{"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":"2025-10-20T18:48:57","modified_gmt":"2025-10-20T09:48:57","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><span class=\"defined\">\uc120\ud0dd \uacf5\ub9ac<\/span>(Axiom of Choice, AC)\ub294 \uc9d1\ud569\ub860\uc5d0\uc11c \ub17c\ub780\uc774 \ub9ce\uc558\ub358 \uacf5\ub9ac\uc774\ub2e4. \uc9c1\uad00\uc801\uc73c\ub85c\ub294 \ub2f9\uc5f0\ud574 \ubcf4\uc774\uc9c0\ub9cc, \uc774\ub85c\ubd80\ud130 \uc9c1\uad00\uc5d0 \ubc18\ud558\ub294 \uacb0\uacfc\ub4e4\uc774 \ub3c4\ucd9c\ub418\uae30\ub3c4 \ud55c\ub2e4. 20\uc138\uae30 \ucd08 \ub9ce\uc740 \uc218\ud559\uc790\ub4e4\uc774 \uc120\ud0dd \uacf5\ub9ac\ub97c \ub2e4\ub978 \uacf5\ub9ac\ub85c\ubd80\ud130 \uc99d\uba85\ud558\uac70\ub098 \ub610\ub294 \ub2e4\ub978 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \ubc18\uc99d\ud558\ub824\uace0 \ud588\uc9c0\ub9cc \uc2e4\ud328\ud558\uc600\ub2e4. 1963\ub144 \ucf54\uc5b8(Paul Cohen)\uc740 \uc120\ud0dd \uacf5\ub9ac\uac00 ZF\uc640 \ub3c5\ub9bd\uc801\uc784\uc744 \uc99d\uba85\ud588\ub2e4.<\/p>\n<h3>1. \uc120\ud0dd \uacf5\ub9ac\uc758 \uc758\ubbf8<\/h3>\n<p>\uc120\ud0dd \uacf5\ub9ac\ub97c \uc774\ud574\ud558\uae30 \uc704\ud574 \uba3c\uc800 <span class=\"defined\">\uc120\ud0dd\ud568\uc218<\/span>(choice function)\ub97c \uc815\uc758\ud558\uc790. \\(\\mathcal{F}\\)\uac00 \uc9d1\ud569\uc871\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \ud568\uc218 \\(f: \\mathcal{F} \\to \\bigcup \\mathcal{F}\\)\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 \ubd80\ub978\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 &#8216;\uc120\ud0dd&#8217;\ud558\ub294 \ud568\uc218\uc774\ub2e4. \uc608\ub97c \ub4e4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\\(\\mathcal{F} = \\{\\{1,\\, 2\\},\\, \\{3,\\, 4,\\, 5\\},\\, \\{6\\}\\}\\)\uc5d0 \ub300\ud574, \\(f(\\{1,\\,2\\}) = 1\\), \\(f(\\{3,\\,4,\\,5\\}) = 4\\), \\(f(\\{6\\}) = 6\\)\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218\ub294 \uc120\ud0dd\ud568\uc218\uc774\ub2e4. \ubb3c\ub860 \\(\\mathcal{F}\\)\uc758 \uc120\ud0dd\ud568\uc218\ub294 \uc774\uac83 \uc678\uc5d0\ub3c4 \ub2e4\uc12f \uac1c\ub97c \ub354 \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/li>\n<li>\uc77c\ubc18\uc801\uc73c\ub85c, \ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uc720\ud55c \uac1c\uc758 \uc9d1\ud569\uc5d0 \ub300\ud574\uc11c\ub294 \uc120\ud0dd\ud568\uc218\ub97c \uad6c\uccb4\uc801\uc73c\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/li>\n<\/ul>\n<h3>2. \uc120\ud0dd \uacf5\ub9ac\uc758 \uc5ec\ub7ec \ud615\ud0dc<\/h3>\n<p>\uc120\ud0dd \uacf5\ub9ac\ub294 \uc5ec\ub7ec \ubc29\uc2dd\uc73c\ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li><strong>\uc9c1\uad00\uc801 \ud45c\ud604<\/strong>: &#8220;\ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \ubaa8\uc784\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uac01 \uc9d1\ud569\uc5d0\uc11c \uc6d0\uc18c\ub97c \ud558\ub098\uc529 \uc120\ud0dd\ud560 \uc218 \uc788\ub2e4.&#8221;<\/li>\n<li><strong>\uc5c4\ubc00\ud55c \ud45c\ud604<\/strong>: \uacf5\uc9d1\ud569\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uc784\uc758\uc758 \uc9d1\ud569\uc871 \\(\\mathcal{F}\\)\uc5d0 \ub300\ud574 \uc120\ud0dd\ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[\\forall \\mathcal{F} \\left( \\varnothing \\notin \\mathcal{F} \\rightarrow \\left( \\exists f : \\mathcal{F} \\to \\bigcup \\mathcal{F},\\,\\, \\forall A \\in \\mathcal{F} (f(A) \\in A) \\right) \\right).\\]<\/li>\n<li><strong>\uacf1\uc9d1\ud569\uc744 \uc774\uc6a9\ud55c \ud45c\ud604<\/strong>: \ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uc871 \\(\\{A_i\\}_{i \\in I}\\)\uc5d0 \ub300\ud574, \uacf1\uc9d1\ud569 \\(\\prod_{i \\in I} A_i\\)\ub294 \ube44\uc5b4\uc788\uc9c0 \uc54a\ub2e4.<\/li>\n<\/ul>\n<h3>3. \uc5ed\uc0ac\uc801 \ubc30\uacbd<\/h3>\n<p>1904\ub144 \uccb4\ub974\uba5c\ub85c(Ernst Zermelo)\uac00 \uc815\ub82c \uc815\ub9ac\ub97c \uc99d\uba85\ud558\uba74\uc11c \uc120\ud0dd \uacf5\ub9ac\ub97c \uba85\uc2dc\uc801\uc73c\ub85c \uc0ac\uc6a9\ud588\ub2e4. \uc774\ud6c4 \ub9ce\uc740 \uc218\ud559\uc790\ub4e4 \uc0ac\uc774\uc5d0\uc11c \ub17c\uc7c1\uc774 \uc77c\uc5b4\ub0ac\ub2e4.<\/p>\n<ul>\n<li>\ud790\ubca0\ub974\ud2b8, \uccb4\ub974\uba5c\ub85c \ub4f1\uc740 \uc120\ud0dd \uacf5\ub9ac\uac00 \uc218\ud559\uc758 \uc911\uc694\ud55c \uc815\ub9ac\ub97c \uc99d\uba85\ud558\ub294 \ub370 \ud544\uc218\uc801\uc774\ub77c\uace0 \uc8fc\uc7a5\ud588\ub2e4.<\/li>\n<li>\ube0c\ub77c\uc6b0\uc5b4, \ubcf4\ub810 \ub4f1\uc740 \uc120\ud0dd \uacf5\ub9ac\uac00 \uad6c\uc131\uc801\uc774\uc9c0 \uc54a\uace0 \uc9c1\uad00\uc5d0 \ubc18\ud558\ub294 \uacb0\uacfc\ub97c \ub0b3\ub294\ub2e4\uace0 \ube44\ud310\ud588\ub2e4.<\/li>\n<li>\ub7ec\uc140, \ud398\uc544\ub178 \ub4f1\uc740 \uc120\ud0dd \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud55c \uc815\ub9ac\uc640 \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uc740 \uc815\ub9ac\ub97c \uad6c\ubd84\ud574\uc57c \ud55c\ub2e4\uace0 \uc8fc\uc7a5\ud588\ub2e4.<\/li>\n<\/ul>\n<p>1938\ub144 \uad34\ub378\uc774 \uc120\ud0dd \uacf5\ub9ac\uac00 ZF\uc640 \ubaa8\uc21c\ub418\uc9c0 \uc54a\uc74c\uc744 \uc99d\uba85\ud588\uace0, 1963\ub144 \ucf54\uc5b8\uc774 \uc120\ud0dd \uacf5\ub9ac\uac00 ZF\ub85c\ubd80\ud130 \uc99d\uba85\ub420 \uc218 \uc5c6\uc74c\uc744 \ubcf4\uc600\ub2e4. \uc774\ub85c\uc368 \uc120\ud0dd \uacf5\ub9ac\ub294 ZF\uc640 \ub3c5\ub9bd\uc801\uc784\uc774 \ubc1d\ud600\uc84c\ub2e4.<\/p>\n<h3>4. \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\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<h4>(1) \uc815\ub82c \uc815\ub9ac (Well-Ordering Theorem)<\/h4>\n<p><strong>&#8220;\ubaa8\ub4e0 \uc9d1\ud569\uc740 \uc815\ub82c \uac00\ub2a5\ud558\ub2e4.&#8221;<\/strong><\/p>\n<p>\uc9d1\ud569 \\(A\\)\uac00 <span class=\"defined\">\uc815\ub82c \uac00\ub2a5<\/span>(well-orderable)\ud558\ub2e4\ub294 \uac83\uc740, \\(A\\) \uc704\uc5d0 \uc815\ub82c \uc21c\uc11c\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4. \uc815\ub82c \uc21c\uc11c\ub780 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc774 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac16\ub294 \uc804\uc21c\uc11c\uc774\ub2e4.<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4, \uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\mathbb{N}\\)\uc740 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\ub85c \uc815\ub82c\ub418\uc5b4 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uc2e4\uc218 \uc9d1\ud569 \\(\\mathbb{R}\\)\uc758 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\ub294 \uc815\ub82c\uc774 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uba74, \uc5f4\ub9b0 \uad6c\uac04 \\((0,\\, 1)\\)\uc740 \ucd5c\uc18c\uc6d0\uc18c\uac00 \uc5c6\ub2e4.<\/p>\n<h4>(2) \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac (Zorn&#8217;s Lemma)<\/h4>\n<p><strong>&#8220;\uc21c\uc11c\uc9d1\ud569\uc5d0\uc11c \ubaa8\ub4e0 \uc0ac\uc2ac\uc774 \uc0c1\uacc4\ub97c \uac00\uc9c0\uba74, \uadf9\ub300\uc6d0\uc18c\uac00 \uc874\uc7ac\ud55c\ub2e4.&#8221;<\/strong><\/p>\n<p>\uc5ec\uae30\uc11c <span class=\"defined\">\uc0ac\uc2ac<\/span>(chain)\uc740 \uc784\uc758\uc758 \ub450 \uc6d0\uc18c\uac00 \ube44\uad50 \uac00\ub2a5\ud55c \ubd80\ubd84\uc9d1\ud569\uc774\uace0, <span class=\"defined\">\uc0c1\uacc4<\/span>(upper bound)\ub294 \uc0ac\uc2ac\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ubcf4\ub2e4 \ud06c\uac70\ub098 \uac19\uc740 \uc6d0\uc18c\uc774\uba70, <span class=\"defined\">\uadf9\ub300\uc6d0\uc18c<\/span>(maximal element)\ub294 \uadf8\ubcf4\ub2e4 \ud070 \uc6d0\uc18c\uac00 \uc5c6\ub294 \uc6d0\uc18c\uc774\ub2e4.<\/p>\n<p>ZF \ud558\uc5d0\uc11c \uc120\ud0dd \uacf5\ub9ac, \uc815\ub82c \uc815\ub9ac, \ucd08\ub978\uc758 \ubcf4\uc870 \uc815\ub9ac\ub294 \ubaa8\ub450 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 10.1.<\/span><br \/>\nZF \ud558\uc5d0\uc11c \uc120\ud0dd \uacf5\ub9ac\uc640 \ub3d9\uce58\uc778 &#8216;\ud558\uc6b0\uc2a4\ub3c4\ub974\ud504\uc758 \uadf9\ub300 \uc6d0\ub9ac'(Hausdorff maximal principle)\ub97c \ucc3e\uc544 \ubcf4\uc790.<\/p>\n<\/div>\n<h3>5. \uc120\ud0dd \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\ub294 \uc815\ub9ac\ub4e4<\/h3>\n<p>\uc120\ud0dd \uacf5\ub9ac\ub294 \uc218\ud559\uc758 \uc5ec\ub7ec \ubd84\uc57c\uc5d0\uc11c \uc911\uc694\ud55c \uc815\ub9ac\ub97c \uc99d\uba85\ud558\ub294 \ub370 \uc0ac\uc6a9\ub41c\ub2e4. \uc5ec\uae30\uc11c\ub294 \uadf8 \uc608\ub85c\uc11c \uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uc815\ub9ac \uba87 \uac1c\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<h4>(1) \ucd94\uc0c1\ub300\uc218\ud559\uc5d0\uc11c\uc758 \uc608<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.1.<\/span><\/p>\n<p>\ub2e8\uc704\uc6d0\uc744 \uac16\ub294 \ubaa8\ub4e0 \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 \uac1c\uc694<\/span><br \/>\n\ud658 \\(R\\)\uc758 \uc9c4\ubd80\ubd84 \uc544\uc774\ub514\uc5bc\ub4e4\uc758 \uc9d1\ud569\uc5d0 \ud3ec\ud568\uad00\uacc4\ub85c \uc21c\uc11c\ub97c \uc8fc\uace0, \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \uc801\uc6a9\ud55c\ub2e4. \uc544\uc774\ub514\uc5bc\ub4e4\uc758 \uc0ac\uc2ac\uc758 \ud569\uc9d1\ud569\uc774 \uc0c1\uacc4\uac00 \ub428\uc744 \ubcf4\uc774\uba74, \uadf9\ub300\uc6d0\uc18c(\uadf9\ub300 \uc544\uc774\ub514\uc5bc)\uc758 \uc874\uc7ac\uac00 \ubcf4\uc7a5\ub41c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<h4>(2) \uc120\ud615\ub300\uc218\ud559\uc5d0\uc11c\uc758 \uc608<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.2.<\/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 \uac1c\uc694<\/span><br \/>\n\ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \uc77c\ucc28\ub3c5\ub9bd \ubd80\ubd84\uc9d1\ud569\ub4e4\uc758 \uc9d1\ud569\uc744 \uc0dd\uac01\ud55c\ub2e4. \ud3ec\ud568\uad00\uacc4\ub85c \uc21c\uc11c\ub97c \uc8fc\uace0 \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74, \uadf9\ub300\uc778 \uc77c\ucc28\ub3c5\ub9bd \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc774 \uc9d1\ud569\uc774 \uae30\uc800\uac00 \ub428\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\ub294 \ubb34\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc5d0\uc11c \ud2b9\ud788 \uc911\uc694\ud558\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc2e4\uc218 \uc704\uc758 \ubaa8\ub4e0 \uc2e4\ud568\uc218\ub4e4\uc758 \ubca1\ud130\uacf5\uac04\uc740 \uae30\uc800\ub97c \uac00\uc9c4\ub2e4. \ube44\ub85d \uadf8 \uae30\uc800\ub97c \uad6c\uccb4\uc801\uc73c\ub85c \uad6c\uc131\ud560 \uc218\ub294 \uc5c6\uc9c0\ub9cc \uae30\uc800\uc758 \uc874\uc7ac\uc131\uc774 \ubcf4\uc7a5\ub41c\ub2e4.<\/p>\n<h4>(3) \uc704\uc0c1\uc218\ud559\uc5d0\uc11c\uc758 \uc608<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.3. (\ud2f0\ud638\ub178\ud504 \uc815\ub9ac)<\/span><\/p>\n<p>\ucef4\ud329\ud2b8 \uacf5\uac04\ub4e4\uc758 \uc784\uc758 \uacf1\uacf5\uac04\uc740 \uacf1\uc704\uc0c1\uc5d0\uc11c \ucef4\ud329\ud2b8\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\\((X_i)_{i \\in I}\\)\uac00 \ucef4\ud329\ud2b8 \uacf5\uac04\ub4e4\uc758 \uc871\uc774\uace0, \\(X = \\prod_{i \\in I} X_i\\)\ub97c \uacf1\uc704\uc0c1\uc744 \uac16\ub294 \uacf1\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \\(X\\)\uc758 \ucef4\ud329\ud2b8\uc131\uc744 \ubcf4\uc774\uae30 \uc704\ud574 \ub2e4\uc74c \ub2e8\uacc4\ub97c \uac70\uce5c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li><strong>\uadf9\ub300 \ud544\ud130\uc758 \uc874\uc7ac<\/strong>(\uc120\ud0dd\uacf5\ub9ac \uc0ac\uc6a9): \\(X\\)\uc758 \uc784\uc758\uc758 \ud544\ud130 \\(\\mathcal{F}\\)\uc5d0 \ub300\ud574, \uc774\ub97c \ud3ec\ud568\ud558\ub294 \uadf9\ub300 \ud544\ud130(ultrafilter) \\(\\mathcal{U}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\uac83\uc740 \ucd08\ub978\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uc5ec \uc5bb\ub294\ub2e4.<\/li>\n<li><strong>\uc218\ub834\uc810 \uad6c\uc131<\/strong>(\uc120\ud0dd\uacf5\ub9ac \uc0ac\uc6a9): \uadf9\ub300 \ud544\ud130 \\(\\mathcal{U}\\)\uc5d0 \ub300\ud574, \uac01 \uc88c\ud45c \\(i \\in I\\)\ub85c\uc758 \uc0ac\uc601 \\(\\pi_i(\\mathcal{U})\\)\ub294 \\(X_i\\)\uc758 \uadf9\ub300 \ud544\ud130\uac00 \ub41c\ub2e4. \\(X_i\\)\uac00 \ucef4\ud329\ud2b8\uc774\ubbc0\ub85c \\(\\pi_i(\\mathcal{U})\\)\ub294 \uc5b4\ub5a4 \uc810 \\(x_i \\in X_i\\)\ub85c \uc218\ub834\ud55c\ub2e4. \uac01 \\(i \\in I\\)\uc5d0 \ub300\ud574 \uc218\ub834\uc810 \\(x_i\\)\ub97c \uc120\ud0dd\ud558\uc5ec \uacf1\uacf5\uac04\uc758 \uc810 \\(x = (x_i)_{i \\in I} \\in X\\)\ub97c \uad6c\uc131\ud560 \ub54c \uc120\ud0dd\uacf5\ub9ac\uac00 \uc0ac\uc6a9\ub41c\ub2e4.<\/li>\n<li><strong>\uc218\ub834\uc131 \uc99d\uba85<\/strong>: \uacf1\uc704\uc0c1\uc758 \uc815\uc758\uc640 \uadf9\ub300 \ud544\ud130\uc758 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(\\mathcal{U}\\)\uac00 \\(x\\)\ub85c \uc218\ub834\ud568\uc744 \ubcf4\uc778\ub2e4.<\/li>\n<li><strong>\ucef4\ud329\ud2b8\uc131 \uacb0\ub860<\/strong>: \ubaa8\ub4e0 \uadf9\ub300 \ud544\ud130\uac00 \uc218\ub834\uc810\uc744 \uac00\uc9c0\ubbc0\ub85c \\(X\\)\ub294 \ucef4\ud329\ud2b8\uc774\ub2e4.<span class=\"qed\"><\/span><\/li>\n<\/ol>\n<\/div>\n<p>\uc2e4\uc81c\ub85c \ud2f0\ud638\ub178\ud504 \uc815\ub9ac\ub294 \uc120\ud0dd \uacf5\ub9ac\uc640 \ub3d9\uce58\uc784\uc774 \uc54c\ub824\uc838 \uc788\ub2e4.<\/p>\n<h3>6. \uc120\ud0dd \uacf5\ub9ac\uc758 \uc5ed\uc124\uc801 \uacb0\uacfc<\/h3>\n<p>\uc120\ud0dd \uacf5\ub9ac\ub85c\ubd80\ud130 \uc9c1\uad00\uc5d0 \ubc18\ud558\ub294 \uacb0\uacfc\ub97c \ub04c\uc5b4\ub0bc \uc218 \uc788\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 10.4.<\/span><\/p>\n<p>\uc2e4\uc218 \uad6c\uac04 \\([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><\/p>\n<ol class=\"parenthesis\">\n<li>\ub3d9\uce58\uad00\uacc4 \\(\\sim\\)\uc744 \\(x \\sim y \\,\\,\\Leftrightarrow\\,\\, x &#8211; y \\in \\mathbb{Q}\\)\ub77c\uace0 \uc815\uc758\ud55c\ub2e4.<\/li>\n<li>\uc120\ud0dd \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uac01 \ub3d9\uce58\ub958\uc5d0\uc11c \ub300\ud45c\uc6d0\uc18c\ub97c \ud558\ub098\uc529 \uc120\ud0dd\ud55c \uc9d1\ud569 \\(S\\)\ub97c \ub9cc\ub4e0\ub2e4.<\/li>\n<li>\\(S\\)\ub97c \uc720\ub9ac\uc218\ub9cc\ud07c \ud3c9\ud589\uc774\ub3d9\ud55c \uc9d1\ud569\ub4e4 \\(\\{S + q \\mid q \\in \\mathbb{Q} \\cap [-1,\\, 1]\\}\\)\uc744 \uc0dd\uac01\ud55c\ub2e4.<\/li>\n<li>\uc774 \uc9d1\ud569\ub4e4\uc774 \uc11c\ub85c\uc18c\uc774\uace0 \\([0,\\, 1]\\)\uc744 \ub36e\uc74c\uc744 \ubcf4\uc778\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(S\\)\uac00 \uac00\uce21\uc774\uba74 \uce21\ub3c4\uc758 \ud3c9\ud589\uc774\ub3d9 \ubd88\ubcc0\uc131\uacfc \uac00\uc0b0\uac00\ubc95\uc131\uc5d0 \ubaa8\uc21c\uc774 \uc0dd\uae34\ub2e4.<span class=\"qed\"><\/span><\/li>\n<\/ol>\n<\/div>\n<h4>(2) \ubc14\ub098\ud750-\ud0c0\ub974\uc2a4\ud0a4 \uc5ed\uc124<\/h4>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 10.5.<\/span><\/p>\n<p>3\ucc28\uc6d0 \ub2e8\uc704\uad6c\ub97c \uc720\ud55c \uac1c\uc758 \uc870\uac01\uc73c\ub85c \ub098\ub208 \ud6c4, \uc774 \uc870\uac01\ub4e4\uc744 \ud68c\uc804\uacfc \ud3c9\ud589\uc774\ub3d9\ub9cc\uc73c\ub85c \uc7ac\ubc30\uc5f4\ud558\uc5ec \uac19\uc740 \ud06c\uae30\uc758 \uad6c \ub450 \uac1c\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><\/p>\n<ol class=\"parenthesis\">\n<li><strong>\uc790\uc720\uad70\uc758 \ubd84\ud574<\/strong>: 2\uac1c\uc758 \uc0dd\uc131\uc6d0\uc744 \uac00\uc9c4 \uc790\uc720\uad70 \\(F_2 = \\langle a,\\, b \\rangle\\)\ub97c \uc0dd\uac01\ud55c\ub2e4. \uc774 \uad70\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ubd84\ud574\ud560 \uc218 \uc788\ub2e4.<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(S_a\\) = \\(a\\)\ub85c \uc2dc\uc791\ud558\ub294 \ubaa8\ub4e0 \uae30\uc57d\uc5b4(reduced word)\ub4e4\uc758 \uc9d1\ud569.<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(S_{a^{-1}}\\) = \\(a^{-1}\\)\ub85c \uc2dc\uc791\ud558\ub294 \ubaa8\ub4e0 \uae30\uc57d\uc5b4\ub4e4\uc758 \uc9d1\ud569.<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(S_b\\) = \\(b\\)\ub85c \uc2dc\uc791\ud558\ub294 \ubaa8\ub4e0 \uae30\uc57d\uc5b4\ub4e4\uc758 \uc9d1\ud569.<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(S_{b^{-1}}\\) = \\(b^{-1}\\)\ub85c \uc2dc\uc791\ud558\ub294 \ubaa8\ub4e0 \uae30\uc57d\uc5b4\ub4e4\uc758 \uc9d1\ud569.<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(\\{e\\}\\) = \ud56d\ub4f1\uc6d0\uc18c\uc758 \uc9d1\ud569.<br \/>\n   \uadf8\ub7ec\uba74 \\(F_2 = S_a \\cup S_{a^{-1}} \\cup S_b \\cup S_{b^{-1}} \\cup \\{e\\}\\)\uc774\uace0, \\(aS_{a^{-1}} \\cup S_a \\cup \\{e\\} = F_2\\)\uc774\ub2e4.<\/li>\n<li><strong>\ud68c\uc804\uad70\uacfc \uc5f0\uacb0<\/strong>: 3\ucc28\uc6d0 \ud68c\uc804\uad70 \\(SO(3)\\)\ub294 \uc790\uc720\ubd80\ubd84\uad70\uc744 \ud3ec\ud568\ud55c\ub2e4. \uad6c\uccb4\uc801\uc73c\ub85c, \uc801\uc808\ud788 \uc120\ud0dd\ub41c \ub450 \ud68c\uc804 \\(\\rho_1\\), \\(\\rho_2\\)\uac00 \uc790\uc720\uad70 \\(F_2\\)\ub97c \uc0dd\uc131\ud55c\ub2e4.<\/li>\n<li><strong>\uad6c\uc758 \uc810\ub4e4\uc5d0 \uc791\uc6a9<\/strong>(\uc120\ud0dd\uacf5\ub9ac \uc0ac\uc6a9): \ub2e8\uc704\uad6c \\(S^2\\)\uc758 \uc810\ub4e4\uc744 \\(\\langle \\rho_1,\\, \\rho_2 \\rangle\\)\uc758 \uada4\ub3c4(orbit)\ub85c \ubd84\ud560\ud55c\ub2e4. \uac01 \uada4\ub3c4\ub294 \uac00\uc0b0\ubb34\ud55c\uac1c\uc758 \uc810\uc744 \ud3ec\ud568\ud558\uba70, \uc120\ud0dd\uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uac01 \uada4\ub3c4\uc5d0\uc11c \ub300\ud45c\uc6d0\uc18c\ub97c \ud558\ub098\uc529 \uc120\ud0dd\ud55c\ub2e4.<\/li>\n<li><strong>\ubd84\ud574 \uad6c\uc131<\/strong>: \ub300\ud45c\uc6d0\uc18c\ub4e4\uc758 \uc9d1\ud569\uc744 \\(D\\)\ub77c \ud558\uc790. \uc790\uc720\uad70\uc758 \ubd84\ud574\ub97c \uc774\uc6a9\ud558\uc5ec<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(A_1 = \\rho_1 S_{a^{-1}}(D) \\cup S_a(D) \\cup D\\),<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(A_2 = S_{a^{-1}}(D)\\),<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(B_1 = \\rho_2 S_{b^{-1}}(D) \\cup S_b(D)\\),<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(B_2 = S_{b^{-1}}(D)\\)<br \/>\n   \ub77c\uace0 \uc815\uc758\ud558\uba74, \\(S^2 = A_1 \\cup A_2 \\cup B_1 \\cup B_2\\)\uc774\ub2e4.<\/li>\n<li><strong>\uc7ac\uc870\ub9bd<\/strong>: \ub2e4\uc74c\uacfc \uac19\uc774 4\uac1c\uc758 \uc870\uac01\uc744 \ud68c\uc804\ub9cc\uc73c\ub85c \uc7ac\ubc30\uc5f4\ud558\uc5ec \ub450 \uac1c\uc758 \uc644\uc804\ud55c \uad6c\ub97c \ub9cc\ub4e0\ub2e4.<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(\\rho_1^{-1}(A_1) \\cup A_2 = S^2\\) (\uccab \ubc88\uc9f8 \uad6c)<br \/>\n   &nbsp;&nbsp;&nbsp;&#8211; \\(\\rho_2^{-1}(B_1) \\cup B_2 = S^2\\) (\ub450 \ubc88\uc9f8 \uad6c)<span class=\"qed\"><\/span>\n<\/li>\n<\/ol>\n<\/div>\n<p>\uc774 \uc99d\uba85\uc5d0\uc11c \uada4\ub3c4\uc758 \ub300\ud45c\uc6d0\uc18c\ub97c \uc120\ud0dd\ud560 \ub54c \uc120\ud0dd\uacf5\ub9ac\uac00 \uc0ac\uc6a9\ub418\uc5c8\ub2e4. \uad6c\uc131\ub41c \uac01 \uc870\uac01\uc774 \ube44\uac00\uce21\uc774\ubbc0\ub85c \ud06c\uae30(\ub113\uc774)\uac00 \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4. \uc774\uac83\uc774 &#8220;\ud558\ub098\uac00 \ub458\uc774 \ub418\ub294&#8221; \uc5ed\uc124\uc801 \uacb0\uacfc\uac00 \ub3c4\ucd9c\ub418\ub294 \uc774\uc720\uc774\ub2e4.<\/p>\n<h3>7. \uc120\ud0dd \uacf5\ub9ac \uc5c6\uc774 \uc99d\uba85 \uac00\ub2a5\ud55c \uac83\ub4e4<\/h3>\n<p>\ub2e4\uc74c \uc815\ub9ac\ub294 \uc120\ud0dd \uacf5\ub9ac \uc5c6\uc774 \uc99d\uba85 \uac00\ub2a5\ud558\uac70\ub098, \uc120\ud0dd \uacf5\ub9ac\ubcf4\ub2e4 \uc57d\ud55c \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<ul>\n<li>\uac00\uc0b0 \uac1c\uc758 \ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uacf1\uc740 \ube44\uc5b4\uc788\uc9c0 \uc54a\ub2e4. (&#8216;\uac00\uc0b0 \uc120\ud0dd \uacf5\ub9ac&#8217;\ub294 ZF\uc5d0\uc11c \uc99d\uba85 \ubd88\uac00\ub2a5\ud558\uc9c0\ub9cc \uc120\ud0dd \uacf5\ub9ac\ubcf4\ub2e4\ub294 \uc57d\ud558\ub2e4.)<\/li>\n<li>\uc720\ud55c \uac1c\uc758 \ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uacf1\uc740 \ube44\uc5b4\uc788\uc9c0 \uc54a\ub2e4.<\/li>\n<li>\ucef4\ud329\ud2b8 \ud558\uc6b0\uc2a4\ub3c4\ub974\ud504 \uacf5\uac04\uc5d0\uc11c \uc5f0\uc18d\ud568\uc218\uc758 \ucd5c\ub313\uac12\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc740 \uae30\uc800\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<\/ul>\n<h3>8. \uc120\ud0dd \uacf5\ub9ac\uc758 \ub300\uc548<\/h3>\n<p>\uc120\ud0dd \uacf5\ub9ac\ubcf4\ub2e4 \uc57d\ud55c \ud615\ud0dc\ub4e4\ub3c4 \uc5f0\uad6c\ub418\uc5b4 \uc654\ub2e4.<\/p>\n<ul>\n<li><strong>\uac00\uc0b0 \uc120\ud0dd \uacf5\ub9ac<\/strong>: \uac00\uc0b0 \uac1c\uc758 \ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc5d0 \ub300\ud574\uc11c\ub9cc \uc120\ud0dd\ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li><strong>\uc885\uc18d \uc120\ud0dd \uacf5\ub9ac<\/strong>: \ud2b9\uc815 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubb34\ud55c \uc218\uc5f4\uc758 \uc874\uc7ac\ub97c \ubcf4\uc7a5\ud55c\ub2e4.<\/li>\n<li><strong>\uc57d\ud55c \uc120\ud0dd \uacf5\ub9ac<\/strong>: \uc720\ud55c \uc9d1\ud569\ub4e4\uc758 \ubb34\ud55c \uc871\uc5d0 \ub300\ud574\uc11c\ub9cc \uc120\ud0dd\ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\ub7ec\ud55c \uc57d\ud55c \ud615\ud0dc\ub4e4\uc740 \ud574\uc11d\ud559\uc758 \ub9ce\uc740 \uc815\ub9ac\ub97c \uc99d\uba85\ud558\uae30\uc5d0 \ucda9\ubd84\ud558\uba74\uc11c\ub3c4, \ubc14\ub098\ud750-\ud0c0\ub974\uc2a4\ud0a4 \uc5ed\uc124 \uac19\uc740 \uadf9\ub2e8\uc801 \uacb0\uacfc\ub294 \ud53c\ud560 \uc218 \uc788\uac8c \ud574\uc900\ub2e4.<\/p>\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>\uc120\ud0dd \uacf5\ub9ac(Axiom of Choice, AC)\ub294 \uc9d1\ud569\ub860\uc5d0\uc11c \ub17c\ub780\uc774 \ub9ce\uc558\ub358 \uacf5\ub9ac\uc774\ub2e4. \uc9c1\uad00\uc801\uc73c\ub85c\ub294 \ub2f9\uc5f0\ud574 \ubcf4\uc774\uc9c0\ub9cc, \uc774\ub85c\ubd80\ud130 \uc9c1\uad00\uc5d0 \ubc18\ud558\ub294 \uacb0\uacfc\ub4e4\uc774 \ub3c4\ucd9c\ub418\uae30\ub3c4 \ud55c\ub2e4. 20\uc138\uae30 \ucd08 \ub9ce\uc740 \uc218\ud559\uc790\ub4e4\uc774 \uc120\ud0dd \uacf5\ub9ac\ub97c \ub2e4\ub978 \uacf5\ub9ac\ub85c\ubd80\ud130 \uc99d\uba85\ud558\uac70\ub098 \ub610\ub294 \ub2e4\ub978 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \ubc18\uc99d\ud558\ub824\uace0 \ud588\uc9c0\ub9cc \uc2e4\ud328\ud558\uc600\ub2e4. 1963\ub144 \ucf54\uc5b8(Paul Cohen)\uc740 \uc120\ud0dd \uacf5\ub9ac\uac00 ZF\uc640 \ub3c5\ub9bd\uc801\uc784\uc744 \uc99d\uba85\ud588\ub2e4. 1. \uc120\ud0dd \uacf5\ub9ac\uc758 \uc758\ubbf8 \uc120\ud0dd \uacf5\ub9ac\ub97c \uc774\ud574\ud558\uae30 \uc704\ud574 \uba3c\uc800 \uc120\ud0dd\ud568\uc218(choice function)\ub97c \uc815\uc758\ud558\uc790. \\(\\mathcal{F}\\)\uac00 \uc9d1\ud569\uc871\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \ud568\uc218 \\(f: \\mathcal{F} \\to \\bigcup \\mathcal{F}\\)\uac00&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":6,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9266\/revisions"}],"predecessor-version":[{"id":9405,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9266\/revisions\/9405"}],"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}]}}