{"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":"2025-10-20T18:48:50","modified_gmt":"2025-10-20T09:48:50","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 &#8216;\ud06c\uae30&#8217;\ub97c \ub098\ud0c0\ub0b8\ub2e4\uba74, <span class=\"defined\">\uc11c\uc218<\/span>(ordinal number)\ub294 \uc9d1\ud569\uc758 \uc6d0\uc18c\ub4e4\uc758 &#8216;\uc21c\uc11c&#8217;\uae4c\uc9c0 \uace0\ub824\ud55c \uac1c\ub150\uc774\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc815\ub82c\uc9d1\ud569\uacfc \uc11c\uc218\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uace0, \uc11c\uc218\uc758 \uc5f0\uc0b0\uacfc \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<p>\uc9d1\ud569 \\(A\\) \uc704\uc758 \uc21c\uc11c\uad00\uacc4 \\(\\leq\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\ud560 \ub54c \uc774 \uc21c\uc11c\uad00\uacc4\ub97c <span class=\"defined\">\uc815\ub82c\uc21c\uc11c<\/span>(well-ordering)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ul>\n<li>\\(\\le\\)\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\uc774 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<\/ul>\n<p>\uc815\ub82c\uc21c\uc11c\ub97c \uac16\ub294 \uc9d1\ud569\uc744 <span class=\"defined\">\uc815\ub82c\uc9d1\ud569<\/span>(well-ordered set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc815\ub82c\uc9d1\ud569\uc758 \ub300\ud45c\uc801\uc778 \uc608\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li>\uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\mathbb{N}\\)\uc740 \ud1b5\uc0c1\uc801\uc778 \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, \uc5f4\ub9b0\uad6c\uac04 \\((0,\\, 1)\\)\uc740 \ucd5c\uc18c\uc6d0\uc18c\uac00 \uc5c6\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\ub294 \uc5ec\ub7ec \uac00\uc9c0 \uc815\ub82c\uc21c\uc11c\ub97c \ubd80\uc5ec\ud560 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(\\mathbb{Z}\\)\uc5d0 \ub2e4\uc74c\uacfc \uac19\uc740 \uc21c\uc11c\ub97c \uc815\uc758\ud558\uba74 \uc815\ub82c\uc9d1\ud569\uc774 \ub41c\ub2e4.<br \/>\n\\[0 < 1 < -1 < 2 < -2 < 3 < -3 < \\cdots\\]<\/p>\n<p>\uc21c\uc11c\uad00\uacc4 \\(\\le\\)\uac00 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569 \\(A\\)\ub97c \uc21c\uc11c\uc9d1\ud569\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\left( A ,\\, \\le \\right)\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \ub450 \uc21c\uc11c\uc9d1\ud569 \\((A,\\, \\leq_A)\\)\uc640 \\((B,\\, \\leq_B)\\)\uac00 <span class=\"defined\">\uc21c\uc11c\ub3d9\ud615<\/span>(order isomorphic)\uc774\ub77c\ub294 \uac83\uc740 \uc21c\uc11c\ub97c \ubcf4\uc874\ud558\ub294 \uc77c\ub300\uc77c\ub300\uc751 \\(f: A \\to B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4. \uc989 \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\ub294 \uc77c\ub300\uc77c\ub300\uc751 \\(f:A \\to B\\)\uac00 \uc874\uc7ac\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc21c\uc11c\ub3d9\ud615\uc778 \uc815\ub82c\uc9d1\ud569\ub4e4\uc740 \ubcf8\uc9c8\uc801\uc73c\ub85c \uac19\uc740 \uc21c\uc11c\uad6c\uc870\ub97c \uac00\uc9c4\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc21c\uc11c\uad00\uacc4<br \/>\n\\[a_1 < a_2 < \\cdots < a_n \\]\n\uc774 \uc8fc\uc5b4\uc9c4 \uc21c\uc11c\uc9d1\ud569 \\(\\{a_1,\\, a_2,\\, a_3,\\, \\ldots,\\, a_n\\}\\)\uc740 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569 \\(\\{1,\\, 2,\\, 3,\\, \\ldots,\\, n\\}\\)\uacfc \uc21c\uc11c\ub3d9\ud615\uc774\ub2e4.<\/p>\n<p>\uc815\ub82c\uc9d1\ud569 \\((A,\\, \\leq)\\)\uc640 \uc6d0\uc18c \\(a \\in A\\)\uc5d0 \ub300\ud558\uc5ec, <span class=\"defined\">\uc808\ud3b8<\/span>(initial segment) \\(A_a\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[A_a = \\{x \\in A \\mid x < a\\}\\]<\/p>\n<p>\ub450 \uc815\ub82c\uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uc5d0 \ub300\ud558\uc5ec, \ub2e4\uc74c \uc911 \uc815\ud655\ud788 \ud558\ub098\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\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<\/ul>\n<p>\uc989 \uc815\ub82c\uc9d1\ud569 \uc0ac\uc774\uc5d0 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \ube44\uad50 \uad00\uacc4\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 8.1.<\/span><br \/>\n\uc21c\uc11c\uc9d1\ud569\uc5d0\uc11c &#8216;\uc0c1\uacc4&#8217;, &#8216;\uc0c1\ud55c&#8217;, &#8216;\ud558\uacc4&#8217;, &#8216;\ud558\ud55c&#8217;\uc758 \uac1c\ub150\uc744 \uc870\uc0ac\ud574 \ubcf4\uc790.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 8.2.<\/span><br \/>\n\uc218\ud559\uc5d0\uc11c &#8216;\ub3d9\ud615&#8217;\uc758 \uac1c\ub150\uc744 \uc870\uc0ac\ud558\uace0, \uc77c\ubc18\uc801\uc778 &#8216;\ub3d9\ud615&#8217;\uc758 \uac1c\ub150\uacfc &#8216;\uc21c\uc11c\ub3d9\ud615&#8217;\uc758 \uac1c\ub150\uc758 \uad00\uacc4\ub97c \uc11c\uc220\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>2. \uc11c\uc218\uc758 \uc815\uc758<\/h3>\n<p><span class=\"defined\">\uc11c\uc218<\/span>(ordinal number)\ub294 \uc21c\uc11c\ub3d9\ud615\uc778 \uc815\ub82c\uc9d1\ud569\ub4e4\uc774 \uacf5\ud1b5\uc73c\ub85c \uac16\ub294 &#8216;\uc21c\uc11c \uc720\ud615&#8217;\uc744 \ub098\ud0c0\ub0b4\ub294 \uac1c\ub150\uc774\ub2e4. \uc11c\uc218\ub97c \uc5c4\ubc00\ud558\uac8c \ub2e4\ub8e8\uae30 \uc704\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 <span class=\"defined\">\uc11c\uc218 \uacf5\ub9ac<\/span>\ub97c \ub3c4\uc785\ud55c\ub2e4.<\/p>\n<div class=\"box\">\n<p>\ubaa8\ub4e0 \uc6d0\uc18c\uac00 \uc815\ub82c\uc9d1\ud569\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 <span class=\"defined\">\uc11c\uc218\uc758 \ubaa8\uc784<\/span> \\(ORD\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<ul>\n<li>(O1) \uc784\uc758\uc758 \uc815\ub82c\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\)\uc640 \uc21c\uc11c\ub3d9\ud615\uc778 \\(\\alpha\\in ORD\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(\\alpha\\)\ub97c \\(A\\)\uc758 <span class=\"defined\">\uc11c\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0, \\(\\operatorname{ord}(A)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>(O2) \ub450 \uc815\ub82c\uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uac00 \uc21c\uc11c\ub3d9\ud615\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\operatorname{ord}(A) = \\operatorname{ord}(B)\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<\/div>\n<p>\uc720\ud55c \uc815\ub82c\uc9d1\ud569\uc758 \uc11c\uc218\ub294 \uc790\uc5f0\uc218\uc774\ub2e4.<\/p>\n<ul>\n<li>\\(\\text{ord}(\\varnothing) = 0\\)<\/li>\n<li>\\(\\text{ord}(\\{a\\}) = 1\\)<\/li>\n<li>\\(\\text{ord}(\\{a < b\\}) = 2\\)<\/li>\n<li>\uc77c\ubc18\uc801\uc73c\ub85c \\(n\\)\uac1c \uc6d0\uc18c\ub97c \uac00\uc9c4 \uc815\ub82c\uc9d1\ud569\uc758 \uc11c\uc218\ub294 \\(n\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ubb34\ud55c \uc815\ub82c\uc9d1\ud569\uc758 \uc11c\uc218\ub97c \ucd08\ud55c\uc11c\uc218(transfinite ordinal)\ub77c\uace0 \ubd80\ub978\ub2e4. \uac00\uc7a5 \uc791\uc740 \ucd08\ud55c\uc11c\uc218\ub294 \\(\\omega\\)\ub85c, \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c4 \uc790\uc5f0\uc218 \uc9d1\ud569\uc758 \uc11c\uc218\uc774\ub2e4.<br \/>\n\\[\\omega = \\text{ord}(\\mathbb{N}) = \\text{ord}(\\{0 < 1 < 2 < 3 < \\cdots\\})\\]<\/p>\n<h3>3. \uc11c\uc218\uc758 \uc21c\uc11c<\/h3>\n<p>\uc11c\uc218\ub4e4 \uc0ac\uc774\uc5d0\ub294 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc21c\uc11c\uac00 \uc788\ub2e4. \uc11c\uc218 \\(\\alpha\\)\uc640 \\(\\beta\\)\uc5d0 \ub300\ud558\uc5ec, \\(\\alpha < \\beta\\)\ub77c\ub294 \uac83\uc740 \uc11c\uc218 \\(\\alpha\\)\ub97c \uac16\ub294 \uc815\ub82c\uc9d1\ud569\uc774 \uc11c\uc218 \\(\\beta\\)\ub97c \uac16\ub294 \uc815\ub82c\uc9d1\ud569\uc758 \uc808\ud3b8\uacfc \uc21c\uc11c\ub3d9\ud615\uc774\ub77c\ub294 \uc758\ubbf8\uc774\ub2e4.<\/p>\n<p>\uc11c\uc218\uc758 \uc21c\uc11c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[0 < 1 < 2 < 3 < \\cdots < \\omega < \\omega + 1 < \\omega + 2 < \\cdots < \\omega \\cdot 2 < \\cdots\\]\n\uc5ec\uae30\uc11c \uc8fc\ubaa9\ud560 \uc810\uc740 \\(\\omega\\) \ub2e4\uc74c\uc5d0\ub3c4 \uc11c\uc218\uac00 \uacc4\uc18d \uc874\uc7ac\ud55c\ub2e4\ub294 \uac83\uc774\ub2e4. \ub610\ud55c \uc11c\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c0\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uba74, \uadf8 \uc9d1\ud569\uc5d0\ub294 \ubc18\ub4dc\uc2dc \uac00\uc7a5 \uc791\uc740 \uc11c\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc989 \uc11c\uc218\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud558\ub294 \uc21c\uc11c \uad00\uacc4\ub294 \uc815\ub82c\uc21c\uc11c\uc774\ub2e4.<\/p>\n<h3>4. \ucd08\ud55c\uadc0\ub0a9\ubc95<\/h3>\n<p>6\uc7a5\uc5d0\uc11c \ub2e4\ub8ec \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc11c\uc218\ub85c \ud655\uc7a5\ud55c \uac83\uc774 <span class=\"defined\">\ucd08\ud55c\uadc0\ub0a9\ubc95<\/span>(transfinite induction)\uc774\ub2e4. \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc9c4\uc220\ud558\uae30 \uc704\ud574\uc11c\ub294 <span class=\"defined\">\uc9c1\uc804\uc6d0\uc18c<\/span>(immediate predecessor)\uc640 <span class=\"defined\">\uadf9\uc6d0\uc18c<\/span>(limit element)\uc758 \uac1c\ub150\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n<p>\\(A\\)\uac00 \uc815\ub82c\uc9d1\ud569\uc774\uace0 \\(x \\in A\\)\ub77c\uace0 \ud558\uc790. \\(A\\)\uc758 \uc6d0\uc18c \uc911 \\(x\\)\ubcf4\ub2e4 \ud070 \uac83\ub4e4\uc758 \uc9d1\ud569\uc774 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac00\uc9c8 \ub54c \uadf8 \ucd5c\uc18c\uc6d0\uc18c\ub97c \\(x\\)\uc758 <span class=\"defined\">\uc9c1\ud6c4\uc6d0\uc18c<\/span>(immediate successor) \ub610\ub294 <span class=\"defined\">\ub530\ub984\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(y\\)\uac00 \\(x\\)\uc758 \uc9c1\ud6c4\uc6d0\uc18c\uc774\uba74, \\(x\\)\ub97c \\(y\\)\uc758 \uc9c1\uc804\uc6d0\uc18c\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc815\ub82c\uc9d1\ud569\uc758 \uc6d0\uc18c \uc911\uc5d0\uc11c\ub294 \uc9c1\uc804\uc6d0\uc18c\ub97c \uac16\uc9c0 \uc54a\ub294 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc790\uc5f0\uc218\ub97c \ud640\uc218\ubd80\ud130 \ub2e4 \ub098\uc5f4\ud55c \ub4a4 \uc9dd\uc218\ub97c \ubc30\uc5f4\ud558\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc21c\uc11c\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\[1 < 3 < 5 < 7 < \\cdots < 0 < 2 < 4 < 6 \\cdots \\]\n\uc5ec\uae30\uc11c \ubaa8\ub4e0 \uc6d0\uc18c\ub294 \uc9c1\ud6c4\uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4. \ud558\uc9c0\ub9cc \\(0\\)\uc740 \uc9c1\uc804\uc6d0\uc18c\ub97c \uac16\uc9c0 \uc54a\ub294\ub2e4. \uc774\uc640 \uac19\uc774 \uc815\ub82c\uc9d1\ud569\uc5d0\uc11c \ucd5c\uc18c\uc6d0\uc18c\uac00 \uc544\ub2c8\uba74\uc11c \uc9c1\uc804\uc6d0\uc18c\ub97c \uac16\uc9c0 \uc54a\ub294 \uc6d0\uc18c\ub97c \uadf9\uc6d0\uc18c\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc11c\uc218\ub97c \ud06c\uae30 \uc21c\uc73c\ub85c \ub098\uc5f4\ud588\uc744 \ub54c \uc9c1\uc804\uc6d0\uc18c\ub97c \uac16\uc9c0 \uc54a\ub294 \uc11c\uc218\ub97c <span class=\"defined\">\uadf9\uc11c\uc218<\/span>(limit ordinal)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(\\omega\\)\ub294 \uac00\uc7a5 \uc791\uc740 \ubb34\ud55c \uadf9\uc11c\uc218\uc774\ub2e4.<\/p>\n<p>\ucd08\ud55c\uadc0\ub0a9\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"box\">\n<p>\uc815\ub82c\uc9d1\ud569 \\(A\\)\uc758 \uc6d0\uc18c\uc5d0 \ub300\ud55c \uba85\uc81c \\(p(x)\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74, \ubaa8\ub4e0 \\(x \\in A\\)\uc5d0 \ub300\ud574 \\(p(x)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p style=\"text-align: center;\">&#8220;\uc784\uc758\uc758 \\(x \\in A\\)\uc5d0 \ub300\ud558\uc5ec, \\(x\\)\ubcf4\ub2e4 \uc791\uc740 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \ub300\ud558\uc5ec \\(p\\)\uac00 \uc131\ub9bd\ud558\uba74 \\(p(x)\\)\ub3c4 \uc131\ub9bd\ud55c\ub2e4.&#8221;<\/p>\n<\/div>\n<p>\ucd08\ud55c\uadc0\ub0a9\ubc95\uc740 \uc99d\uba85\ud560 \ub54c\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc815\uc758\ud560 \ub54c\ub3c4 \uc0ac\uc6a9\ub41c\ub2e4. \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\uc5ed\uc774 \uc815\ub82c\uc9d1\ud569\uc778 \ud568\uc218\ub97c \uc815\uc758\ud558\ub294 \uac83\uc744 <span class=\"defined\">\ucd08\ud55c\uc7ac\uadc0<\/span>(transfinite recursion)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.3.<\/span><br \/>\n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc815\ub82c\uc9d1\ud569\uc5d0\uc11c \ucd5c\uc18c\uc6d0\uc18c\uac00 \uc544\ub2cc \ubaa8\ub4e0 \uc6d0\uc18c \\(x\\)\ub294 \uc9c1\uc804\uc6d0\uc18c\ub97c \uac16\uac70\ub098 \uadf9\ud55c\uc6d0\uc18c\uc774\ub2e4.<\/li>\n<li>\uc11c\ub85c \ub2e4\ub978 \uc11c\uc218 \\(\\alpha\\), \\(\\beta\\)\uc5d0 \ub300\ud558\uc5ec \\(\\alpha < \\beta\\)\uc774\uac70\ub098 \\(\\beta < \\alpha\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<h3>5. \uc11c\uc218\uc758 \ub367\uc148<\/h3>\n<p>\ub450 \uc11c\uc218 \\(\\alpha\\)\uc640 \\(\\beta\\)\uc758 <span class=\"defined\">\ud569<\/span> \\(\\alpha + \\beta\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4. \uc11c\uc218 \\(\\alpha\\)\ub97c \uac16\ub294 \uc815\ub82c\uc9d1\ud569 \\(A\\)\uc640 \uc11c\uc218 \\(\\beta\\)\ub97c \uac16\ub294 \uc815\ub82c\uc9d1\ud569 \\(B\\)\ub97c \ud0dd\ud558\ub418 \ub450 \uc9d1\ud569\uc774 \uc11c\ub85c\uc18c\uac00 \ub418\ub3c4\ub85d \ud0dd\ud558\uace0, \\(A \\cup B\\)\uc5d0 \ub2e4\uc74c \uc21c\uc11c\ub97c \uc900\ub2e4.<\/p>\n<ul>\n<li>\\(A\\)\uc758 \uc6d0\uc18c\ub4e4\uc740 \uc6d0\ub798 \uc21c\uc11c\ub97c \uc720\uc9c0\ud55c\ub2e4.<\/li>\n<li>\\(B\\)\uc758 \uc6d0\uc18c\ub4e4\uc740 \uc6d0\ub798 \uc21c\uc11c\ub97c \uc720\uc9c0\ud55c\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc784\uc758\uc758 \uc6d0\uc18c\ub294 \\(B\\)\uc758 \uc784\uc758\uc758\uc758 \uc6d0\uc18c\ubcf4\ub2e4 \uc791\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\ub54c \\(\\alpha + \\beta = \\text{ord}(A \\cup B)\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\uc608\ub97c \ub4e4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\\(2 + 3 = 5\\)<\/li>\n<li>\\(\\omega + 1\\) (\uc790\uc5f0\uc218\ub4e4 \ub2e4\uc74c\uc5d0 \ud558\ub098\uc758 \uc6d0\uc18c\uac00 \ub354 \uc788\ub294 \uc21c\uc11c)<\/li>\n<li>\\(1 + \\omega = \\omega\\)<\/li>\n<\/ul>\n<p>\uc11c\uc218 \ub367\uc148\uc758 \uc131\uc9c8\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\uacb0\ud569\ubc95\uce59: \\((\\alpha + \\beta) + \\gamma = \\alpha + (\\beta + \\gamma)\\)<\/li>\n<li>\uc67c\ucabd \ud56d\ub4f1\uc6d0: \\(0 + \\alpha = \\alpha\\)<\/li>\n<li>\uc624\ub978\ucabd \uc18c\uac70\ubc95\uce59: \\(\\alpha + \\gamma = \\beta + \\gamma\\)\uc774\uba74 \\(\\alpha = \\beta\\)<\/li>\n<li>\uad50\ud658\ubc95\uce59\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc598\ub97c \ub4e4\uc5b4 \\(1 + \\omega \\neq \\omega + 1\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<h3>6. \uc11c\uc218\uc758 \uacf1\uc148<\/h3>\n<p>\ub450 \uc11c\uc218 \\(\\alpha\\)\uc640 \\(\\beta\\)\uc758 <span class=\"defined\">\uacf1<\/span> \\(\\alpha \\cdot \\beta\\)\ub294 \ub370\uce74\ub974\ud2b8 \uacf1 \\(B \\times A\\)\uc5d0 <span class=\"defined\">\uc0ac\uc804\uc2dd \uc21c\uc11c<\/span>\ub97c \uc900 \uc815\ub82c\uc9d1\ud569\uc758 \uc11c\uc218\uc774\ub2e4. \uc5ec\uae30\uc11c \\((b_1,\\, a_1) < (b_2,\\, a_2)\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[b_1 < b_2 \\quad \\text{\ub610\ub294} \\quad (b_1 = b_2 \\,\\text{ \uadf8\ub9ac\uace0 }\\, a_1 < a_2)\\]<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4 \ub2e4\uc74c\uacfc \uac19\uc740 \uc11c\uc218\uc758 \uacf1\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li>\\(2 \\cdot 3 = 6\\)<\/li>\n<li>\\(\\omega \\cdot 2\\) (\uc790\uc5f0\uc218\uac00 \ub450 \ubc8c \uc788\ub294 \uc21c\uc11c: \\(0 < 1 < 2 < \\cdots < 0' < 1' < 2' < \\cdots\\))<\/li>\n<li>\\(2 \\cdot \\omega = \\omega\\)<\/li>\n<\/ul>\n<p>\uc11c\uc218 \uacf1\uc148\uc758 \uc131\uc9c8\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\uacb0\ud569\ubc95\uce59: \\((\\alpha \\cdot \\beta) \\cdot \\gamma = \\alpha \\cdot (\\beta \\cdot \\gamma)\\)<\/li>\n<li>\uc67c\ucabd \ubd84\ubc30\ubc95\uce59: \\(\\alpha \\cdot (\\beta + \\gamma) = \\alpha \\cdot \\beta + \\alpha \\cdot \\gamma\\)<\/li>\n<li>\uc624\ub978\ucabd \ubd84\ubc30\ubc95\uce59\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<li>\uad50\ud658\ubc95\uce59\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4, \uc608\ub97c \ub4e4\uc5b4 \\(2 \\cdot \\omega \\neq \\omega \\cdot 2\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<h3>7. \uc11c\uc218\uc758 \uac70\ub4ed\uc81c\uacf1<\/h3>\n<p>\uc11c\uc218\uc758 <span class=\"defined\">\uac70\ub4ed\uc81c\uacf1<\/span> \\(\\alpha^\\beta\\)\ub294 \ucd08\ud55c\uc7ac\uadc0\ub85c \uc815\uc758\ub41c\ub2e4.<\/p>\n<ul>\n<li>\\(\\alpha^0 = 1\\)<\/li>\n<li>\\(\\alpha^{\\beta+1} = \\alpha^\\beta \\cdot \\alpha\\)<\/li>\n<li>\\(\\beta\\)\uac00 \uadf9\ud55c\uc11c\uc218\uc77c \ub54c, \\(\\alpha^\\beta = \\operatorname{sup}\\{\\alpha^\\gamma \\mid \\gamma < \\beta\\}\\). [\\(\\sup\\)\uc740 '\uc0c1\ud55c'(supremum)\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc0c1\ud55c\uc744 '\ucd5c\uc18c\uc0c1\uacc4'(least upper bound)\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.]<\/li>\n<\/ul>\n<p>\uc608\ub97c \ub4e4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\\(2^\\omega\\): \uc774\uc9c4\uc218\uc5f4\uc744 \uc0ac\uc804\uc2dd \uc21c\uc11c\ub85c \ub098\uc5f4\ud55c \uac83.<\/li>\n<li>\\(\\omega^\\omega\\): \uc790\uc5f0\uc218 \uc218\uc5f4\uc744 \uc0ac\uc804\uc2dd \uc21c\uc11c\ub85c \ub098\uc5f4\ud55c \uac83.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 8.4.<\/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 \\cdot 2\\)\ub97c \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<li>\uc11c\uc218 \\((\\omega + 1) \\cdot 2\\)\uc640 \\(\\omega \\cdot 2 + 2\\)\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/li>\n<li>\\(2^3\\)\uacfc \\(3^2\\)\uc744 \uc11c\uc218\ub85c\uc11c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/li>\n<li>\uc11c\uc218 \\(\\omega^2\\)\uc758 \uc758\ubbf8\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>8. \uc11c\uc218\uc640 \uae30\uc218\uc758 \uad00\uacc4<\/h3>\n<p>\ubaa8\ub4e0 \uae30\uc218\ub294 \ud2b9\ubcc4\ud55c \uc11c\uc218\ub85c \ubcfc \uc218 \uc788\ub2e4. \uad6c\uccb4\uc801\uc73c\ub85c, \uae30\uc218\ub294 \uadf8 \ud06c\uae30\ub97c \uac16\ub294 \uac00\uc7a5 \uc791\uc740 \uc11c\uc218\uc774\ub2e4. \uc774\ub7ec\ud55c \uc11c\uc218\ub97c <span class=\"defined\">\uae30\uc218\uc11c\uc218<\/span>(cardinal ordinal) \ub610\ub294 <span class=\"defined\">\ucd08\uae30\uc11c\uc218<\/span>(initial ordinal)\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4 \ub2e4\uc74c\uacfc \uac19\uc740 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li>\uc720\ud55c\uae30\uc218 \\(n\\)\uc740 \uc11c\uc218 \\(n\\)\uacfc \uac19\ub2e4.<\/li>\n<li>\\(\\aleph_0\\)\uc740 \uc11c\uc218 \\(\\omega\\)\uc640 \uac19\ub2e4.<\/li>\n<li>\\(\\omega + 1\\), \\(\\omega \\cdot 2\\), \\(\\omega^2\\) \ub4f1\uc740 \ubaa8\ub450 \uac00\uc0b0\ubb34\ud55c\uc774\uc9c0\ub9cc \uc11c\ub85c \ub2e4\ub978 \uc11c\uc218\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc989, \uac19\uc740 \uae30\uc218\ub97c \uac16\ub294 \uc5ec\ub7ec \uc11c\uc218\uac00 \uc874\uc7ac\ud560 \uc218 \uc788\ub2e4. \uc11c\uc218\ub294 \uc9d1\ud569\uc758 \ud06c\uae30\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc21c\uc11c \uad6c\uc870\uae4c\uc9c0 \uad6c\ubcc4\ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<h3>9. \uc11c\uc218\uc640 \uad00\ub828\ub41c \uc5ed\uc124<\/h3>\n<p>\ubaa8\ub4e0 \uc11c\uc218\uc758 \ubaa8\uc784\uc744 \uc0dd\uac01\ud558\uba74, \uc774\uac83\ub3c4 \uc815\ub82c\uc21c\uc11c\ub97c \uac00\uc9c0\ubbc0\ub85c \uc11c\uc218\ub97c \uac00\uc838\uc57c \ud55c\ub2e4. \uadf8\ub7f0\ub370 \uc774 \uc11c\uc218\ub294 \ubaa8\ub4e0 \uc11c\uc218\ubcf4\ub2e4 \ucee4\uc57c \ud558\ubbc0\ub85c \ubaa8\uc21c\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \uc5ed\uc124\uc744 <span class=\"defined\">\ubd80\ub784\ub9ac-\ud3ec\ub974\ud2f0\uc758 \uc5ed\uc124<\/span>(Burali-Forti paradox)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc774 \uc5ed\uc124\uc740 &#8220;\ubaa8\ub4e0 \uc11c\uc218\uc758 \ubaa8\uc784&#8221;\uc774 \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c \uace0\uc720\ud074\ub798\uc2a4\uc784\uc744 \ubcf4\uc5ec\uc900\ub2e4. <a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\/\">ZFC \uacf5\ub9ac\uacc4<\/a>\ub294 \uc774\ub7ec\ud55c \uc5ed\uc124\uc744 \ud53c\ud558\ub3c4\ub85d \uc124\uacc4\ub418\uc5b4 \uc788\ub2e4.<\/p>\n<h3>10. \uc11c\uc218\uc758 \uc9d1\ud569\ub860\uc801 \uad6c\uc131<\/h3>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uc11c\uc218\ub97c \uacf5\ub9ac\uc801\uc73c\ub85c \ub3c4\uc785\ud588\uc9c0\ub9cc, \uc0ac\uc2e4 \uc11c\uc218\ub294 \uc9d1\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc9c1\uc811 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \uad6c\uc131\uc740 \ud3f0 \ub178\uc774\ub9cc(von Neumann)\uc774 \uc81c\uc548\ud55c \ubc29\ubc95\uc73c\ub85c, 6\uc7a5\uc5d0\uc11c \uc790\uc5f0\uc218\ub97c \uc9d1\ud569\uc73c\ub85c \uc815\uc758\ud55c \ubc29\ubc95\uc744 \ud655\uc7a5\ud55c \uac83\uc774\ub2e4.<\/p>\n<p>\ud3f0 \ub178\uc774\ub9cc\uc758 \ubc29\ubc95\uc5d0\uc11c \uc11c\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4.<\/p>\n<ul>\n<li>\uc11c\uc218 \\(0 = \\varnothing\\)<\/li>\n<li>\uc11c\uc218 \\(1 = \\{0\\} = \\{\\varnothing\\}\\)<\/li>\n<li>\uc11c\uc218 \\(2 = \\{0,\\, 1\\} = \\{\\varnothing,\\, \\{\\varnothing\\}\\}\\)<\/li>\n<li>\uc11c\uc218 \\(3 = \\{0,\\, 1,\\, 2\\}\\)<\/li>\n<li>\uc77c\ubc18\uc801\uc73c\ub85c, \uc11c\uc218 \\(\\alpha\\)\uc758 \ub530\ub984\uc218\ub294 \\(\\alpha \\cup \\{\\alpha\\}\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<li>\uadf9\ud55c\uc11c\uc218\ub294 \uadf8\ubcf4\ub2e4 \uc791\uc740 \ubaa8\ub4e0 \uc11c\uc218\uc758 \ud569\uc9d1\ud569\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \uc815\uc758\uc5d0\uc11c \uac01 \uc11c\uc218\ub294 \uc790\uae30\ubcf4\ub2e4 \uc791\uc740 \ubaa8\ub4e0 \uc11c\uc218\ub4e4\uc758 \uc9d1\ud569\uc774 \ub41c\ub2e4. \uc989 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\alpha = \\{\\beta \\mid \\beta < \\alpha\\}\\]\n\ud2b9\ud788 \uac00\uc7a5 \uc791\uc740 \ubb34\ud55c\uc11c\uc218 \\(\\omega\\)\ub294 \ubaa8\ub4e0 \uc720\ud55c\uc11c\uc218(\uc790\uc5f0\uc218)\uc758 \uc9d1\ud569\uc774\ub2e4.\n\\[\\omega = \\{0,\\, 1,\\, 2,\\, 3,\\, \\ldots\\} = \\mathbb{N}\\]<\/p>\n<p>\uc774\ub7ec\ud55c \uad6c\uc131\uc758 \ud2b9\uc9d5\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\uac01 \uc11c\uc218\uac00 \uadf8 \uc790\uc2e0\uc774 \uc815\ub82c\uc9d1\ud569\uc774 \ub41c\ub2e4. (\uc6d0\uc18c \uad00\uacc4 \\(\\in\\)\uc774 \uc21c\uc11c\uad00\uacc4\uac00 \ub41c\ub2e4.)<\/li>\n<li>\uc11c\uc218 \\(\\alpha\\)\ub294 \uc790\uae30 \uc790\uc2e0\uc758 \uc11c\uc218\ub97c \uac16\ub294 \uc815\ub82c\uc9d1\ud569\uc758 \ud45c\uc900\uc801\uc778 \ub300\ud45c\uc6d0\uc18c\uac00 \ub41c\ub2e4.<\/li>\n<li>\ub450 \uc11c\uc218 \\(\\alpha\\), \\(\\beta\\)\uc5d0 \ub300\ud558\uc5ec, \\(\\alpha < \\beta\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\alpha \\in \\beta\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\uc640 \uac19\uc740 \uad6c\uc131\uc744 \ud1b5\ud574 \uc11c\uc218 \uacf5\ub9ac (O1)\uacfc (O2)\uac00 \ub9cc\uc871\ub428\uc744 \uc54c \uc218 \uc788\ub2e4. \uc989, \uc11c\uc218\ub97c \ubcc4\ub3c4\uc758 \uacf5\ub9ac\ub85c \ub3c4\uc785\ud560 \ud544\uc694 \uc5c6\uc774 ZFC \uacf5\ub9ac\uacc4 \ub0b4\uc5d0\uc11c \uc9c1\uc811 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \ub2e4\ub9cc \uc774\uc640 \uac19\uc740 \uad6c\uc131 \ubc29\ubc95\uc740 \uc9d1\ud569\ub860\uc758 \uacf5\ub9ac\ub4e4, \ud2b9\ud788 \ubb34\ud55c\uacf5\ub9ac\uc640 \uce58\ud658\uacf5\ub9ac\ub97c \ud544\uc694\ub85c \ud55c\ub2e4.<\/p>\n<p>\uc774\ub807\uac8c \uad6c\uc131\ub41c \uc11c\uc218 \uc804\uccb4\uc758 \ubaa8\uc784 \\(ORD\\)\ub294 \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c \uace0\uc720\ud074\ub798\uc2a4\uac00 \ub41c\ub2e4. \ub9cc\uc57d \uc774\uac83\uc774 \uc9d1\ud569\uc774\ub77c\uba74 \ubd80\ub784\ub9ac-\ud3ec\ub974\ud2f0\uc758 \uc5ed\uc124\uc774 \ubc1c\uc0dd\ud558\uae30 \ub54c\ubb38\uc774\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>\uae30\uc218\uac00 \uc9d1\ud569\uc758 &#8216;\ud06c\uae30&#8217;\ub97c \ub098\ud0c0\ub0b8\ub2e4\uba74, \uc11c\uc218(ordinal number)\ub294 \uc9d1\ud569\uc758 \uc6d0\uc18c\ub4e4\uc758 &#8216;\uc21c\uc11c&#8217;\uae4c\uc9c0 \uace0\ub824\ud55c \uac1c\ub150\uc774\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc815\ub82c\uc9d1\ud569\uacfc \uc11c\uc218\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uace0, \uc11c\uc218\uc758 \uc5f0\uc0b0\uacfc \ucd08\ud55c\uadc0\ub0a9\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. 1. \uc815\ub82c\uc21c\uc11c\uc640 \uc815\ub82c\uc9d1\ud569 \uc9d1\ud569 \\(A\\) \uc704\uc758 \uc21c\uc11c\uad00\uacc4 \\(\\leq\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\ud560 \ub54c \uc774 \uc21c\uc11c\uad00\uacc4\ub97c \uc815\ub82c\uc21c\uc11c(well-ordering)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\le\\)\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. \\(A\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc774 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4. \uc815\ub82c\uc21c\uc11c\ub97c \uac16\ub294 \uc9d1\ud569\uc744 \uc815\ub82c\uc9d1\ud569(well-ordered set)\uc774\ub77c\uace0&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":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9261\/revisions"}],"predecessor-version":[{"id":9438,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9261\/revisions\/9438"}],"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}]}}