{"id":9261,"date":"2025-10-17T20:09:33","date_gmt":"2025-10-17T11:09:33","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9261"},"modified":"2026-09-27T12:04:11","modified_gmt":"2026-09-27T03:04:11","slug":"ch08-ordinal-numbers","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch08-ordinal-numbers\/","title":{"rendered":"\uc9d1\ud569\uc758 \uc11c\uc218"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>8. \uc9d1\ud569\uc758 \uc11c\uc218<\/h2>\n\n --><\/p>\n<p>\uae30\uc218\uac00 \uc9d1\ud569\uc758 \u2018\ud06c\uae30\u2019\ub97c \ub098\ud0c0\ub0b8\ub2e4\uba74, <span class=\"defined\">\uc11c\uc218<\/span>(ordinal number)\ub294 \uc815\ub82c\ub41c \uc9d1\ud569\uc758 \u2018\uc21c\uc11c \uc720\ud615\u2019\uc744 \ub098\ud0c0\ub0b8\ub2e4. <a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">6\uc7a5<\/a>\uc5d0\uc11c \uc790\uc5f0\uc218\ub97c \ud3f0 \ub178\uc774\ub9cc \ubc29\uc2dd\uc744 \uc0ac\uc6a9\ud558\uc5ec \uad6c\uc131\ud558\uace0, <a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">7\uc7a5<\/a>\uc5d0\uc11c \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \uae30\uc218\ub85c \ub098\ud0c0\ub0b4\uc5c8\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc815\ub82c\uc9d1\ud569\uacfc \uc11c\uc218\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uace0, \uc11c\uc218\uc758 \uc21c\uc11c\uc640 \uc5f0\uc0b0, \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h3>1. \uc815\ub82c\uc21c\uc11c\uc640 \uc815\ub82c\uc9d1\ud569<\/h3>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\uc758 8.1. (\uc815\ub82c\uc21c\uc11c\uc640 \uc815\ub82c\uc9d1\ud569)<\/span><\/p>\n<p>\uc9d1\ud569 \\(A\\) \uc704\uc758 \uc21c\uc11c\uad00\uacc4 \\(\\leq\\)\uac00 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c \\(\\leq\\)\ub97c \\(A\\) \uc704\uc758 <span class=\"defined\">\uc815\ub82c\uc21c\uc11c<\/span>(well-ordering)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\leq\\)\ub294 \uc804\uc21c\uc11c\uc774\ub2e4. \uc989, \uc784\uc758\uc758 \\(a,b\\in A\\)\uc5d0 \ub300\ud558\uc5ec \\(a\\leq b\\)\uc774\uac70\ub098 \\(b\\leq a\\)\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc740 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<\/ol>\n<p>\uc815\ub82c\uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569\uc744 <span class=\"defined\">\uc815\ub82c\uc9d1\ud569<\/span>(well-ordered set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<p>\uc815\ub82c\uc9d1\ud569\uc758 \ub300\ud45c\uc801\uc778 \uc608\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\mathbb N\\)\uc740 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">6\uc7a5<\/a>\uc5d0\uc11c \uc815\uc758\ud55c \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \uc815\ub82c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\uc2e4\uc218 \uc9d1\ud569 \\(\\mathbb R\\)\uc740 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \uc815\ub82c\uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\((0,1)\\)\uc740 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<li>\uc804\uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \uc720\ud55c\uc9d1\ud569\uc740 \ud56d\uc0c1 \uc815\ub82c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ubb34\ud55c\uc9d1\ud569\uc5d0\ub3c4 \uc5ec\ub7ec \uac00\uc9c0 \uc21c\uc11c\ub97c \ubd80\uc5ec\ud560 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(\\mathbb Z\\)\uc5d0<br \/>\n\\[0<1<-1<2<-2<3<-3<\\cdots\\]\n\ub77c\ub294 \uc21c\uc11c\ub97c \uc8fc\uba74 \uc815\ub82c\uc9d1\ud569\uc774 \ub41c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 8.2. (\uc21c\uc11c\ub3d9\ud615)<\/span><\/p>\n<p>\ub450 \uc21c\uc11c\uc9d1\ud569 \\((A,\\leq_A)\\)\uc640 \\((B,\\leq_B)\\) \uc0ac\uc774\uc758 \uc77c\ub300\uc77c\ub300\uc751 \\(f\\colon A\\to B\\)\uac00 \uc784\uc758\uc758 \\(a_1,a_2\\in A\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[a_1\\leq_A a_2\\quad\\Longleftrightarrow\\quad f(a_1)\\leq_B f(a_2)\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(f\\)\ub97c <span class=\"defined\">\uc21c\uc11c\ub3d9\ud615<\/span>(order isomorphism)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(A\\)\uc640 \\(B\\)\uac00 \uc21c\uc11c\ub3d9\ud615\uc774\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.1.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \\(\\mathbb N\\), \\(\\mathbb Z\\), \\(\\mathbb R\\) \uac00\uc6b4\ub370 \uc815\ub82c\uc9d1\ud569\uc778 \uac83\uc744 \ubaa8\ub450 \ucc3e\uc73c\uc2dc\uc624. \uc815\ub82c\uc9d1\ud569\uc774 \uc544\ub2cc \uacbd\uc6b0\uc5d0\ub294 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac16\uc9c0 \uc54a\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubd80\ubd84\uc9d1\ud569\uc744 \ud558\ub098 \uc81c\uc2dc\ud558\uc2dc\uc624.<\/li>\n<li>\uc704\uc5d0\uc11c \uc815\uc758\ud55c \uc21c\uc11c<br \/>\n\\[0<1<-1<2<-2<3<-3<\\cdots\\]\n\uac00 \uc8fc\uc5b4\uc9c4 \\(\\mathbb Z\\)\uc5d0\uc11c \uc9d1\ud569 \\(\\{-3,2,-2,3\\}\\)\uc758 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\ud568\uc218 \\(f\\colon\\mathbb N\\to\\mathbb Z\\)\ub97c<br \/>\n\\[<br \/>\nf(0)=0,\\quad f(2n-1)=n,\\quad f(2n)=-n\\quad(n\\geq1)<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud558\uc790. \\(f(0),\\,\\ldots,\\,f(6)\\)\uc744 \uad6c\ud558\uace0, \\(f\\)\uac00 \\(\\mathbb N\\)\uc758 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uc640 \uc704\uc758 \\(\\mathbb Z\\)\uc758 \uc21c\uc11c \uc0ac\uc774\uc758 \uc21c\uc11c\ub3d9\ud615\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\uc21c\uc11c\ub3d9\ud615\uc778 \uc815\ub82c\uc9d1\ud569\ub4e4\uc740 \uac19\uc740 \uc21c\uc11c\uad6c\uc870\ub97c \uac00\uc9c4\ub2e4. \uc608\ub97c \ub4e4\uc5b4,<br \/>\n\\[a_1<a_2<\\cdots<a_n\\]\n\uc774\ub77c\ub294 \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569 \\(\\{a_1,\\ldots,a_n\\}\\)\uc740 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \\(\\{1,\\ldots,n\\}\\)\uacfc \uc21c\uc11c\ub3d9\ud615\uc774\ub2e4.<\/p>\n<p>\uc815\ub82c\uc9d1\ud569 \\((A,\\leq)\\)\uc640 \\(a\\in A\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[A_a=\\{x\\in A\\mid x<a\\}\\]\n\ub97c \\(a\\)\uc5d0\uc11c\uc758 <span class=\"defined\">\uc808\ud3b8<\/span>(initial segment)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ucc45\uc5d0\uc11c \uc808\ud3b8\uc774\ub77c \ud558\uba74 \uc774\uc640 \uac19\uc740 \uc9c4\uc808\ud3b8\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.2.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \\(A=\\{1,2,3,4,5\\}\\)\uc5d0\uc11c \\(A_1\\), \\(A_3\\), \\(A_5\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \\(\\mathbb N\\)\uc5d0\uc11c \\(\\mathbb N_n\\)\uc744 \uad6c\ud558\uc2dc\uc624. \uc774\ub97c \ud3f0 \ub178\uc774\ub9cc \uc790\uc5f0\uc218 \\(n\\)\uacfc \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<li>\ubb38\uc81c 8.1(3)\uc758 \uc21c\uc11c\ub3d9\ud615 \\(f\\)\uc5d0 \ub300\ud558\uc5ec \\(\\mathbb N_4\\)\uc640 \\(\\mathbb Z_{f(4)}\\)\ub97c \uac01\uac01 \uad6c\ud558\uace0 \\(f(\\mathbb N_4)=\\mathbb Z_{f(4)}\\)\uc784\uc744 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 8.3. (\uc0c1\uacc4\uc640 \uc0c1\ud55c)<\/span><\/p>\n<p>\uc21c\uc11c\uc9d1\ud569 \\((A,\\leq)\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(S\\subseteq A\\)\uc5d0 \ub300\ud558\uc5ec \\(u\\in A\\)\uac00 \ubaa8\ub4e0 \\(s\\in S\\)\uc5d0 \ub300\ud558\uc5ec \\(s\\leq u\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(u\\)\ub97c \\(S\\)\uc758 <span class=\"defined\">\uc0c1\uacc4<\/span>(upper bound)\ub77c\uace0 \ud55c\ub2e4. \uc0c1\uacc4\ub4e4 \uac00\uc6b4\ub370 \ucd5c\uc18c\uc778 \uc6d0\uc18c\uac00 \uc874\uc7ac\ud558\uba74 \uc774\ub97c \\(S\\)\uc758 <span class=\"defined\">\uc0c1\ud55c<\/span>(supremum)\uc774\ub77c \ud558\uace0 \\(\\sup S\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub9c8\ucc2c\uac00\uc9c0\ub85c \ubaa8\ub4e0 \\(s\\in S\\)\uc5d0 \ub300\ud558\uc5ec \\(l\\leq s\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(l\\in A\\)\ub97c <span class=\"defined\">\ud558\uacc4<\/span>(lower bound)\ub77c\uace0 \ud55c\ub2e4. \ud558\uacc4\ub4e4 \uac00\uc6b4\ub370 \ucd5c\ub300\uc778 \uc6d0\uc18c\uac00 \uc874\uc7ac\ud558\uba74 \uc774\ub97c \\(S\\)\uc758 <span class=\"defined\">\ud558\ud55c<\/span>(infimum)\uc774\ub77c \ud558\uace0 \\(\\inf S\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n<p>\uc0c1\ud55c\uacfc \ud558\ud55c\uc740 \uc8fc\uc5b4\uc9c4 \uc21c\uc11c\uc9d1\ud569 \uc548\uc5d0 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\((0,1)\\cap\\mathbb Q\\)\ub97c \\(\\mathbb Q\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c \ubcfc \ub54c \uc0c1\ud55c\uc740 \\(1\\)\uc774\ubbc0\ub85c \uc874\uc7ac\ud558\uc9c0\ub9cc, \\(\\{q\\in\\mathbb Q\\mid q^2<2\\}\\)\ub294 \\(\\mathbb Q\\) \uc548\uc5d0\uc11c \uc0c1\ud55c\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.3.<\/span><br \/>\n\ub2e4\uc74c \ubd80\ubd84\uc9d1\ud569\uc758 \uc0c1\uacc4, \ud558\uacc4, \uc0c1\ud55c, \ud558\ud55c\uc744 \uad6c\ud558\uc2dc\uc624. \ub2e8, (1), (3)\uc740 \\(\\mathbb R\\)\uc758 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\ub97c \uc0ac\uc6a9\ud558\uace0 (2)\ub294 \\(\\mathbb N\\)\uc758 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\ub97c \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\((0,1)\\)<\/li>\n<li>\\(\\{n\\in\\mathbb N\\mid n<5\\}\\)<\/li>\n<li>\\(\\left\\{\\dfrac{1}{n+1}\\;\\middle|\\;n\\in\\mathbb N\\right\\}\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 8.4.<\/span><br \/>\n\uc21c\uc11c\ub3d9\ud615\uc774\ub77c\ub294 \uad00\uacc4\uac00 \uc21c\uc11c\uc9d1\ud569\ub4e4 \uc0ac\uc774\uc758 \ub3d9\uce58\uad00\uacc4\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624. \ub610\ud55c \\(f\\colon A\\to B\\)\uac00 \ub450 \uc815\ub82c\uc9d1\ud569 \uc0ac\uc774\uc758 \uc21c\uc11c\ub3d9\ud615\uc774\uba74 \ubaa8\ub4e0 \\(a\\in A\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[f(A_a)=B_{f(a)}\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h3>2. \uc11c\uc218\uc758 \uc815\uc758<\/h3>\n<p>\uc9d1\ud569 \\(T\\)\uac00 \\(x\\in y\\in T\\)\uc77c \ub54c\ub9c8\ub2e4 \\(x\\in T\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(T\\)\ub97c <span class=\"defined\">\ucd94\uc774\uc801 \uc9d1\ud569<\/span>(transitive set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. <a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">6\uc7a5<\/a>\uc5d0\uc11c \uc790\uc5f0\uc218\ub294 \uc774\ub7ec\ud55c \uc131\uc9c8\uc744 \uac16\ub294\ub2e4\ub294 \uac83\uc744 \ubcf4\uc558\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 8.4. (\ud3f0 \ub178\uc774\ub9cc \uc11c\uc218)<\/span><\/p>\n<p>\uc9d1\ud569 \\(\\alpha\\)\uac00 \ucd94\uc774\uc801 \uc9d1\ud569\uc774\uace0, \\(\\alpha\\)\uc758 \uc6d0\uc18c\ub4e4 \uc0ac\uc774\uc5d0<br \/>\n\\[\\beta<\\gamma\\quad\\Longleftrightarrow\\quad\\beta\\in\\gamma\\]\n\ub85c \uc815\uc758\ud55c \uad00\uacc4\uac00 \uc815\ub82c\uc21c\uc11c\uc758 \ud615\ud0dc\ub97c \uc774\ub8e8\uba74 \\(\\alpha\\)\ub97c <span class=\"defined\">\uc11c\uc218<\/span>(ordinal number)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989<br \/>\n\\[\\beta\\leq\\gamma\\quad\\Longleftrightarrow\\quad\\beta\\in\\gamma\\;\\text{ \ub610\ub294 }\\;\\beta=\\gamma\\]<br \/>\n\ub85c \ub450\uba74 \\(\\alpha\\)\uac00 \uc815\ub82c\uc9d1\ud569\uc774 \ub41c\ub2e4.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 8.5. (\uc11c\uc218\uc758 \uae30\ubcf8 \uc131\uc9c8)<\/span><\/p>\n<p>\uc11c\uc218 \\(\\alpha\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\beta\\in\\alpha\\)\uc774\uba74 \\(\\beta\\)\ub3c4 \uc11c\uc218\uc774\ub2e4.<\/li>\n<li>\\(\\alpha\\notin\\alpha\\)\uc774\ub2e4.<\/li>\n<li>\\(S(\\alpha)=\\alpha\\cup\\{\\alpha\\}\\)\ub3c4 \uc11c\uc218\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(\\beta\\in\\alpha\\)\ub77c\uace0 \ud558\uc790. \\(\\alpha\\)\uac00 \ucd94\uc774\uc801\uc774\ubbc0\ub85c \\(\\beta\\subseteq\\alpha\\)\uc774\ub2e4. \\(x\\in y\\in\\beta\\)\uc774\uba74 \\(x,y,\\beta\\)\ub294 \\(\\alpha\\)\uc758 \uc6d0\uc18c\uc774\uace0, \\(\\alpha\\)\uc5d0\uc11c\uc758 \uc21c\uc11c \\(\\in\\)\uc758 \ucd94\uc774\uc131\uc5d0 \uc758\ud574 \\(x\\in\\beta\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\beta\\)\ub294 \ucd94\uc774\uc801\uc774\ub2e4. \ub610\ud55c \\(\\beta\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubd80\ubd84\uc9d1\ud569\uc740 \\(\\alpha\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ubbc0\ub85c \\(\\in\\)\uc5d0 \ub300\ud55c \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac16\ub294\ub2e4. \ub530\ub77c\uc11c \\(\\beta\\)\ub294 \uc11c\uc218\uc774\ub2e4.<\/p>\n<p>\ub458\uc9f8\ub294 \\(\\in\\)\uc774 \\(\\alpha\\)\uc5d0\uc11c \uc5c4\uaca9\ud55c \uc21c\uc11c\uc774\ubbc0\ub85c \\(\\alpha\\in\\alpha\\)\uac00 \uc131\ub9bd\ud560 \uc218 \uc5c6\ub2e4\ub294 \uc0ac\uc2e4\uc5d0\uc11c \ub530\ub978\ub2e4.<\/p>\n<p>\ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(S(\\alpha)\\)\uac00 \ucd94\uc774\uc801\uc784\uc740 \uc815\uc758\uc5d0\uc11c \ubc14\ub85c \ud655\uc778\ub41c\ub2e4. \\(S(\\alpha)\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubd80\ubd84\uc9d1\ud569 \\(C\\)\uc5d0 \ub300\ud558\uc5ec \\(C\\cap\\alpha\\ne\\varnothing\\)\uc774\uba74 \\(C\\cap\\alpha\\)\uc758 \ucd5c\uc18c\uc6d0\uc18c\uac00 \\(C\\)\uc758 \ucd5c\uc18c\uc6d0\uc18c\uc774\uace0, \\(C\\cap\\alpha=\\varnothing\\)\uc774\uba74 \\(C=\\{\\alpha\\}\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(S(\\alpha)\\)\ub3c4 \uc11c\uc218\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc790\uc5f0\uc218\uc758 \ud3f0 \ub178\uc774\ub9cc \uad6c\uc131\uacfc <a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">6\uc7a5\uc758 \uc790\uc5f0\uc218 \uc21c\uc11c \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \ubaa8\ub4e0 \uc790\uc5f0\uc218\ub294 \uc11c\uc218\uc774\ub2e4. \ud2b9\ud788<br \/>\n\\[<br \/>\n0=\\varnothing,\\quad1=\\{0\\},\\quad2=\\{0,1\\},\\quad3=\\{0,1,2\\},\\quad\\ldots<br \/>\n\\]<br \/>\n\uc774\uace0, \uac00\uc7a5 \uc791\uc740 \ubb34\ud55c\uc11c\uc218\ub294<br \/>\n\\[\\omega=\\mathbb N=\\{0,1,2,3,\\ldots\\}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.5.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(0,1,2,3,4\\)\ub97c \uac01\uac01 \uc9d1\ud569\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n<li>\\(2\\in4\\), \\(2\\subseteq4\\), \\(4\\in4\\), \\(3<4\\)\uc758 \ucc38\uacfc \uac70\uc9d3\uc744 \ud310\uc815\ud558\uc2dc\uc624.<\/li>\n<li>\ub2e4\uc74c \uc9d1\ud569 \uac00\uc6b4\ub370 \ucd94\uc774\uc801 \uc9d1\ud569\uc778 \uac83\uc744 \ubaa8\ub450 \ucc3e\uace0, \uc11c\uc218\uc778 \uac83\uc744 \ubaa8\ub450 \ucc3e\uc73c\uc2dc\uc624.<br \/>\n\\[<br \/>\nA=\\{0,1,2\\},\\quad B=\\{0,2\\},\\quad C=\\{0,1,2,3\\}.<br \/>\n\\]\n<\/li>\n<li>\\(S(3)\\)\uc744 \uc9d1\ud569\uc73c\ub85c \uc9c1\uc811 \uacc4\uc0b0\ud558\uace0 \\(S(3)=4\\)\uc784\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 8.6. (\uc815\ub82c\uc9d1\ud569\uc758 \uc11c\uc218)<\/span><\/p>\n<p>\ubaa8\ub4e0 \uc815\ub82c\uc9d1\ud569\uc740 \uc815\ud655\ud788 \ud558\ub098\uc758 \uc11c\uc218\uc640 \uc21c\uc11c\ub3d9\ud615\uc774\ub2e4. \uc815\ub82c\uc9d1\ud569 \\(A\\)\uc640 \uc21c\uc11c\ub3d9\ud615\uc778 \uc720\uc77c\ud55c \uc11c\uc218\ub97c \\(\\operatorname{ord}(A)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\ub294 \uc11c\uc218\uac00 \uc815\ub82c\uc9d1\ud569\uc758 \uc21c\uc11c \uc720\ud615\uc744 \ub098\ud0c0\ub0b8\ub2e4\ub294 \uac83\uc744 \uc5c4\ubc00\ud558\uac8c \ud45c\ud604\ud55c\ub2e4. \uc99d\uba85\uc758 \ud575\uc2ec\uc740 \ucd08\ud55c\uc7ac\uadc0\uc774\uba70, \uc774 \uc7a5\uc758 \ub9c8\uc9c0\ub9c9 \uc808\uc5d0\uc11c \uc124\uba85\ud55c\ub2e4.<\/p>\n<h3>3. \uc11c\uc218\uc758 \uc21c\uc11c<\/h3>\n<p>\uc11c\uc218 \\(\\alpha\\), \\(\\beta\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\alpha<\\beta\\quad\\Longleftrightarrow\\quad\\alpha\\in\\beta\\]\n\ub85c \uc815\uc758\ud55c\ub2e4. \ub610\ud55c \\(\\alpha\\leq\\beta\\)\ub294 \\(\\alpha<\\beta\\)\uc774\uac70\ub098 \\(\\alpha=\\beta\\)\ub77c\ub294 \ub73b\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\ubcf4\uc870\uc815\ub9ac 8.7. (\uc11c\uc218\uc758 \uc808\ud3b8)<\/span><\/p>\n<p>\uc11c\uc218 \\(\\alpha\\)\uc640 \\(\\beta\\in\\alpha\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\beta=\\{\\gamma\\in\\alpha\\mid\\gamma<\\beta\\}\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\alpha\\)\uc758 \uc808\ud3b8\uc740 \uc815\ud655\ud788 \\(\\alpha\\)\uc758 \uc6d0\uc18c\uc778 \uc11c\uc218\ub4e4\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(\\gamma\\in\\beta\\)\uc774\uba74 \\(\\alpha\\)\uc758 \ucd94\uc774\uc131\uc5d0 \uc758\ud574 \\(\\gamma\\in\\alpha\\)\uc774\uace0 \uc815\uc758\uc0c1 \\(\\gamma<\\beta\\)\uc774\ub2e4. \uc5ed\uc740 \uc815\uc758\uc5d0\uc11c \ubc14\ub85c \ub530\ub978\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 8.8. (\uc11c\uc218\uc758 \ube44\uad50\uc640 \uc815\ub82c\uc131)<\/span><\/p>\n<p>\uc11c\uc218 \\(\\alpha\\), \\(\\beta\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc14b \uc911 \uc815\ud655\ud788 \ud558\ub098\uac00 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\alpha<\\beta,\\quad\\alpha=\\beta,\\quad\\beta<\\alpha.\\]\n\ub610\ud55c \uc11c\uc218\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc9d1\ud569\uc740 \uac00\uc7a5 \uc791\uc740 \uc6d0\uc18c\ub97c \uac16\ub294\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(C=\\alpha\\cap\\beta\\)\ub77c\uace0 \ud558\uc790. \\(C\\)\uac00 \\(\\alpha\\)\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc774\uba74 \\(\\alpha\\setminus C\\)\uc758 \ucd5c\uc18c\uc6d0\uc18c\ub97c \\(\\delta\\)\ub77c\uace0 \ub458 \uc218 \uc788\uace0, \uc815\ub82c\uc131\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[C=\\{\\gamma\\in\\alpha\\mid\\gamma<\\delta\\}=\\delta\\]\n\ub97c \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \\(C\\)\ub294 \\(\\alpha\\)\uc758 \uc808\ud3b8\uc774\ub2e4. \\(C\\)\uac00 \\(\\beta\\)\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc77c \ub54c\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\uc774\ub2e4. \ub9cc\uc57d \\(C\\)\uac00 \\(\\alpha\\)\uc640 \\(\\beta\\) \ubaa8\ub450\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc774\uba74 \\(C\\in\\alpha\\)\uc774\uace0 \\(C\\in\\beta\\)\uc774\ubbc0\ub85c \\(C\\in\\alpha\\cap\\beta=C\\)\uac00 \ub418\uc5b4 \uc815\ub9ac 8.5(2)\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(C=\\alpha\\)\uc774\uac70\ub098 \\(C=\\beta\\)\uc774\ub2e4. \uc774\ub85c\ubd80\ud130 \uc138 \uacbd\uc6b0 \uac00\uc6b4\ub370 \uc815\ud655\ud788 \ud558\ub098\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \\(S\\)\uac00 \uc11c\uc218\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(\\alpha\\in S\\)\ub97c \ud558\ub098 \ud0dd\ud55c\ub2e4. \\(\\alpha\\)\ubcf4\ub2e4 \uc791\uc740 \\(S\\)\uc758 \uc6d0\uc18c\uac00 \uc5c6\uc73c\uba74 \\(\\alpha\\)\uac00 \ucd5c\uc18c\uc6d0\uc18c\uc774\ub2e4. \uadf8\ub807\uc9c0 \uc54a\uc73c\uba74<br \/>\n\\[\\{\\beta\\in S\\mid\\beta<\\alpha\\}\\subseteq\\alpha\\]\n\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ubbc0\ub85c \\(\\alpha\\)\uc758 \uc815\ub82c\uc131\uc5d0 \uc758\ud574 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac16\uace0, \uc774 \uc6d0\uc18c\uac00 \\(S\\) \uc804\uccb4\uc758 \ucd5c\uc18c\uc6d0\uc18c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\ub530\ub984\uc815\ub9ac 8.9. (\uc815\ub82c\uc9d1\ud569\uc758 \ube44\uad50)<\/span><\/p>\n<p>\ub450 \uc815\ub82c\uc9d1\ud569 \\(A\\), \\(B\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc14b \uc911 \uc815\ud655\ud788 \ud558\ub098\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A\\)\uc640 \\(B\\)\uac00 \uc21c\uc11c\ub3d9\ud615\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uac00 \\(B\\)\uc758 \uc5b4\ub5a4 \uc808\ud3b8\uacfc \uc21c\uc11c\ub3d9\ud615\uc774\ub2e4.<\/li>\n<li>\\(B\\)\uac00 \\(A\\)\uc758 \uc5b4\ub5a4 \uc808\ud3b8\uacfc \uc21c\uc11c\ub3d9\ud615\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(\\alpha=\\operatorname{ord}(A)\\), \\(\\beta=\\operatorname{ord}(B)\\)\ub85c \ub450\uace0 \uc815\ub9ac 8.8\uc744 \uc801\uc6a9\ud558\uba74 \ub41c\ub2e4. \\(\\alpha<\\beta\\)\uc774\uba74 \\(\\alpha\\)\ub294 \\(\\beta\\)\uc758 \uc808\ud3b8\uc774\ubbc0\ub85c \ub450 \ubc88\uc9f8 \uacbd\uc6b0\uc774\uace0, \\(\\beta<\\alpha\\)\uc774\uba74 \uc138 \ubc88\uc9f8 \uacbd\uc6b0\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub530\ub77c\uc11c \uc11c\uc218\uc758 \uc21c\uc11c\ub294<br \/>\n\\[0<1<2<3<\\cdots<\\omega<\\omega+1<\\omega+2<\\cdots<\\omega\\cdot2<\\cdots\\]\n\ucc98\ub7fc \uacc4\uc18d \uc774\uc5b4\uc9c4\ub2e4. \uc5ec\uae30\uc11c \\(+\\)\uc640 \\(\\cdot\\)\uc740 \ub4a4\uc5d0\uc11c \uc815\uc758\ud560 \uc11c\uc218\uc758 \uc5f0\uc0b0\uc774\ub2e4.<\/p>\n<p>\uc11c\uc218\ub4e4\uc758 \uc9d1\ud569\uc5d0\uc11c\ub294 \uc0c1\ud55c\uc774 \ud56d\uc0c1 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 8.10. (\uc11c\uc218 \uc9d1\ud569\uc758 \uc0c1\ud55c)<\/span><\/p>\n<p>\uc11c\uc218\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569 \\(S\\)\uc5d0 \ub300\ud558\uc5ec \\(\\bigcup S\\)\ub294 \uc11c\uc218\uc774\uace0<br \/>\n\\[\\sup S=\\bigcup S\\]<br \/>\n\uc774\ub2e4. \ud2b9\ud788 \\(S=\\varnothing\\)\uc774\uba74 \\(\\sup S=0\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(\\bigcup S\\)\ub294 \ucd94\uc774\uc801\uc774\ub2e4. \ub610\ud55c \uadf8 \uc6d0\uc18c\ub4e4\uc740 \ubaa8\ub450 \uc11c\uc218\uc774\ubbc0\ub85c \uc815\ub9ac 8.8\uc5d0 \uc758\ud574 \uc11c\ub85c \ube44\uad50 \uac00\ub2a5\ud558\uace0, \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubd80\ubd84\uc9d1\ud569\uc740 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac16\ub294\ub2e4. \ub530\ub77c\uc11c \\(\\bigcup S\\)\ub294 \uc11c\uc218\uc774\ub2e4.<\/p>\n<p>\uac01 \\(\\alpha\\in S\\)\uc5d0 \ub300\ud558\uc5ec \\(\\alpha\\subseteq\\bigcup S\\)\uc774\ubbc0\ub85c \\(\\alpha\\leq\\bigcup S\\)\uc774\ub2e4. \ud55c\ud3b8 \ubaa8\ub4e0 \\(\\alpha\\in S\\)\uc758 \uc0c1\uacc4\uc778 \uc11c\uc218 \\(\\gamma\\)\uac00 \uc788\uc73c\uba74 \uac01 \\(\\alpha\\subseteq\\gamma\\)\uc774\ubbc0\ub85c \\(\\bigcup S\\subseteq\\gamma\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\bigcup S\\)\ub294 \ucd5c\uc18c\uc0c1\uacc4\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<h3>4. \ucd08\ud55c\uadc0\ub0a9\ubc95\uacfc \ucd08\ud55c\uc7ac\uadc0<\/h3>\n<p>\uc11c\uc218 \\(\\alpha\\)\uc758 <span class=\"defined\">\ub530\ub984\uc11c\uc218<\/span>(successor ordinal)\ub97c<br \/>\n\\[S(\\alpha)=\\alpha\\cup\\{\\alpha\\}\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \\(0\\)\uc774 \uc544\ub2c8\uba74\uc11c \uc5b4\ub5a4 \uc11c\uc218\uc758 \ub530\ub984\uc11c\uc218\ub3c4 \uc544\ub2cc \uc11c\uc218\ub97c <span class=\"defined\">\uadf9\ud55c\uc11c\uc218<\/span>(limit ordinal)\ub77c\uace0 \ubd80\ub978\ub2e4. \uadf9\ud55c\uc11c\uc218\ub294 \ucd5c\ub300\uc6d0\uc18c\ub97c \uac16\uc9c0 \uc54a\ub294\ub2e4. \uac00\uc7a5 \uc791\uc740 \ubb34\ud55c \uadf9\ud55c\uc11c\uc218\ub294 \\(\\omega\\)\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.6.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(0\\), \\(4\\), \\(\\omega\\), \\(S(\\omega)\\), \\(S(S(\\omega))\\) \uac00\uc6b4\ub370 \ub530\ub984\uc11c\uc218\uc640 \uadf9\ud55c\uc11c\uc218\ub97c \uac01\uac01 \ucc3e\uc73c\uc2dc\uc624.<\/li>\n<li>\\(\\sup\\{0,1,2,3\\}\\)\uacfc \\(\\sup\\omega\\)\ub97c \uad6c\ud558\uc2dc\uc624. \uc5ec\uae30\uc11c \\(\\omega\\)\ub294 \uc11c\uc218\ub4e4\uc758 \uc9d1\ud569\uc73c\ub85c \ubcf8\ub2e4.<\/li>\n<li>\ubaa8\ub4e0 \\(n\\in\\mathbb N\\setminus\\{0\\}\\)\uac00 \ub530\ub984\uc11c\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\omega\\)\uac00 \ub530\ub984\uc11c\uc218\uac00 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 8.11. (\ucd08\ud55c\uadc0\ub0a9\ubc95)<\/span><\/p>\n<p>\\(\\theta\\)\uac00 \uc11c\uc218\uc774\uace0, \uac01 \\(\\alpha<\\theta\\)\uc5d0 \ub300\ud55c \uba85\uc81c \\(P(\\alpha)\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ubaa8\ub4e0 \\(\\alpha<\\theta\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\bigl(\\text{\ubaa8\ub4e0 }\\beta<\\alpha\\text{\uc5d0 \ub300\ud558\uc5ec }P(\\beta)\\bigr)\n\\quad\\Longrightarrow\\quad P(\\alpha)\n\\]\n\uac00 \uc131\ub9bd\ud558\uba74 \ubaa8\ub4e0 \\(\\alpha<\\theta\\)\uc5d0 \ub300\ud558\uc5ec \\(P(\\alpha)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(P\\)\uac00 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294 \\(\\alpha<\\theta\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\ud55c \uc11c\uc218\ub4e4\uc758 \uc9d1\ud569\uc740 \uc815\ub9ac 8.8\uc5d0 \uc758\ud574 \ucd5c\uc18c\uc6d0\uc18c \\(\\alpha_0\\)\ub97c \uac16\ub294\ub2e4. \\(\\alpha_0\\)\ubcf4\ub2e4 \uc791\uc740 \ubaa8\ub4e0 \\(\\beta\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(P(\\beta)\\)\uac00 \uc131\ub9bd\ud558\ubbc0\ub85c \uac00\uc815\uc5d0 \uc758\ud574 \\(P(\\alpha_0)\\)\ub3c4 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub294 \\(\\alpha_0\\)\uc758 \uc120\ud0dd\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ucd08\ud55c\uadc0\ub0a9\ubc95\uc740 \ud754\ud788 \ub2e4\uc74c \uc138 \ub2e8\uacc4\ub85c \ub098\ub204\uc5b4 \uc0ac\uc6a9\ud55c\ub2e4. \\(P(0)\\)\uc744 \ubcf4\uc774\uace0, \\(P(\\alpha)\\)\uc5d0\uc11c \\(P(S(\\alpha))\\)\ub97c \ubcf4\uc774\uba70, \uadf9\ud55c\uc11c\uc218 \\(\\lambda\\)\uc5d0 \ub300\ud574\uc11c\ub294 \ubaa8\ub4e0 \\(\\beta<\\lambda\\)\uc5d0\uc11c \\(P(\\beta)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc5ec \\(P(\\lambda)\\)\ub97c \ubcf4\uc778\ub2e4.<\/p>\n<p>\ucd08\ud55c\uadc0\ub0a9\ubc95\uc5d0 \ub300\uc751\ud558\ub294 \uc815\uc758 \uc6d0\ub9ac\ub97c <span class=\"defined\">\ucd08\ud55c\uc7ac\uadc0<\/span>(transfinite recursion)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 8.12. (\ucd08\ud55c\uc7ac\uadc0)<\/span><\/p>\n<p>\uc815\ub82c\uc9d1\ud569 \\(A\\) \uc704\uc5d0\uc11c \uac01 \\(a\\in A\\)\uc758 \uac12\uc744 \uadf8\ubcf4\ub2e4 \uc791\uc740 \uc6d0\uc18c\ub4e4\uc5d0\uc11c \uc774\ubbf8 \uc815\ud574\uc9c4 \uac12\ub4e4\ub85c\ubd80\ud130 \uc720\uc77c\ud558\uac8c \uc815\ud558\ub294 \uaddc\uce59\uc774 \uc8fc\uc5b4\uc9c0\uba74, \uadf8 \uaddc\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\uc758 \uc644\uc804\ud55c \uc9d1\ud569\ub860\uc801 \uc9c4\uc220\uacfc \uc99d\uba85\uc5d0\ub294 \uce58\ud658 \uacf5\ub9ac \ub4f1\uc758 \uc9d1\ud569 \uc874\uc7ac \uc6d0\ub9ac\uac00 \ud544\uc694\ud558\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4. \uc544\ub798\uc758 \uc11c\uc218 \ub367\uc148, \uacf1\uc148, \uac70\ub4ed\uc81c\uacf1\uc740 \uc21c\uc11c \uc720\ud615 \ub610\ub294 \ucd08\ud55c\uc7ac\uadc0\ub97c \uc774\uc6a9\ud558\uc5ec \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.7.<\/span><br \/>\n\ud568\uc218 \\(F\\colon S(\\omega)\\to S(\\omega)\\)\ub97c \ub2e4\uc74c \uaddc\uce59\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ud55c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[<br \/>\nF(0)=0,\\quad F(S(n))=S(S(F(n)))\\quad(n<\\omega),\n\\]\n\uadf8\ub9ac\uace0 \uadf9\ud55c \ub2e8\uacc4\uc5d0\uc11c\ub294\n\\[\nF(\\omega)=\\bigcup_{n<\\omega}F(n)\n\\]\n\uc73c\ub85c \ub454\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(F(0)\\), \\(F(1)\\), \\(F(2)\\), \\(F(3)\\), \\(F(4)\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\ubaa8\ub4e0 \\(n<\\omega\\)\uc5d0 \ub300\ud558\uc5ec \\(F(n)=2n\\)\uc784\uc744 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(F(\\omega)\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>(3)\uc758 \uacb0\uacfc\ub97c \uc774\uc6a9\ud558\uc5ec \uc720\ud55c \ub2e8\uacc4\uc758 \uac12\ub4e4\uc774 \uc544\ubb34\ub9ac \ucee4\uc838\ub3c4 \uadf9\ud55c \ub2e8\uacc4\uc5d0\uc11c \uc5bb\ub294 \uc0c1\ud55c\uc740 \ub2e4\uc2dc \\(\\omega\\)\uac00 \ub428\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.8.<\/span><br \/>\n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ubaa8\ub4e0 \uc11c\uc218\ub294 \\(0\\), \ub530\ub984\uc11c\uc218, \uadf9\ud55c\uc11c\uc218 \uac00\uc6b4\ub370 \uc815\ud655\ud788 \ud558\ub098\uc774\ub2e4.<\/li>\n<li>\uc815\ub9ac 8.11\ub85c\ubd80\ud130 \uc704\uc5d0\uc11c \uc124\uba85\ud55c \uae30\ucd08 \ub2e8\uacc4\u2013\ub530\ub984\uc11c\uc218 \ub2e8\uacc4\u2013\uadf9\ud55c\uc11c\uc218 \ub2e8\uacc4\uc758 \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc720\ub3c4\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>5. \uc11c\uc218\uc758 \ub367\uc148<\/h3>\n<p>\uc11c\uc218\uc758 \ub367\uc148\uc740 \ub450 \uc815\ub82c\uc9d1\ud569\uc744 \ucc28\ub840\ub85c \uc774\uc5b4 \ubd99\uc774\ub294 \uc5f0\uc0b0\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 8.13. (\uc11c\uc218\uc758 \ub367\uc148)<\/span><\/p>\n<p>\\(\\operatorname{ord}(A)=\\alpha\\), \\(\\operatorname{ord}(B)=\\beta\\)\ub77c\uace0 \ud558\uc790. \uc11c\ub85c\uc18c\uc778 \ub450 \ubcf5\uc0ac\ubcf8 \\(A\\times\\{0\\}\\)\uacfc \\(B\\times\\{1\\}\\)\uc744 \ub9cc\ub4e4\uace0, \uac01 \ubcf5\uc0ac\ubcf8 \uc548\uc5d0\uc11c\ub294 \uc6d0\ub798 \uc21c\uc11c\ub97c \uc720\uc9c0\ud558\uba70 \\(A\\times\\{0\\}\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \\(B\\times\\{1\\}\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ubcf4\ub2e4 \uc791\ub3c4\ub85d \uc21c\uc11c\ub97c \uc900\ub2e4. \uc774 \uc815\ub82c\uc9d1\ud569\uc758 \uc11c\uc218\ub97c<br \/>\n\\[\\alpha+\\beta\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\ub300\ud45c \uc815\ub82c\uc9d1\ud569\uc744 \uc21c\uc11c\ub3d9\ud615\uc778 \ub2e4\ub978 \uc815\ub82c\uc9d1\ud569\uc73c\ub85c \ubc14\uafb8\uc5b4\ub3c4 \ub450 \ubcf5\uc0ac\ubcf8\uc5d0 \uc131\ubd84\ubcc4 \uc21c\uc11c\ub3d9\ud615\uc744 \uc801\uc6a9\ud558\uba74 \uac19\uc740 \uc21c\uc11c \uc720\ud615\uc744 \uc5bb\uc73c\ubbc0\ub85c \uc774 \uc815\uc758\ub294 \ub300\ud45c\uc6d0\uc758 \uc120\ud0dd\uacfc \ubb34\uad00\ud558\ub2e4.<\/p>\n<p>\uc608\ub97c \ub4e4\uba74<br \/>\n\\[2+3=5,\\quad1+\\omega=\\omega\\]<br \/>\n\uc774\uace0, \\(\\omega+1\\)\uc740 \uc790\uc5f0\uc218\ub4e4\uc744 \ubaa8\ub450 \ub193\uc740 \ub4a4 \ub9c8\uc9c0\ub9c9\uc5d0 \uc0c8 \uc6d0\uc18c \ud558\ub098\ub97c \ubd99\uc778 \uc21c\uc11c \uc720\ud615\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[1+\\omega=\\omega<\\omega+1\\]\n\uc774\ubbc0\ub85c \uc11c\uc218\uc758 \ub367\uc148\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uad50\ud658\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 8.14. (\uc11c\uc218 \ub367\uc148\uc758 \uae30\ubcf8 \ubc95\uce59)<\/span><\/p>\n<p>\uc11c\uc218 \\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\((\\alpha+\\beta)+\\gamma=\\alpha+(\\beta+\\gamma)\\).<\/li>\n<li>\\(0+\\alpha=\\alpha=\\alpha+0\\).<\/li>\n<li>\\(\\beta<\\gamma\\)\uc774\uba74 \\(\\alpha+\\beta<\\alpha+\\gamma\\)\uc774\ub2e4.<\/li>\n<li>\\(\\alpha+\\beta=\\alpha+\\gamma\\)\uc774\uba74 \\(\\beta=\\gamma\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab\uc9f8\uc640 \ub458\uc9f8\ub294 \uc815\ub82c\uc9d1\ud569\uc744 \uc774\uc5b4 \ubd99\uc774\ub294 \uc815\uc758\uc5d0\uc11c \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc21c\uc11c\ub3d9\ud615\uc744 \uc4f0\uba74 \ub41c\ub2e4. \\(\\beta<\\gamma\\)\uc774\uba74 \\(\\beta\\)\ub294 \\(\\gamma\\)\uc758 \uc808\ud3b8\uc774\ubbc0\ub85c \\(\\alpha+\\beta\\)\ub3c4 \\(\\alpha+\\gamma\\)\uc758 \uc808\ud3b8\uc774 \ub41c\ub2e4. \ub530\ub77c\uc11c \uc14b\uc9f8\uac00 \uc131\ub9bd\ud558\uace0, \ub137\uc9f8\ub294 \uc815\ub9ac 8.8\uacfc \uc14b\uc9f8\uc5d0\uc11c \ub530\ub978\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uacf5\ud1b5\ub41c \uc67c\ucabd \ud56d\uc740 \uc18c\uac70\ud560 \uc218 \uc788\uc9c0\ub9cc \uacf5\ud1b5\ub41c \uc624\ub978\ucabd \ud56d\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc18c\uac70\ud560 \uc218 \uc5c6\ub2e4. \uc2e4\uc81c\ub85c<br \/>\n\\[1+\\omega=2+\\omega=\\omega\\]<br \/>\n\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.9.<\/span><br \/>\n\ub2e4\uc74c \uc11c\uc218\uc758 \ud569\uc744 \uacc4\uc0b0\ud558\uac70\ub098 \uc21c\uc11c \uc720\ud615\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(3+4\\)<\/li>\n<li>\\(3+\\omega\\)<\/li>\n<li>\\(\\omega+3\\)<\/li>\n<li>\\(2+\\omega\\)\uc640 \\(\\omega+2\\)\ub97c \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<li>\\(\\alpha+\\gamma=\\beta+\\gamma\\)\uc774\uc5b4\ub3c4 \\(\\alpha=\\beta\\)\uac00 \ub530\ub974\uc9c0 \uc54a\ub294 \uad6c\uccb4\uc801\uc778 \uc608\ub97c \ud558\ub098 \uc81c\uc2dc\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>6. \uc11c\uc218\uc758 \uacf1\uc148<\/h3>\n<p>\uc11c\uc218\uc758 \uacf1\uc148\uc740 \uac19\uc740 \uc21c\uc11c \uc720\ud615\uc758 \ube14\ub85d\uc744 \ucc28\ub840\ub85c \ub193\ub294 \uc5f0\uc0b0\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 8.15. (\uc11c\uc218\uc758 \uacf1\uc148)<\/span><\/p>\n<p>\\(\\operatorname{ord}(A)=\\alpha\\), \\(\\operatorname{ord}(B)=\\beta\\)\ub77c\uace0 \ud558\uc790. \\(B\\times A\\)\uc5d0<br \/>\n\\[(b_1,a_1)<(b_2,a_2)\\]\n\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc744\n\\[b_1<b_2\\quad\\text{\ub610\ub294}\\quad\\bigl(b_1=b_2\\text{\uc774\uace0 }a_1<a_2\\bigr)\\]\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc774 \uc815\ub82c\uc9d1\ud569\uc758 \uc11c\uc218\ub97c \\(\\alpha\\cdot\\beta\\)\ub77c\uace0 \uc815\uc758\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\ub530\ub77c\uc11c \\(\\alpha\\cdot\\beta\\)\ub294 \\(\\alpha\\)\ud615\uc758 \ube14\ub85d\uc744 \\(\\beta\\)\uc758 \uc21c\uc11c\ub300\ub85c \ub298\uc5b4\ub193\uc740 \uc21c\uc11c \uc720\ud615\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[2\\cdot3=6,\\quad2\\cdot\\omega=\\omega\\]<br \/>\n\uc774\uace0, \\(\\omega\\cdot2\\)\ub294 \uc790\uc5f0\uc218\ud615\uc758 \ube14\ub85d \ub450 \uac1c\ub97c \ucc28\ub840\ub85c \ub193\uc740 \uc21c\uc11c \uc720\ud615\uc774\ubbc0\ub85c<br \/>\n\\[2\\cdot\\omega=\\omega<\\omega\\cdot2\\]\n\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 8.16. (\uc11c\uc218 \uacf1\uc148\uc758 \uae30\ubcf8 \ubc95\uce59)<\/span><\/p>\n<p>\uc11c\uc218 \\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\((\\alpha\\cdot\\beta)\\cdot\\gamma=\\alpha\\cdot(\\beta\\cdot\\gamma)\\).<\/li>\n<li>\\(1\\cdot\\alpha=\\alpha=\\alpha\\cdot1\\)\uc774\uace0 \\(0\\cdot\\alpha=0=\\alpha\\cdot0\\)\uc774\ub2e4.<\/li>\n<li>\\(\\alpha\\cdot(\\beta+\\gamma)=\\alpha\\cdot\\beta+\\alpha\\cdot\\gamma\\).<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uacf1\uc148\uc744 \\(\\alpha\\)\ud615 \ube14\ub85d\uc744 \\(\\beta\\)\uc758 \uc21c\uc11c\ub300\ub85c \ub298\uc5b4\ub193\ub294 \uc5f0\uc0b0\uc73c\ub85c \ud574\uc11d\ud558\uba74 \uccab\uc9f8\uc640 \ub458\uc9f8\uac00 \ubc14\ub85c \ub530\ub978\ub2e4. \\(\\beta+\\gamma\\)\uac1c\uc758 \ube14\ub85d\uc744 \ub298\uc5b4\ub193\ub294 \uac83\uc740 \uba3c\uc800 \\(\\beta\\)\uac1c\uc758 \ube14\ub85d\uc744 \ub193\uace0 \uc774\uc5b4\uc11c \\(\\gamma\\)\uac1c\uc758 \ube14\ub85d\uc744 \ub193\ub294 \uac83\uacfc \uc21c\uc11c\ub3d9\ud615\uc774\ubbc0\ub85c \uc14b\uc9f8\ub3c4 \uc131\ub9bd\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc624\ub978\ucabd \ubd84\ubc30\ubc95\uce59\uacfc \uad50\ud658\ubc95\uce59\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[(1+1)\\cdot\\omega=2\\cdot\\omega=\\omega\\]<br \/>\n\uc774\uc9c0\ub9cc<br \/>\n\\[1\\cdot\\omega+1\\cdot\\omega=\\omega+\\omega=\\omega\\cdot2\\]<br \/>\n\uc774\ubbc0\ub85c \uc624\ub978\ucabd \ubd84\ubc30\ubc95\uce59\uc774 \uc2e4\ud328\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.10.<\/span><br \/>\n\ub2e4\uc74c \uc11c\uc218\uc758 \uacf1\uc744 \uacc4\uc0b0\ud558\uac70\ub098 \uc21c\uc11c \uc720\ud615\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(3\\cdot4\\)<\/li>\n<li>\\(3\\cdot\\omega\\)<\/li>\n<li>\\(\\omega\\cdot3\\)<\/li>\n<li>\\(2\\cdot\\omega\\)\uc640 \\(\\omega\\cdot2\\)\ub97c \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<li>\\(2\\cdot(\\omega+1)\\)\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>7. \uc11c\uc218\uc758 \uac70\ub4ed\uc81c\uacf1<\/h3>\n<p>\uc11c\uc218\uc758 \uac70\ub4ed\uc81c\uacf1\uc740 \ucd08\ud55c\uc7ac\uadc0\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\uc758 8.17. (\uc11c\uc218\uc758 \uac70\ub4ed\uc81c\uacf1)<\/span><\/p>\n<p>\uc11c\uc218 \\(\\alpha\\), \\(\\beta\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \\(\\alpha^\\beta\\)\ub97c \uc815\uc758\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\alpha^0=1\\).<\/li>\n<li>\\(\\alpha^{S(\\beta)}=\\alpha^\\beta\\cdot\\alpha\\).<\/li>\n<li>\\(\\lambda\\)\uac00 \\(0\\)\uc774 \uc544\ub2cc \uadf9\ud55c\uc11c\uc218\uc774\uba74<br \/>\n\\[<br \/>\n\\alpha^\\lambda=<br \/>\n\\begin{cases}<br \/>\n0 &#038; (\\alpha=0),\\\\[6pt]<br \/>\n\\displaystyle\\sup_{\\gamma<\\lambda}\\alpha^\\gamma &#038; (\\alpha>0).<br \/>\n\\end{cases}<br \/>\n\\]\n<\/li>\n<\/ol>\n<\/div>\n<p>\ub530\ub77c\uc11c \\(0^0=1\\)\uc774\uace0 \\(\\beta>0\\)\uc774\uba74 \\(0^\\beta=0\\)\uc774\ub2e4. \ub610\ud55c \uc815\ub9ac 8.10\uc5d0 \uc758\ud574 \uadf9\ud55c \ub2e8\uacc4\uc758 \uc0c1\ud55c\uc740 \ud56d\uc0c1 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.11.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(2^0,2^1,2^2,2^3,2^4\\)\ub97c \uc11c\uc218\uc758 \uac70\ub4ed\uc81c\uacf1 \uc815\uc758\ub85c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/li>\n<li>\\(0^0,0^1,0^\\omega\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(1^\\alpha=1\\)\uc774 \ubaa8\ub4e0 \uc11c\uc218 \\(\\alpha\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud568\uc744 \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(2^\\omega=\\sup_{n<\\omega}2^n\\)\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[2^\\omega=\\sup_{n<\\omega}2^n=\\omega\\]\n\uc774\uace0,\n\\[\\omega^\\omega=\\sup_{n<\\omega}\\omega^n\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\omega^\\omega\\)\ub294 \ubaa8\ub4e0 \\(\\omega^n\\)\ubcf4\ub2e4 \ud070 \uac00\uc0b0\uc11c\uc218\uc774\ub2e4.<\/p>\n<p>\uc5ec\uae30\uc11c \uc11c\uc218\uc758 \uac70\ub4ed\uc81c\uacf1\uacfc <a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">7\uc7a5<\/a>\uc758 \uae30\uc218 \uac70\ub4ed\uc81c\uacf1\uc744 \uad6c\ubcc4\ud574\uc57c \ud55c\ub2e4. \uac19\uc740 \uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc9c0\ub9cc \uc11c\ub85c \ub2e4\ub978 \uc5f0\uc0b0\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[2^\\omega=\\omega\\quad\\text{(\uc11c\uc218 \uc5f0\uc0b0)}\\]<br \/>\n\uc778 \ubc18\uba74<br \/>\n\\[2^{\\aleph_0}=\\mathfrak c\\quad\\text{(\uae30\uc218 \uc5f0\uc0b0)}\\]<br \/>\n\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc11c\uc218 \\(\\omega^\\omega\\)\ub294 \uac00\uc0b0\uc774\uc9c0\ub9cc \uae30\uc218 \\(\\aleph_0^{\\aleph_0}\\)\uc740 \\(\\mathfrak c\\)\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.12.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc11c\uc218 \\(\\omega+\\omega\\)\uc640 \\(\\omega\\cdot2\\)\ub97c \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<li>\uc11c\uc218 \\((\\omega+1)\\cdot2\\)\uc640 \\(\\omega\\cdot2+2\\)\ub97c \uacc4\uc0b0\ud558\uc5ec \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<li>\\(2^3\\)\uacfc \\(3^2\\)\ub97c \uc11c\uc218\ub85c\uc11c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/li>\n<li>\uc11c\uc218 \\(\\omega^2\\)\uc758 \uc21c\uc11c \uc720\ud615\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>8. \uc11c\uc218\uc640 \uae30\uc218\uc758 \uad00\uacc4<\/h3>\n<p>\uc11c\uc218 \\(\\kappa\\)\uac00 \uc790\uae30\ubcf4\ub2e4 \uc791\uc740 \uc5b4\ub5a4 \uc11c\uc218\uc640\ub3c4 \ub300\ub4f1\ud558\uc9c0 \uc54a\uc73c\uba74 \\(\\kappa\\)\ub97c <span class=\"defined\">\ucd08\uae30\uc11c\uc218<\/span>(initial ordinal)\ub77c\uace0 \ubd80\ub978\ub2e4. <span class=\"defined\">\uae30\uc218\uc11c\uc218<\/span>(cardinal ordinal)\ub77c\uace0\ub3c4 \ud55c\ub2e4. \ucd08\uae30\uc11c\uc218\ub294 \uac19\uc740 \ud06c\uae30\ub97c \uac16\ub294 \uc11c\uc218\ub4e4 \uac00\uc6b4\ub370 \uac00\uc7a5 \uc791\uc740 \uc11c\uc218\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 8.18. (\ucd08\uae30\uc11c\uc218\uc640 \uae30\uc218)<\/span><\/p>\n<p>\uc815\ub82c \uac00\ub2a5\ud55c \uc9d1\ud569 \\(A\\)\uc758 \uae30\uc218 \\(|A|\\)\ub9c8\ub2e4 \uadf8 \uae30\uc218\ub97c \uac16\ub294 \ucd08\uae30\uc11c\uc218\uac00 \uc815\ud655\ud788 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub77c\uc11c \uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uba74 \ubaa8\ub4e0 \uae30\uc218\ub97c \ucd08\uae30\uc11c\uc218\ub85c \ub300\ud45c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\uc758 \uc874\uc7ac \ubd80\ubd84\uc740 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">9\uc7a5<\/a>\uc758 \ucd08\uae30\uc11c\uc218 \uc874\uc7ac \uc815\ub9ac\uc5d0\uc11c \uc99d\uba85\ud55c\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uc9c0 \uc54a\ub294 ZF\uc5d0\uc11c\ub294 \uc815\ub82c \uac00\ub2a5\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569\uc774 \uc788\uc744 \uc218 \uc788\uc73c\ubbc0\ub85c, \ubaa8\ub4e0 \uae30\uc218\ub97c \uc11c\uc218\ub85c \ub300\ud45c\ud560 \uc218 \uc788\ub2e4\uace0 \ub9d0\ud574\uc11c\ub294 \uc548 \ub41c\ub2e4.<\/p>\n<p>\uc720\ud55c \ucd08\uae30\uc11c\uc218\ub294 \\(0,1,2,\\ldots\\)\uc774\uace0, \uac00\uc7a5 \uc791\uc740 \ubb34\ud55c \ucd08\uae30\uc11c\uc218\ub294 \\(\\omega\\)\uc774\ub2e4. \ubb34\ud55c \ucd08\uae30\uc11c\uc218\ub97c \ucc28\ub840\ub85c<br \/>\n\\[\\omega_0,\\omega_1,\\omega_2,\\ldots\\]<br \/>\n\ubc0f \ucd08\ud55c \ucca8\uc790\ub85c \uacc4\uc18d \ub098\ud0c0\ub0b4\uace0, \uadf8 \uae30\uc218\ub97c<br \/>\n\\[\\aleph_\\alpha=|\\omega_\\alpha|\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud2b9\ud788<br \/>\n\\[\\omega_0=\\omega,\\quad\\aleph_0=|\\omega|\\]<br \/>\n\uc774\uace0, \\(\\omega_1\\)\uc740 \uac00\uc7a5 \uc791\uc740 \ube44\uac00\uc0b0\uc11c\uc218\uc774\uba70<br \/>\n\\[\\aleph_1=|\\omega_1|\\]<br \/>\n\uc774\ub2e4. \uc774 \uc54c\ub808\ud504 \uacc4\uce35\uc758 \uc77c\ubc18\uc801\uc778 \uc874\uc7ac\uc640 \uad6c\uc131\uc740 \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc758 \ub3c4\uad6c\ub97c \ud544\uc694\ub85c \ud55c\ub2e4.<\/p>\n<p>\ud55c\ud3b8 \\(\\omega+1\\), \\(\\omega\\cdot2\\), \\(\\omega^2\\), \\(\\omega^\\omega\\)\ub294 \uc11c\ub85c \ub2e4\ub978 \uc11c\uc218\uc774\uc9c0\ub9cc \ubaa8\ub450 \uac00\uc0b0\uc774\ubbc0\ub85c \uae30\uc218\ub294 \\(\\aleph_0\\)\uc774\ub2e4. \uc989 \uae30\uc218\ub294 \ud06c\uae30\ub9cc \uad6c\ubcc4\ud558\uc9c0\ub9cc \uc11c\uc218\ub294 \uc21c\uc11c \uc720\ud615\uae4c\uc9c0 \uad6c\ubcc4\ud55c\ub2e4. \uc120\ud0dd\uacf5\ub9ac \uc544\ub798\uc5d0\uc11c <a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">7\uc7a5<\/a>\uc758 \uc5f0\uc18d\uccb4 \uac00\uc124\uc740<br \/>\n\\[2^{\\aleph_0}=\\aleph_1\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.13.<\/span><br \/>\n\ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ud568\uc218 \\(f\\colon\\omega\\to S(\\omega)\\)\ub97c<br \/>\n\\[<br \/>\nf(0)=\\omega,\\quad f(n+1)=n<br \/>\n\\]<br \/>\n\uc73c\ub85c \uc815\uc758\ud558\uc790. \\(f\\)\uac00 \uc77c\ub300\uc77c\ub300\uc751\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>(1)\uc758 \\(f\\)\uac00 \uc21c\uc11c\ub3d9\ud615\uc740 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\omega\\), \\(\\omega+1\\), \\(\\omega\\cdot2\\), \\(\\omega^2\\)\uc758 \uae30\uc218\ub97c \uac01\uac01 \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>(3)\uc758 \uacb0\uacfc\ub97c \uc774\uc6a9\ud558\uc5ec \uac19\uc740 \uae30\uc218\ub97c \uac00\uc9c4 \uc11c\ub85c \ub2e4\ub978 \uc11c\uc218\uac00 \uc874\uc7ac\ud568\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>9. \uc11c\uc218\uc640 \uad00\ub828\ub41c \uc5ed\uc124<\/h3>\n<p>\ubaa8\ub4e0 \uc11c\uc218\uc758 \ubaa8\uc784\uc744 \\(\\mathrm{Ord}\\)\ub77c\uace0 \ud558\uc790. \\(\\mathrm{Ord}\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \uac00\uc815\ud558\uba74, \ubaa8\ub4e0 \uc11c\uc218\uc758 \uc6d0\uc18c\ub294 \ub2e4\uc2dc \uc11c\uc218\uc774\uace0 \uc11c\uc218\ub4e4\uc740 \\(\\in\\)\uc5d0 \uc758\ud558\uc5ec \uc815\ub82c\ub418\ubbc0\ub85c \\(\\mathrm{Ord}\\) \uc790\uccb4\ub3c4 \uc11c\uc218\uac00 \ub41c\ub2e4. \uadf8\ub7ec\uba74 \uc11c\uc218 \uc804\uccb4\uc758 \ubaa8\uc784\uc774\ub77c\ub294 \uc815\uc758\uc5d0 \uc758\ud574<br \/>\n\\[\\mathrm{Ord}\\in\\mathrm{Ord}\\]<br \/>\n\uc774\uc5b4\uc57c \ud558\ub294\ub370, \uc774\ub294 \uc815\ub9ac 8.5(2)\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uc774\ub97c <span class=\"defined\">\ubd80\ub784\ub9ac-\ud3ec\ub974\ud2f0\uc758 \uc5ed\uc124<\/span>(Burali\u2013Forti paradox)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ub530\ub77c\uc11c \ubaa8\ub4e0 \uc11c\uc218\uc758 \ubaa8\uc784\uc740 \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c <span class=\"defined\">\uace0\uc720\ud074\ub798\uc2a4<\/span>(proper class)\uc774\ub2e4. <a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">9\uc7a5<\/a>\uc5d0\uc11c \ub2e4\ub8f0 \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \u201c\ubaa8\ub4e0 \uc11c\uc218\uc758 \uc9d1\ud569\u201d\uc744 \ub9cc\ub4e4 \uc218 \uc5c6\uc73c\uba70, \uc774\ub7ec\ud55c \uad6c\ubd84\uc774 \uc5ed\uc124\uc744 \ub9c9\ub294\ub2e4.<\/p>\n<h3>10. \uc11c\uc218\uc758 \uc9d1\ud569\ub860\uc801 \uad6c\uc131<\/h3>\n<p>\uc55e\uc5d0\uc11c\ub294 \ud3f0 \ub178\uc774\ub9cc \uc11c\uc218\ub97c \uc815\uc758\ud558\uace0 \uc815\ub9ac 8.6\uc744 \uc0ac\uc6a9\ud558\uc600\ub2e4. \uc774\uc81c \uc774 \uc815\ub9ac\uc758 \ud575\uc2ec \uad6c\uc131\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc815\ub82c\uc9d1\ud569 \\((A,<)\\)\uc5d0 \ub300\ud558\uc5ec \ucd08\ud55c\uc7ac\uadc0\ub85c\n\\[F(a)=\\{F(x)\\mid x<a\\}\\]\n\uac00 \ub418\ub3c4\ub85d \\(F\\)\ub97c \uc815\uc758\ud55c\ub2e4. \uc774\ub54c \uac01 \\(F(a)\\)\ub294 \uc55e\uc5d0\uc11c \uc774\ubbf8 \ub9cc\ub4e4\uc5b4\uc9c4 \uac12\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<\/p>\n<p>\ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uba74 \uac01 \\(F(a)\\)\uac00 \uc11c\uc218\uc774\uace0<br \/>\n\\[x<a\\quad\\Longleftrightarrow\\quad F(x)\\in F(a)\\]\n\uc784\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \\(F\\)\ub294 \\(A\\)\uc5d0\uc11c \uadf8 \uce58\uc5ed\n\\[\\alpha=\\{F(a)\\mid a\\in A\\}\\]\n\uc73c\ub85c \uac00\ub294 \uc21c\uc11c\ub3d9\ud615\uc774\ub2e4. \ub610\ud55c \\(\\alpha\\)\ub294 \ucd94\uc774\uc801\uc774\uace0 \\(\\in\\)\uc5d0 \uc758\ud574 \uc815\ub82c\ub418\ubbc0\ub85c \uc11c\uc218\uc774\ub2e4. \uc774\uac83\uc774 \uc815\ub9ac 8.6\uc758 \uc874\uc7ac \ubd80\ubd84\uc774\ub2e4.<\/p>\n<p>\uc720\uc77c\uc131\uc744 \ubcf4\uc774\uae30 \uc704\ud558\uc5ec \ub450 \uc11c\uc218 \\(\\alpha\\), \\(\\beta\\) \uc0ac\uc774\uc5d0 \uc21c\uc11c\ub3d9\ud615 \\(f\\)\uac00 \uc788\ub2e4\uace0 \ud558\uc790. \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec<br \/>\n\\[f(\\gamma)=\\gamma\\quad(\\gamma\\in\\alpha)\\]<br \/>\n\uc784\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \uc2e4\uc81c\ub85c \ubaa8\ub4e0 \\(\\delta<\\gamma\\)\uc5d0\uc11c \\(f(\\delta)=\\delta\\)\ub77c\uace0 \uac00\uc815\ud558\uba74 \uc21c\uc11c\ub3d9\ud615\uc740 \uc808\ud3b8\uc744 \uc808\ud3b8\uc73c\ub85c \ubcf4\ub0b4\ubbc0\ub85c\n\\[f(\\gamma)=\\{f(\\delta)\\mid\\delta<\\gamma\\}=\\{\\delta\\mid\\delta<\\gamma\\}=\\gamma\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(f\\)\ub294 \ud56d\ub4f1\ud568\uc218\uc774\uace0 \\(\\alpha=\\beta\\)\uc774\ub2e4.<\/p>\n<p>\uc774 \uad6c\uc131\uc758 \uc5c4\ubc00\ud55c \uc9d1\ud569\ub860\uc801 \uc815\ub2f9\ud654\uc5d0\ub294 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">9\uc7a5<\/a>\uc5d0\uc11c \ub2e4\ub8f0 \uce58\ud658 \uacf5\ub9ac\uaf34 \ub4f1\uc774 \uc0ac\uc6a9\ub41c\ub2e4. \ub610\ud55c \\(\\omega\\)\uc640 \uac19\uc740 \ubb34\ud55c\uc11c\uc218\uc758 \uc874\uc7ac\uc5d0\ub294 \ubb34\ud55c \uacf5\ub9ac\uac00 \ud544\uc694\ud558\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.14.<\/span><br \/>\n\uc815\ub82c\uc9d1\ud569<br \/>\n\\[A=\\{a<b<c<d\\}\\]\n\uc5d0 \ub300\ud558\uc5ec \ubcf8\ubb38\uc758 \ucd08\ud55c\uc7ac\uadc0\n\\[F(x)=\\{F(y)\\mid y<x\\}\\]\n\ub97c \uc801\uc6a9\ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(F(a),F(b),F(c),F(d)\\)\ub97c \ucc28\ub840\ub85c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(\\alpha=\\{F(x)\\mid x\\in A\\}\\)\ub97c \uad6c\ud558\uace0, \\(\\alpha\\)\uac00 \uc5b4\ub5a4 \uc11c\uc218\uc778\uc9c0 \ubc1d\ud788\uc2dc\uc624.<\/li>\n<li>\\(F\\colon A\\to\\alpha\\)\uac00 \uc21c\uc11c\ub3d9\ud615\uc784\uc744 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/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>\uae30\uc218\uac00 \uc9d1\ud569\uc758 \u2018\ud06c\uae30\u2019\ub97c \ub098\ud0c0\ub0b8\ub2e4\uba74, \uc11c\uc218(ordinal number)\ub294 \uc815\ub82c\ub41c \uc9d1\ud569\uc758 \u2018\uc21c\uc11c \uc720\ud615\u2019\uc744 \ub098\ud0c0\ub0b8\ub2e4. 6\uc7a5\uc5d0\uc11c \uc790\uc5f0\uc218\ub97c \ud3f0 \ub178\uc774\ub9cc \ubc29\uc2dd\uc744 \uc0ac\uc6a9\ud558\uc5ec \uad6c\uc131\ud558\uace0, 7\uc7a5\uc5d0\uc11c \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \uae30\uc218\ub85c \ub098\ud0c0\ub0b4\uc5c8\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc815\ub82c\uc9d1\ud569\uacfc \uc11c\uc218\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uace0, \uc11c\uc218\uc758 \uc21c\uc11c\uc640 \uc5f0\uc0b0, \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. 1. \uc815\ub82c\uc21c\uc11c\uc640 \uc815\ub82c\uc9d1\ud569 \uc815\uc758 8.1. (\uc815\ub82c\uc21c\uc11c\uc640 \uc815\ub82c\uc9d1\ud569) \uc9d1\ud569 \\(A\\) \uc704\uc758 \uc21c\uc11c\uad00\uacc4 \\(\\leq\\)\uac00 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c \\(\\leq\\)\ub97c \\(A\\) \uc704\uc758 \uc815\ub82c\uc21c\uc11c(well-ordering)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\leq\\)\ub294 \uc804\uc21c\uc11c\uc774\ub2e4. \uc989, \uc784\uc758\uc758 \\(a,b\\in A\\)\uc5d0 \ub300\ud558\uc5ec \\(a\\leq&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":108,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9261","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9261","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=9261"}],"version-history":[{"count":10,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9261\/revisions"}],"predecessor-version":[{"id":10083,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9261\/revisions\/10083"}],"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=9261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}