\ub974\ud504\uc758 \uadf9\ub300 \uc6d0\ub9ac\ub97c \uc790\uc2e0\uc758 \ub9d0\ub85c \uc11c\uc220\ud558\uace0, \uc704 \ub17c\uc99d\uc744 \uc644\uc131\ud558\uc5ec \ud558\uc6b0\uc2a4\ub3c4\ub974\ud504\uc758 \uadf9\ub300 \uc6d0\ub9ac\ub85c\ubd80\ud130 \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\uac00 \ub530\ub984\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>3. \ubb34\ud55c\uae30\uc218 \uc0b0\uc220\uc5d0\uc758 \uc801\uc6a9<\/h3>\n<p>\uc815\ub82c \uc815\ub9ac\uc5d0 \uc758\ud574 \uc120\ud0dd\uacf5\ub9ac \uc544\ub798\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \uc9d1\ud569\uc744 \uc5b4\ub5a4 \ucd08\uae30\uc11c\uc218\uc640 \ub300\ub4f1\ud558\uac8c \ub193\uc744 \uc218 \uc788\ub2e4. \uc774\ub97c \uc774\uc6a9\ud558\uba74 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">7\uc7a5<\/a>\uc5d0\uc11c \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c \uc815\ub9ac 7.10\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uba3c\uc800 \ubaa8\ub4e0 \ubb34\ud55c\uae30\uc218 \\(\\kappa\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\kappa\\cdot\\kappa=\\kappa\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc790. \uc120\ud0dd\uacf5\ub9ac\uc5d0 \uc758\ud574 \\(\\kappa\\)\ub97c \ubb34\ud55c \ucd08\uae30\uc11c\uc218\uc640 \ub3d9\uc77c\uc2dc\ud55c\ub2e4. \ubb34\ud55c \ucd08\uae30\uc11c\uc218\uc5d0 \ub300\ud55c \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \\(\\kappa\\times\\kappa\\)\uc5d0 \ub2e4\uc74c \uc21c\uc11c\ub97c \uc900\ub2e4. \ub450 \uc21c\uc11c\uc30d \\((\\alpha,\\beta)\\)\uc640 \\((\\gamma,\\delta)\\)\uc5d0 \ub300\ud558\uc5ec \uba3c\uc800<br \/>\n\\[\\max\\{\\alpha,\\beta\\}\\quad\\text{\uc640}\\quad\\max\\{\\gamma,\\delta\\}\\]<br \/>\n\ub97c \ube44\uad50\ud558\uace0, \uac19\uc73c\uba74 \uccab\uc9f8 \uc131\ubd84, \ub2e4\uc2dc \uac19\uc73c\uba74 \ub458\uc9f8 \uc131\ubd84\uc744 \ube44\uad50\ud55c\ub2e4. \uc774 \uc21c\uc11c\ub294 \\(\\kappa\\times\\kappa\\)\uc758 \uc815\ub82c\uc21c\uc11c\uc774\ub2e4.<\/p>\n<p>\\((\\alpha,\\beta)\\)\ubcf4\ub2e4 \uc55e\uc5d0 \uc624\ub294 \uc21c\uc11c\uc30d\ub4e4\uc740 \\(\\theta=\\max\\{\\alpha,\\beta\\}\\)\ub77c \ud560 \ub54c \\((\\theta+1)\\times(\\theta+1)\\) \uc548\uc5d0 \ub4e4\uc5b4 \uc788\ub2e4. \\(|\\theta+1|&lt;\\kappa\\)\uc774\uace0 \uadc0\ub0a9\uac00\uc815\uc5d0 \uc758\ud574 \uc774 \uc9d1\ud569\uc758 \uae30\uc218\ub3c4 \\(\\kappa\\)\ubcf4\ub2e4 \uc791\ub2e4. \ub530\ub77c\uc11c \uc704 \uc815\ub82c\uc21c\uc11c\uc758 \ubaa8\ub4e0 \uc9c4\uc808\ud3b8\uc758 \uae30\uc218\ub294 \\(\\kappa\\)\ubcf4\ub2e4 \uc791\ub2e4. \uc774 \uc815\ub82c\uc9d1\ud569\uc758 \uc21c\uc11c\uc720\ud615\uc744 \\(\\eta\\)\ub77c \ud558\uba74 \\(\\eta&gt;\\kappa\\)\uc77c \uacbd\uc6b0 \\(\\kappa\\)\uc5d0\uc11c\uc758 \uc808\ud3b8\uc774 \uae30\uc218 \\(\\kappa\\)\ub97c \uac00\uc838 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(\\eta\\leq\\kappa\\)\uc774\ub2e4. \ud55c\ud3b8 \\(\\alpha\\mapsto(\\alpha,0)\\)\uc740 \\(\\kappa\\to\\kappa\\times\\kappa\\)\uc778 \uc77c\ub300\uc77c\ud568\uc218\uc774\ubbc0\ub85c \uce78\ud1a0\uc5b4-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(\\kappa\\cdot\\kappa=\\kappa\\)\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.7.<\/span><br \/>\n\ubcf8\ubb38\uc5d0\uc11c \\(\\kappa\\times\\kappa\\)\uc5d0 \uc900 \uc815\ub82c\uc21c\uc11c\ub97c \\(\\omega\\times\\omega\\)\uc5d0 \uc801\uc6a9\ud558\uc790. \uc989 \uba3c\uc800 \ub450 \uc88c\ud45c\uc758 \ucd5c\ub313\uac12\uc744 \ube44\uad50\ud558\uace0, \uac19\uc73c\uba74 \uccab\uc9f8 \uc88c\ud45c, \ub2e4\uc2dc \uac19\uc73c\uba74 \ub458\uc9f8 \uc88c\ud45c\ub97c \ube44\uad50\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc774 \uc21c\uc11c\uc5d0\uc11c \ucc98\uc74c \uc544\ud649 \uac1c\uc758 \uc6d0\uc18c\ub97c \ucc28\ub840\ub85c \uc4f0\uc2dc\uc624.<\/li>\n<li>\\((m,n)\\)\ubcf4\ub2e4 \uc55e\uc5d0 \uc624\ub294 \uc6d0\uc18c\uac00 \uc720\ud55c \uac1c\ubfd0\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\uc774 \uc815\ub82c\uc21c\uc11c\uc758 \uc21c\uc11c\uc720\ud615\uc774 \\(\\omega\\)\uc784\uc744 \uc124\uba85\ud558\uace0, \ub530\ub77c\uc11c \\(\\omega\\times\\omega\\sim\\omega\\)\uc784\uc744 \ub2e4\uc2dc \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\uc774\uc81c \\(\\kappa\\), \\(\\lambda\\)\uac00 \ubb34\ud55c\uae30\uc218\uc774\uace0 \\(\\mu=\\max\\{\\kappa,\\lambda\\}\\)\ub77c \ud558\uc790. \ud55c\ucabd \uae30\uc218\uac00 \\(\\mu\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\mu\\leq\\kappa+\\lambda\\leq\\mu+\\mu\\leq\\mu\\cdot\\mu=\\mu\\]<br \/>\n\uc774\uace0,<br \/>\n\\[\\mu\\leq\\kappa\\cdot\\lambda\\leq\\mu\\cdot\\mu=\\mu\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\kappa+\\lambda=\\kappa\\cdot\\lambda=\\mu\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.2. (\ud0c0\ub974\uc2a4\ud0a4\uc758 \uae30\uc218\uc81c\uacf1 \uc815\ub9ac)<\/span><\/p>\n<p>ZF \ud558\uc5d0\uc11c \ub2e4\uc74c \ub450 \uba85\uc81c\ub294 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc120\ud0dd\uacf5\ub9ac.<\/li>\n<li>\ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\times A\\sim A\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc815\ub9ac 7.10\uc5d0\uc11c \uccab\uc9f8 \uba85\uc81c\ub85c\ubd80\ud130 \ub458\uc9f8 \uba85\uc81c\uac00 \ub530\ub974\ub294 \uac83\uc740 \uc774\ubbf8 \ubcf4\uc558\ub2e4. \uc5ed\ubc29\ud5a5\uc758 \ud0c0\ub974\uc2a4\ud0a4 \uc815\ub9ac\ub294 \uc774 \ucc45\uc758 \ubc94\uc704\ub97c \ub118\uc5b4\uac00\ubbc0\ub85c \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4. \ub530\ub77c\uc11c \ubb38\uc81c 7.10\uc758 \uba85\uc81c\ub97c \ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569\uc5d0 \ub300\ud558\uc5ec \uc120\ud0dd\uacf5\ub9ac \uc5c6\uc774 \uc99d\uba85\ud560 \uc218\ub294 \uc5c6\ub2e4.<\/p>\n<h3>4. \uc120\ud0dd \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\ub294 \uc815\ub9ac\ub4e4<\/h3>\n<p>\uc120\ud0dd\uacf5\ub9ac\ub294 \uc5ec\ub7ec \ubd84\uc57c\uc758 \uc874\uc7ac\uc815\ub9ac\ub97c \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub098 \uc815\ub82c \uc815\ub9ac\uc758 \ud615\ud0dc\ub85c \uc99d\uba85\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<h4>(1) \uadf9\ub300 \uc544\uc774\ub514\uc5bc<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.3. (\uadf9\ub300 \uc544\uc774\ub514\uc5bc\uc758 \uc874\uc7ac)<\/span><\/p>\n<p>\uc601\ud658\uc774 \uc544\ub2cc \ub2e8\uc704\uc6d0\uc744 \uac16\ub294 \uac00\ud658\ud658\uc740 \uadf9\ub300 \uc544\uc774\ub514\uc5bc\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\ud658 \\(R\\)\uc758 \uc9c4\uc544\uc774\ub514\uc5bc\ub4e4\uc758 \uc9d1\ud569\uc744 \ud3ec\ud568\uad00\uacc4\ub85c \uc21c\uc11c\ud654\ud55c\ub2e4. \uc9c4\uc544\uc774\ub514\uc5bc\ub4e4\uc758 \uc0ac\uc2ac \\(\\mathcal C\\)\uc5d0 \ub300\ud558\uc5ec \\(\\bigcup\\mathcal C\\)\ub294 \uc544\uc774\ub514\uc5bc\uc774\ub2e4. \ub9cc\uc57d \\(1\\in\\bigcup\\mathcal C\\)\uc774\uba74 \uc5b4\ub5a4 \\(I\\in\\mathcal C\\)\uc5d0 \\(1\\in I\\)\uc774\ubbc0\ub85c \\(I=R\\)\uac00 \ub418\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\bigcup\\mathcal C\\)\ub3c4 \uc9c4\uc544\uc774\ub514\uc5bc\uc774\uba70 \uc0ac\uc2ac\uc758 \uc0c1\uacc4\uc774\ub2e4. \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\uc5d0 \uc758\ud574 \uadf9\ub300\uc778 \uc9c4\uc544\uc774\ub514\uc5bc\uc774 \uc874\uc7ac\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<h4>(2) \ubca1\ud130\uacf5\uac04\uc758 \uae30\uc800<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.4. (\ubca1\ud130\uacf5\uac04\uc758 \uae30\uc800)<\/span><\/p>\n<p>\ubaa8\ub4e0 \ubca1\ud130\uacf5\uac04\uc740 \uae30\uc800\ub97c \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \uc77c\ucc28\ub3c5\ub9bd \ubd80\ubd84\uc9d1\ud569\ub4e4\uc744 \ud3ec\ud568\uad00\uacc4\ub85c \uc21c\uc11c\ud654\ud55c\ub2e4. \uc0ac\uc2ac \\(\\mathcal C\\)\uc758 \ud569\uc9d1\ud569 \\(L=\\bigcup\\mathcal C\\)\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc784\uc744 \ubcf4\uc774\uc790. \\(L\\)\uc758 \uc720\ud55c\ud55c \uc6d0\uc18c\ub4e4\uc774 \uc120\ud615\uad00\uacc4\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74, \uc0ac\uc2ac\uc758 \uc131\uc9c8 \ub54c\ubb38\uc5d0 \uadf8 \uc720\ud55c\ud55c \uc6d0\uc18c\ub4e4\uc744 \ubaa8\ub450 \ud3ec\ud568\ud558\ub294 \ud558\ub098\uc758 \\(C\\in\\mathcal C\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(C\\)\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774\ubbc0\ub85c \ubaa8\ub4e0 \uacc4\uc218\uac00 \\(0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(L\\)\uc740 \uc0ac\uc2ac\uc758 \uc0c1\uacc4\uc774\ub2e4.<\/p>\n<p>\ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub85c \uadf9\ub300 \uc77c\ucc28\ub3c5\ub9bd \uc9d1\ud569 \\(B\\)\ub97c \uc5bb\ub294\ub2e4. \ub9cc\uc57d \\(\\operatorname{span}(B)\\ne V\\)\uc774\uba74 \\(v\\in V\\setminus\\operatorname{span}(B)\\)\ub97c \ud0dd\ud560 \uc218 \uc788\uace0 \\(B\\cup\\{v\\}\\)\ub3c4 \uc77c\ucc28\ub3c5\ub9bd\uc774\ubbc0\ub85c \uadf9\ub300\uc131\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(B\\)\ub294 \\(V\\)\uc758 \uae30\uc800\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.8.<\/span><br \/>\n\\(V=\\mathbb R^3\\)\uc774\uace0<br \/>\n\\[<br \/>\nL=\\{(1,0,0),(1,1,0)\\}<br \/>\n\\]<br \/>\n\ub77c \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(L\\)\uc774 \uc77c\ucc28\ub3c5\ub9bd\uc784\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<li>\\(v\\in V\\)\ub97c \ud558\ub098 \ucc3e\uc544 \\(L\\cup\\{v\\}\\)\uac00 \\(V\\)\uc758 \uae30\uc800\uac00 \ub418\uac8c \ud558\uc2dc\uc624.<\/li>\n<li>\uc720\ud55c\ucc28\uc6d0\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uacfc\uc815\uc744 \uc720\ud55c \ubc88 \ubc18\ubcf5\ud558\uc5ec \uae30\uc800\ub97c \uc5bb\uc744 \uc218 \uc788\uc74c\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\ubcf8\ubb38\uc758 \uc77c\ubc18\uc801\uc778 \uae30\uc800 \uc874\uc7ac \uc99d\uba85\uc5d0\uc11c \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\uac00 \ud544\uc694\ud55c \uc9c0\uc810\uc774 \ubb34\uc5c7\uc778\uc9c0 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h4>(3) \ud2f0\ud638\ub178\ud504 \uc815\ub9ac<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.5. (\ud2f0\ud638\ub178\ud504 \uc815\ub9ac)<\/span><\/p>\n<p>\ucef4\ud329\ud2b8 \uacf5\uac04\ub4e4\uc758 \uc784\uc758\uc758 \uacf1\uc740 \uacf1\uc704\uc0c1\uc5d0\uc11c \ucef4\ud329\ud2b8\uc774\ub2e4.<\/p>\n<\/div>\n<p>ZF \ud558\uc5d0\uc11c \uc774 \uc77c\ubc18\ud615\uc758 \ud2f0\ud638\ub178\ud504 \uc815\ub9ac\ub294 \uc120\ud0dd\uacf5\ub9ac\uc640 \ub3d9\uce58\uc774\ub2e4. \ud55c \ud45c\uc900\uc801\uc778 \uc99d\uba85\uc740 \ud544\ud130\ub97c \uadf9\ub300\ud544\ud130\ub85c \ud655\uc7a5\ud55c \ub4a4 \uac01 \uc88c\ud45c\uacf5\uac04\uc5d0\uc11c \uc218\ub834\uc810\uc744 \uc5bb\uace0, \uadf8 \uc218\ub834\uc810\ub4e4\uc744 \uc88c\ud45c\ubcc4\ub85c \uc120\ud0dd\ud558\uc5ec \uacf1\uacf5\uac04\uc758 \uc810\uc744 \uad6c\uc131\ud55c\ub2e4. \uc774 \uacfc\uc815\uc758 \uc0c1\uc138\ud55c \uc704\uc0c1\uc218\ud559\uc801 \uc99d\uba85\uc740 \uc774 \ucc45\uc758 \ubc94\uc704\ub97c \ub118\uc5b4\uac00\ubbc0\ub85c \uc0dd\ub7b5\ud55c\ub2e4. \ucef4\ud329\ud2b8 \ud558\uc6b0\uc2a4\ub3c4\ub974\ud504 \uacf5\uac04\uc73c\ub85c \uc81c\ud55c\ud55c \ud2f0\ud638\ub178\ud504 \uc815\ub9ac\ub294 \uc804\uccb4 \uc120\ud0dd\uacf5\ub9ac\ubcf4\ub2e4 \uc57d\ud55c \ubd88 \uc18c\uc544\uc774\ub514\uc5bc \uc815\ub9ac(Boolean prime ideal theorem)\uc640 \ub3d9\uce58\uc774\ubbc0\ub85c \ub450 \ud615\ud0dc\ub97c \uad6c\ubcc4\ud574\uc57c \ud55c\ub2e4.<\/p>\n<h3>5. \uc120\ud0dd\uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc5bb\ub294 \ube44\uad6c\uc131\uc801 \uacb0\uacfc<\/h3>\n<p>\uc120\ud0dd\uacf5\ub9ac\ub294 \ub9e4\uc6b0 \uac15\ud55c \uc874\uc7ac\uc815\ub9ac\ub97c \uc81c\uacf5\ud558\uc9c0\ub9cc, \uc120\ud0dd\ub41c \ub300\uc0c1\uc744 \uba85\uc2dc\uc801\uc73c\ub85c \uae30\uc220\ud558\ub294 \ubc29\ubc95\uc744 \uc8fc\uc9c0\ub294 \uc54a\ub294\ub2e4. \ub2e4\uc74c \ub450 \uacb0\uacfc\uac00 \ub300\ud45c\uc801\uc774\ub2e4.<\/p>\n<h4>(1) \ube44\uac00\uce21 \uc9d1\ud569\uc758 \uc874\uc7ac<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.6. (\ube44\ud0c8\ub9ac \uc9d1\ud569)<\/span><\/p>\n<p>\uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uba74 \\([0,1]\\)\uc758 \ub974\ubca0\uadf8 \ube44\uac00\uce21 \ubd80\ubd84\uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\\([0,1]\\)\uc5d0\uc11c<br \/>\n\\[x\\sim y\\quad\\Longleftrightarrow\\quad x-y\\in\\mathbb Q\\]<br \/>\n\ub85c \ub450\uba74 \ub3d9\uce58\uad00\uacc4\uac00 \ub41c\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uac01 \ub3d9\uce58\ub958\uc5d0\uc11c \ub300\ud45c\uc6d0\uc18c\ub97c \ud558\ub098\uc529 \ud0dd\ud55c \uc9d1\ud569 \\(V\\subseteq[0,1]\\)\ub97c \ub9cc\ub4e0\ub2e4. \\(q\\in\\mathbb Q\\cap[-1,1]\\)\uc5d0 \ub300\ud55c \ud3c9\ud589\uc774\ub3d9 \\(V+q\\)\ub4e4\uc740 \uc11c\ub85c\uc18c\uc774\uace0,<br \/>\n\\[[0,1]\\subseteq\\bigcup_{q\\in\\mathbb Q\\cap[-1,1]}(V+q)\\subseteq[-1,2]\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\\(V\\)\uac00 \uac00\uce21\uc774\ub77c\uace0 \ud558\uc790. \uce21\ub3c4\uc758 \ud3c9\ud589\uc774\ub3d9 \ubd88\ubcc0\uc131 \ub54c\ubb38\uc5d0 \ubaa8\ub4e0 \\(V+q\\)\ub294 \uac19\uc740 \uce21\ub3c4\ub97c \uac16\ub294\ub2e4. \uadf8 \uce21\ub3c4\uac00 \\(0\\)\uc774\uba74 \uc704 \uac00\uc0b0\ud569\uc9d1\ud569\ub3c4 \uce21\ub3c4 \\(0\\)\uc774 \ub418\uc5b4 \\([0,1]\\)\uc744 \ud3ec\ud568\ud560 \uc218 \uc5c6\ub2e4. \uc591\uc218\uc774\uba74 \uc11c\ub85c\uc18c\uc778 \uac00\uc0b0 \uac1c\uc758 \ud3c9\ud589\uc774\ub3d9\ub4e4\uc758 \ud569\uc9d1\ud569\uc740 \ubb34\ud55c\ud55c \uce21\ub3c4\ub97c \uac00\uc838\uc57c \ud558\uc9c0\ub9cc \\([-1,2]\\) \uc548\uc5d0 \ub4e4\uc5b4 \uc788\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(V\\)\ub294 \ube44\uac00\uce21\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.9.<\/span><br \/>\n\\([0,1]\\)\uc5d0\uc11c \\(x\\sim y\\Longleftrightarrow x-y\\in\\mathbb Q\\)\ub85c \uc815\uc758\ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\sim\\)\uc774 \ub3d9\uce58\uad00\uacc4\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\uac01 \ub3d9\uce58\ub958\uc5d0\uc11c \ub300\ud45c\uc6d0\uc18c \ud558\ub098\uc529\uc744 \ud0dd\ud55c \uc9d1\ud569\uc744 \\(V\\)\ub77c \ud558\uc790. \uc11c\ub85c \ub2e4\ub978 \\(q_1,q_2\\in\\mathbb Q\\cap[-1,1]\\)\uc5d0 \ub300\ud558\uc5ec \\((V+q_1)\\cap(V+q_2)=\\varnothing\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\ubaa8\ub4e0 \\(x\\in[0,1]\\)\uc774 \uc5b4\ub5a4 \\(q\\in\\mathbb Q\\cap[-1,1]\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\in V+q\\)\ub97c \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h4>(2) \ubc14\ub098\ud750-\ud0c0\ub974\uc2a4\ud0a4 \uc815\ub9ac<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.7. (\ubc14\ub098\ud750-\ud0c0\ub974\uc2a4\ud0a4 \uc815\ub9ac)<\/span><\/p>\n<p>\uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uba74 \\(\\mathbb R^3\\)\uc758 \ub2e8\uc704\uad6c\ub97c \uc720\ud55c \uac1c\uc758 \uc11c\ub85c\uc18c\uc778 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c \ubd84\ud560\ud55c \ub4a4, \uac01 \uc870\uac01\uc744 \ud68c\uc804\uacfc \ud3c9\ud589\uc774\ub3d9\ud558\uc5ec \uc6d0\ub798 \ub2e8\uc704\uad6c\uc640 \ud569\ub3d9\uc778 \ub450 \uac1c\uc758 \uad6c\ub85c \uc7ac\uc870\ub9bd\ud560 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n<p>\uc99d\uba85\uc758 \ud575\uc2ec\uc740 \\(SO(3)\\) \uc548\uc758 \ub450 \ud68c\uc804\uc774 \uc0dd\uc131\ud558\ub294 \uc790\uc720\ubd80\ubd84\uad70\uc758 \uc5ed\uc124\uc801 \ubd84\ud574\uc774\ub2e4. \uc774 \uc790\uc720\uad70\uc774 \uad6c\uba74\uc5d0 \uc791\uc6a9\ud560 \ub54c \ube44\uc790\uba85\ud55c \uc548\uc815\uc790\ub97c \uac16\ub294 \uc608\uc678\uc801\uc778 \uc810\ub4e4\uc744 \uba3c\uc800 \uc81c\uac70\ud558\uba74 \ub0a8\uc740 \uac01 \uada4\ub3c4\ub294 \uc790\uc720\uad70 \uc790\uccb4\uc640 \uc77c\ub300\uc77c\ub85c \ub300\uc751\ud55c\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uac01 \uada4\ub3c4\uc5d0\uc11c \ub300\ud45c\uc6d0\uc18c\ub97c \ud558\ub098\uc529 \uc120\ud0dd\ud558\uace0, \uc790\uc720\uad70\uc758 \uc5ed\uc124\uc801 \ubd84\ud574\ub97c \uac01 \uada4\ub3c4\uc5d0 \ub3d9\uc2dc\uc5d0 \uc62e\uae34\ub2e4. \uc608\uc678\uc801\uc778 \uac00\uc0b0\uc9d1\ud569\uc744 \ub2e4\uc2dc \ud761\uc218\ud558\uace0 \uad6c\uba74\uc758 \ubd84\ud574\ub97c \ubc18\uc9c0\ub984 \ubc29\ud5a5\uc73c\ub85c \uc5f0\uc7a5\ud558\uba74 \ub2e8\uc704\uad6c\uc758 \ubd84\ud574\ub97c \uc5bb\ub294\ub2e4. \uc138\ubd80 \uad6c\uc131\uc740 \uad70\ub860\uacfc \uce21\ub3c4\ub860\uc744 \ud544\uc694\ub85c \ud558\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uc0dd\ub7b5\ud55c\ub2e4.<\/p>\n<p>\uc774 \uacb0\uacfc\ub294 \uc870\uac01\ub4e4\uc774 \ubcf4\ud1b5\uc758 \ubd80\ud53c\ub97c \uac16\ub294 \uac00\uce21\uc9d1\ud569\uc77c \uc218 \uc5c6\uae30 \ub54c\ubb38\uc5d0 \ubd80\ud53c\uc758 \uac00\uc0b0\uac00\ubc95\uc131\uacfc \ubaa8\uc21c\ub418\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<h3>6. \uc57d\ud55c \uc120\ud0dd\uc6d0\ub9ac<\/h3>\n<p>\uc120\ud0dd\uacf5\ub9ac\uc758 \uc77c\ubd80\ub9cc\uc744 \uc694\uad6c\ud558\ub294 \uc5ec\ub7ec \uc57d\ud55c \uc6d0\ub9ac\uac00 \uc788\ub2e4. \uc774\ub4e4\uc744 ZF\uc5d0\uc11c \uc99d\uba85\ub418\ub294 \uc0ac\uc2e4\uacfc \uad6c\ubcc4\ud574\uc57c \ud55c\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\uc720\ud55c \uc120\ud0dd<\/span>: \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569 \uc720\ud55c \uac1c\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \uc120\ud0dd\ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\uac83\uc740 ZF\uc5d0\uc11c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ub41c\ub2e4.<\/li>\n<li><span class=\"defined\">\uac00\uc0b0 \uc120\ud0dd \uacf5\ub9ac<\/span>(Axiom of Countable Choice, \\(\\mathrm{AC}_\\omega\\)): \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0 \uc9d1\ud569\uc871\uc5d0\ub294 \uc120\ud0dd\ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub294 ZF\uc5d0\uc11c\ub294 \uc77c\ubc18\uc801\uc73c\ub85c \uc99d\uba85\ub418\uc9c0 \uc54a\uc73c\uba70 \uc120\ud0dd\uacf5\ub9ac\ubcf4\ub2e4 \uc57d\ud558\ub2e4. \ub530\ub77c\uc11c \u201c\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uac00\uc0b0 \uac1c\uc758 \uc9d1\ud569\uc758 \uacf1\uc740 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4\u201d\ub77c\ub294 \uba85\uc81c\ub294 \uc120\ud0dd\uacf5\ub9ac \uc5c6\uc774 \uc790\ub3d9\uc73c\ub85c \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<li><span class=\"defined\">\uc885\uc18d \uc120\ud0dd \uacf5\ub9ac<\/span>(Axiom of Dependent Choice, DC): \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569 \\(X\\)\uc758 \uad00\uacc4 \\(R\\)\uc774 \ubaa8\ub4e0 \\(x\\in X\\)\uc5d0 \ub300\ud574 \uc5b4\ub5a4 \\(y\\in X\\)\uc640 \\(xRy\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(x_0\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(x_nRx_{n+1}\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218\uc5f4 \\((x_n)_{n\\in\\mathbb N}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. DC\ub294 \ud574\uc11d\ud559\uc5d0\uc11c \uc790\uc8fc \ucda9\ubd84\ud55c \uc120\ud0dd\uc6d0\ub9ac\uc774\uba70 \uc804\uccb4 \uc120\ud0dd\uacf5\ub9ac\ubcf4\ub2e4 \uc57d\ud558\ub2e4.<\/li>\n<li><span class=\"defined\">\uc720\ud55c\uc9d1\ud569 \uc120\ud0dd\uacf5\ub9ac<\/span>(Axiom of Choice for Finite Sets): \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc720\ud55c\uc9d1\ud569\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc784\uc758\uc758 \uc9d1\ud569\uc871\uc5d0 \uc120\ud0dd\ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\uac83\ub3c4 \uc720\ud55c\ud55c \uc9d1\ud569\uc871\uc5d0 \ub300\ud55c \uc120\ud0dd\uacfc\ub294 \ub2e4\ub974\uba70 ZF\uc5d0\uc11c \uc77c\ubc18\uc801\uc73c\ub85c \uc99d\uba85\ub418\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ul>\n<p>\uc608\ub97c \ub4e4\uc5b4 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc5d0\uc11c \uc720\ud55c\ud55c \uc0dd\uc131\uc9d1\ud569\uc73c\ub85c\ubd80\ud130 \uae30\uc800\ub97c \ucd94\ucd9c\ud558\ub294 \uacfc\uc815\uc740 \uc804\uccb4 \uc120\ud0dd\uacf5\ub9ac\ub97c \ud544\uc694\ub85c \ud558\uc9c0 \uc54a\ub294\ub2e4. \ubc18\uba74 \uc784\uc758\uc758 \ubca1\ud130\uacf5\uac04\uc758 \uae30\uc800 \uc874\uc7ac\ub294 \uc704\uc5d0\uc11c \ubcf8 \uac83\ucc98\ub7fc \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc5bb\ub294\ub2e4. \uc5b4\ub5a4 \uc218\ud559 \uc815\ub9ac\uc5d0 \uc5b4\ub290 \uc815\ub3c4\uc758 \uc120\ud0dd\uc6d0\ub9ac\uac00 \ud544\uc694\ud55c\uc9c0\ub294 \uc815\ub9ac\ub9c8\ub2e4 \ubcc4\ub3c4\ub85c \ud655\uc778\ud574\uc57c \ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 10.10.<\/span><br \/>\n\ub2e4\uc74c \uc0c1\ud669\uc5d0\uc11c \ubcf8\ubb38\uc5d0 \uc18c\uac1c\ud55c \uc120\ud0dd\uc6d0\ub9ac \uac00\uc6b4\ub370 \uc9c1\uc811 \ub300\uc751\ud558\ub294 \uac83\uc744 \ucc3e\uc73c\uc2dc\uc624. \uc120\ud0dd\uacf5\ub9ac\uac00 \uc804\ud600 \ud544\uc694\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0\uc5d0\ub294 \uadf8\ub807\uac8c \ubc1d\ud788\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569 \ub2e4\uc12f \uac1c\uc5d0\uc11c \uac01\uac01 \uc6d0\uc18c \ud558\ub098\ub97c \uc120\ud0dd\ud55c\ub2e4.<\/li>\n<li>\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0 \uc9d1\ud569\uc871\uc5d0\uc11c \uac01\uac01 \uc6d0\uc18c \ud558\ub098\ub97c \uc120\ud0dd\ud55c\ub2e4.<\/li>\n<li>\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc720\ud55c\uc9d1\ud569\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc784\uc758\uc758 \uc9d1\ud569\uc871\uc5d0\uc11c \uac01\uac01 \uc6d0\uc18c \ud558\ub098\ub97c \uc120\ud0dd\ud55c\ub2e4.<\/li>\n<li>\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569 \\(X\\)\uc758 \uad00\uacc4 \\(R\\)\uc774 \ubaa8\ub4e0 \\(x\\in X\\)\uc5d0 \ub300\ud574 \uc5b4\ub5a4 \\(y\\in X\\)\uc640 \\(xRy\\)\ub97c \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(x_nRx_{n+1}\\)\uc778 \uc218\uc5f4\uc744 \ub9cc\ub4e0\ub2e4.<\/li>\n<li>\uc544\ubb34 \uc81c\ud55c\uc774 \uc5c6\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\ub4e4\uc758 \uc784\uc758\uc758 \uc9d1\ud569\uc871\uc5d0\uc11c \uac01\uac01 \uc6d0\uc18c \ud558\ub098\ub97c \uc120\ud0dd\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-logic\/\">\uc9d1\ud569\uacfc \uc218\ub9ac\ub17c\ub9ac \uccab\uac78\uc74c \ubaa9\ucc28 \ubcf4\uae30<\/a><\/p>\n<p><span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch01-naive-logic\/\">\uba85\uc81c\uc640 \ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch02-sets\">\uc9d1\ud569\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch03-algebra-of-classes\">\ub2e4\uc591\ud55c \uc9d1\ud569\uc758 \uc5f0\uc0b0<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch04-relations-and-functions\">\uad00\uacc4\uc640 \ud568\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">\uc790\uc5f0\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">\uc9d1\ud569\uc758 \uae30\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch08-ordinal-numbers\">\uc9d1\ud569\uc758 \uc11c\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">\uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch10-axiom-of-choice\">\uc120\ud0dd \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch11-formal-logic\">\ud615\uc2dd\ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch12-propositional-logic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch13-soundness-completeness-proplogic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac74\uc804\uc131\uacfc \uc644\uc804\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch15-semantics-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch16-inference-rule-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucd94\ub860\uaddc\uce59<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch17-compactness-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch18-peano-arithmetics\">\ud398\uc544\ub178 \uc0b0\uc220<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch19-incompleteness-theorem\">\ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/a><\/span>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>9\uc7a5\uc5d0\uc11c ZF\uc5d0 \uc120\ud0dd\uacf5\ub9ac\ub97c \ub354\ud55c \uccb4\uacc4\ub97c ZFC\ub77c\uace0 \uc815\uc758\ud558\uc600\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub294 \uc720\ud55c\ud55c \uc120\ud0dd\uc5d0\uc11c\ub294 \ub4dc\ub7ec\ub098\uc9c0 \uc54a\uc9c0\ub9cc, \uc11c\ub85c \uc544\ubb34 \uad00\ub828\uc774 \uc5c6\uace0 \ubb34\ud55c\ud788 \ub9ce\uc73c\uba70 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc5d0\uc11c \uc6d0\uc18c\ub97c \ud558\ub098\uc529 \ub3d9\uc2dc\uc5d0 \uace8\ub77c\uc57c \ud560 \ub54c \ud575\uc2ec\uc801\uc778 \uc5ed\ud560\uc744 \ud55c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uacf5\ub9ac 9.12\uc758 \uc5ec\ub7ec \ub3d9\uce58\ud615\ud0dc\uc640 \ub300\ud45c\uc801\uc778 \uc751\uc6a9\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uc120\ud0dd\uacf5\ub9ac\ubcf4\ub2e4 \uc57d\ud55c \uc120\ud0dd\uc6d0\ub9ac\ub3c4 \uad6c\ubd84\ud55c\ub2e4. \uad34\ub378\uc740 ZF\uac00 \ubb34\ubaa8\uc21c\uc774\ub77c\uba74 ZFC\ub3c4 \ubb34\ubaa8\uc21c\uc784\uc744 \ubcf4\uc600\uace0, \ucf54\uc5b8\uc740 ZF\uac00 \ubb34\ubaa8\uc21c\uc774\ub77c\uba74 \\(\\mathrm{ZF}+\\neg\\mathrm{AC}\\)\ub3c4 \ubb34\ubaa8\uc21c\uc784\uc744 \ubcf4\uc600\ub2e4. \ub530\ub77c\uc11c ZF\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \uc804\uc81c\ub85c \ud558\uba74 \uc120\ud0dd\uacf5\ub9ac\ub294 ZF\ub85c\ubd80\ud130 \uc99d\uba85\ud560 \uc218\ub3c4&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":110,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9266","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9266","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9266"}],"version-history":[{"count":7,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9266\/revisions"}],"predecessor-version":[{"id":10086,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9266\/revisions\/10086"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9246"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9266"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}