{"id":9253,"date":"2025-10-17T19:57:43","date_gmt":"2025-10-17T10:57:43","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9253"},"modified":"2025-10-20T18:48:37","modified_gmt":"2025-10-20T09:48:37","slug":"ch05-infinite-sets","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\/","title":{"rendered":"\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>5. \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/h2>\n\n --><\/p>\n<p>\uc9d1\ud569\uc740 \uc6d0\uc18c\uc758 \uac1c\uc218\uc5d0 \ub530\ub77c \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569\uc73c\ub85c \ub098\ub25c\ub2e4. \ubb34\ud55c\uc9d1\ud569\uc740 \ub2e4\uc2dc \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uacfc \ube44\uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc73c\ub85c \ubd84\ub958\ub41c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \ubb34\ud55c\uc758 \uac1c\ub150\uc744 \uc5c4\ubc00\ud558\uac8c \ub2e4\ub8e8\uace0, \uc11c\ub85c \ub2e4\ub978 \ud06c\uae30\uc758 \ubb34\ud55c\uc774 \uc874\uc7ac\ud568\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h3>1. \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569\uc758 \uc815\uc758<\/h3>\n<p>\uc9d1\ud569 \\(A\\)\uac00 <span class=\"defined\">\uc720\ud55c\uc9d1\ud569<\/span>(finite set)\uc774\ub77c\ub294 \uac83\uc740 \\(A\\)\uac00 \uacf5\uc9d1\ud569\uc774\uac70\ub098, \ub610\ub294 \uc5b4\ub5a4 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\)\uc758 \uc6d0\uc18c\ub97c \\(a_1,\\, a_2,\\, \\ldots,\\, a_k\\)\uc640 \uac19\uc774 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4. \uc774\ub54c \\(k\\)\ub97c \uc9d1\ud569 \\(A\\)\uc758 <span class=\"defined\">\uc6d0\uc18c\uc758 \uac1c\uc218<\/span> \ub610\ub294 <span class=\"defined\">\ud06c\uae30<\/span>\ub77c\uace0 \ud558\uace0 \\(|A| = k\\) \ub610\ub294 \\(n(A) = k\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub2e8, \uacf5\uc9d1\ud569\uc758 \uacbd\uc6b0 \\(|\\varnothing | = 0\\)\uc778 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n<p>\uc9d1\ud569 \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774 \uc544\ub2d0 \ub54c \\(A\\)\ub97c <span class=\"defined\">\ubb34\ud55c\uc9d1\ud569<\/span>(infinite set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ubb34\ud55c\uc9d1\ud569\uc758 \uc6d0\uc18c\ub294 \ub05d\uc5c6\uc774 \ub9ce\uc73c\ubbc0\ub85c \ubaa8\ub450 \ub098\uc5f4\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n<p>\ubb34\ud55c\uc9d1\ud569\uc758 \ud2b9\uc9d5 \uc911 \ud558\ub098\ub294 \uc790\uae30 \uc790\uc2e0\uacfc \uc77c\ub300\uc77c\ub85c \ub300\uc751\ud558\ub294 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4\ub294 \uac83\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\mathbb{N} = \\{0,\\, 1,\\, 2,\\, 3,\\, \\ldots\\}\\)\uacfc \uadf8 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc778 \uc9dd\uc218 \uc9d1\ud569 \\(\\{0,\\, 2,\\, 4,\\, 6,\\, \\ldots\\}\\) \uc0ac\uc774\uc5d0\ub294 \\(n \\mapsto 2n\\)\uc774\ub77c\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud55c\ub2e4. [\\(n\\mapsto 2n\\)\uc740 \\(n\\)\uc744 \\(2n\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub77c\ub294 \ub73b\uc774\ub2e4.]<\/p>\n<h3>2. \uc77c\ub300\uc77c \ub300\uc751\uacfc \uc9d1\ud569\uc758 \ud06c\uae30<\/h3>\n<p>\ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\) \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751 \\(f: A \\to B\\)\uac00 \uc874\uc7ac\ud560 \ub54c, \\(A\\)\uc640 \\(B\\)\ub294 <span class=\"defined\">\ub300\ub4f1<\/span>(equipotent)\ud558\ub2e4\uace0 \ud558\uba70 \\(A \\sim B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub300\ub4f1 \uad00\uacc4\ub294 \ub2e4\uc74c \uc131\uc9c8\uc744 \ub9cc\uc871\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ubc18\uc0ac\uc801 \uc131\uc9c8: \\(A \\sim A\\)<\/li>\n<li>\ub300\uce6d\uc801 \uc131\uc9c8: \\(A \\sim B\\)\uc774\uba74 \\(B \\sim A\\)\uc774\ub2e4.<\/li>\n<li>\ucd94\uc774\uc801 \uc131\uc9c8: \\(A \\sim B\\)\uc774\uace0 \\(B \\sim C\\)\uc774\uba74 \\(A \\sim C\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc720\ud55c\uc9d1\ud569\uc758 \uacbd\uc6b0, \ub450 \uc9d1\ud569\uc774 \ub300\ub4f1\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc6d0\uc18c\uc758 \uac1c\uc218\uac00 \uac19\uc740 \uac83\uc774\ub2e4. \ubb34\ud55c\uc9d1\ud569\uc758 \uacbd\uc6b0\uc5d0\ub294 \ub300\ub4f1 \uad00\uacc4\ub97c \ud1b5\ud574 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.1.<\/span><br \/>\n<span class=\"defined\">\uc288\ub8b0\ub354-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac<\/span>(Schr\u00f6der\u2013Bernstein theorem)\ub97c \uc870\uc0ac\ud574 \ubcf4\uc790.<\/p>\n<\/div>\n<h3>3. \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569<\/h3>\n<p>\uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\mathbb{N}\\)\uacfc \ub300\ub4f1\ud55c \ubb34\ud55c\uc9d1\ud569\uc744 <span class=\"defined\">\uac00\uc0b0\ubb34\ud55c\uc9d1\ud569<\/span>(countably infinite set) \ub610\ub294 <span class=\"defined\">\uac00\ubd80\ubc88\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc720\ud55c\uc9d1\ud569\uacfc \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc744 \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\uac00\uc0b0\uc9d1\ud569<\/span>(countable set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc758 \uc6d0\uc18c\ub294 \\(a_0,\\, a_1,\\, a_2,\\, a_3,\\, \\ldots\\)\uc640 \uac19\uc774 \uc790\uc5f0\uc218\ub85c \ubc88\ud638\ub97c \ub9e4\uaca8 \ube60\uc9d0 \uc5c6\uc774 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774\uc81c \uc815\uc218 \uc9d1\ud569 \\(\\mathbb{Z}\\)\uc640 \uc720\ub9ac\uc218 \uc9d1\ud569 \\(\\mathbb{Q}\\)\uac00 \uac00\uc0b0\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc790.<\/p>\n<p>\uc6b0\uc120 \uc815\uc218 \uc9d1\ud569 \\(\\mathbb{Z}\\)\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ub2e4\uc74c\uacfc \uac19\uc740 \ub300\uc751\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[f: \\mathbb{N} \\to \\mathbb{Z}, \\quad f(n) = \\begin{cases}<br \/>\n\\frac{n}{2} &#038; (n\\text{\uc774 \uc9dd\uc218\uc77c \ub54c}) \\\\[6pt]<br \/>\n-\\frac{n+1}{2} &#038; (n\\text{\uc774 \ud640\uc218\uc77c \ub54c})<br \/>\n\\end{cases}\\]<br \/>\n\uc774 \ud568\uc218\ub294 \\(\\mathbb{N}\\)\uc758 \uc6d0\uc18c\ub97c \\(0,\\, -1,\\, 1,\\, -2,\\, 2,\\, -3,\\, 3,\\, \\ldots\\) \uc21c\uc11c\ub85c \\(\\mathbb{Z}\\)\uc758 \uc6d0\uc18c\uc640 \ub300\uc751\uc2dc\ud0a4\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\mathbb{Z}\\)\ub294 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \uc720\ub9ac\uc218 \uc9d1\ud569 \\(\\mathbb{Q}\\)\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ubaa8\ub4e0 \uc720\ub9ac\uc218\ub97c \ub450 \uc815\uc218\uc758 \ube44 \\(\\frac{p}{q}\\) \ud615\ud0dc\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4\ub294 \uc131\uc9c8\uc744 \ubc14\ud0d5\uc73c\ub85c \uc720\ub9ac\uc218 \uc9d1\ud569\uc774 \uac00\uc0b0\uc784\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. (\ub2e8, \\(p \\in \\mathbb{Z}\\), \\(q \\in \\mathbb{N} \\setminus \\{0\\}\\), \\(\\gcd(p,\\,q) = 1\\).)<\/p>\n<p>\uc720\ub9ac\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\uc5f4\ud558\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\n&#038;\\frac{0}{1}, \\\\[6pt]<br \/>\n&#038;\\frac{-1}{1},\\,\\, \\frac{1}{1}, \\\\[6pt]<br \/>\n&#038;\\frac{-1}{2},\\,\\, \\frac{1}{2},\\,\\, \\frac{-2}{1},\\,\\, \\frac{2}{1}, \\\\[6pt]<br \/>\n&#038;\\frac{-1}{3},\\,\\, \\frac{1}{3},\\,\\, \\frac{-3}{1},\\,\\, \\frac{3}{1}, \\\\[6pt]<br \/>\n&#038;\\frac{-1}{4},\\,\\, \\frac{1}{4},\\,\\, \\frac{-2}{3},\\,\\, \\frac{2}{3},\\,\\, \\frac{-3}{2},\\,\\, \\frac{3}{2},\\,\\, \\frac{-4}{1},\\,\\, \\frac{4}{1}, \\\\[6pt]<br \/>\n&#038;\\frac{-1}{5},\\,\\, \\frac{1}{5},\\,\\, \\frac{-5}{1},\\,\\, \\frac{5}{1}, \\\\[6pt]<br \/>\n&#038; \\,\\, \\vdots<br \/>\n\\end{aligned}\\]<br \/>\n\uc989 \\(k\\)\ubc88\uc9f8 \uc904\uc5d0 \\(|p| + q = k\\)\uc778 \uae30\uc57d\ubd84\uc218\ub4e4\uc744 \ub098\uc5f4\ud55c \uac83\uc774\ub2e4. \uac01 \uc904\uc740 \uc720\ud55c\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c0\ubbc0\ub85c, \uc774 \ubc29\ubc95\uc73c\ub85c \ubaa8\ub4e0 \uc720\ub9ac\uc218\ub97c \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4. \ubaa8\ub4e0 \uc720\ub9ac\uc218\ub97c \uc790\uc5f0\uc218\ucc98\ub7fc \ub098\uc5f4\ud558\uc600\uc73c\ubbc0\ub85c, \uc720\ub9ac\uc218 \uc9d1\ud569 \\(\\mathbb{Q}\\)\ub294 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.2.<\/span><br \/>\n\ub2e4\uc74c \uc9d1\ud569\uc774 \uac00\uc0b0\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ubaa8\ub4e0 \uc9dd\uc218\uc758 \uc9d1\ud569<\/li>\n<li>\ubaa8\ub4e0 \ud640\uc218 \uc815\uc218\uc758 \uc9d1\ud569<\/li>\n<li>\\(\\mathbb{N} \\times \\mathbb{N}\\)<\/li>\n<li>\uc720\ud55c \uac1c\uc758 \uac00\uc0b0\uc9d1\ud569\uc758 \ud569\uc9d1\ud569<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.3.<\/span><br \/>\n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\uba74, \\(A\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \uc911\uc5d0\uc11c \\(\\mathbb{N}\\)\uacfc \ub300\ub4f1\ud55c \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\\(B\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\uba74, \\(B\\)\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569 \uc911\uc5d0\uc11c \\(B\\)\uc640 \ub300\ub4f1\ud55c \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\\(C\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uba74, \\(C\\)\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569 \uc911\uc5d0\uc11c \\(C\\)\uc640 \ub300\ub4f1\ud55c \uac83\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<h3>4. \ube44\uac00\uc0b0\uc9d1\ud569<\/h3>\n<p>\uc790\uc5f0\uc218 \uc9d1\ud569\uacfc \ub300\ub4f1\ud558\uc9c0 \uc54a\uc740 \ubb34\ud55c\uc9d1\ud569\uc744 <span class=\"defined\">\ube44\uac00\uc0b0\uc9d1\ud569<\/span>(uncountable set) \ub610\ub294 <span class=\"defined\">\ube44\uac00\ubd80\ubc88 \ubb34\ud55c\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc2e4\uc218 \uc9d1\ud569\uc774 \ube44\uac00\uc0b0\uc9d1\ud569\uc784\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc5f4\ub9b0\uad6c\uac04 \\((0,\\,1)\\)\uc774 \ube44\uac00\uc0b0\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uba74 \ucda9\ubd84\ud558\ub2e4.<\/p>\n<p>\uadc0\ub958\ubc95\uc744 \uc0ac\uc6a9\ud558\uc790. \uc5f4\ub9b0\uad6c\uac04 \\((0,\\,1)\\)\uc774 \uac00\uc0b0\ubb34\ud55c\uc774\ub77c\uace0 \uac00\uc815\ud558\uba74, \uc774 \uad6c\uac04\uc758 \ubaa8\ub4e0 \uc2e4\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc790\uc5f0\uc218\ucc98\ub7fc \ud55c \uc904\ub85c, \ube60\uc9d0\uc5c6\uc774 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[r_1,\\, r_2,\\, r_3,\\, r_4,\\, \\ldots .\\]<br \/>\n\uac01 \uc2e4\uc218\ub97c \uc2ed\uc9c4\ubc95 \uc18c\uc218 \uc804\uac1c\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b4\uc5c8\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\nr_1 &#038;= 0.d_{11}\\,d_{12}\\,d_{13}\\,d_{14}\\cdots \\\\[6pt]<br \/>\nr_2 &#038;= 0.d_{21}\\,d_{22}\\,d_{23}\\,d_{24}\\cdots \\\\[6pt]<br \/>\nr_3 &#038;= 0.d_{31}\\,d_{32}\\,d_{33}\\,d_{34}\\cdots \\\\[6pt]<br \/>\nr_4 &#038;= 0.d_{41}\\,d_{42}\\,d_{43}\\,d_{44}\\cdots \\\\[6pt]<br \/>\n&#038;\\vdots<br \/>\n\\end{aligned}\\]<br \/>\n\uc989, \\(d_{ij}\\)\ub294 \uc2e4\uc218 \\(r_i\\)\uc758 \uc18c\uc218\uc810 \uc544\ub798 \\(j\\)\uc9f8 \uc790\ub9ac\uc758 \uc22b\uc790\uc774\ub2e4. [\ub9cc\uc57d \\(r_i\\)\uac00 \uc720\ud55c\uc18c\uc218\ub85c \ud45c\ud604\ub41c\ub2e4\uba74, \ub9c8\uc9c0\ub9c9 \uc790\ub9ac\uc758 \uc22b\uc790\uc5d0\uc11c \\(1\\)\uc744 \ube7c\uace0 \uadf8 \ub4a4\uc5d0 \\(9\\)\ub97c \uc5f0\ub2ec\uc544 \ubd99\uc784\uc73c\ub85c\uc368 \\(d_{ij}\\)\ub97c \uc720\uc77c\ud558\uac8c \uc815\uc758\ud55c\ub2e4.]<\/p>\n<p>\uc774\uc81c \ub2e4\uc74c\uacfc \uac19\uc740 \uc2e4\uc218 \\(s\\)\ub97c \uad6c\uc131\ud55c\ub2e4.<br \/>\n\\[s = 0.s_1\\,s_2\\,s_3\\,s_4\\cdots\\]<br \/>\n\uc5ec\uae30\uc11c \\(s_i\\)\ub294 \\(d_{ii} \\neq 5\\)\uc774\uba74 \\(s_i = 5\\), \\(d_{ii} = 5\\)\uc774\uba74 \\(s_i = 6\\)\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\uadf8\ub7ec\uba74 \\(s \\in (0,\\, 1)\\)\uc774\uc9c0\ub9cc, \ubaa8\ub4e0 \\(i\\)\uc5d0 \ub300\ud574 \\(s \\neq r_i\\)\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc18c\uc218\uc810 \uc544\ub798 \\(i\\)\uc9f8 \uc790\ub9ac \uc22b\uc790\uac00 \ub2e4\ub974\uae30 \ub54c\ubb38\uc774\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc5f4\ub9b0\uad6c\uac04 \\((0,\\,1)\\)\uc740 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.[\uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(\\mathbb{R}\\)\uc774 \ube44\uac00\uc0b0\uc784\uc744 \ubcf4\uc774\ub294 \ubc29\ubc95\uc744 &#8216;\uce78\ud1a0\uc5b4\uc758 \ub300\uac01\uc120 \ub17c\ubc95&#8217;\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.]<\/p>\n<p>\ud55c\ud3b8 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\ub294 \\((0,\\, 1)\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}\\) \uc704\ub85c\uc758 \uc77c\ub300\uc77c\ub300\uc751\uc774\ub2e4.<br \/>\n\\[f(x) = \\tan\\left(\\pi x &#8211; \\frac{\\pi}{2}\\right)\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(\\mathbb{R}\\)\ub3c4 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.4.<\/span><br \/>\n\uc9d1\ud569 \\(A\\)\uac00 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\uace0 \uc9d1\ud569 \\(B\\)\uac00 \\(A\\)\uc640 \ub300\ub4f1\ud560 \ub54c, \uc9d1\ud569 \\(B\\)\ub3c4 \ube44\uac00\uc0b0\uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ub354 \uc0b4\ud3b4\ubcf4\uc790. \ub180\ub78d\uac8c\ub3c4 \ub2e4\uc74c \uc9d1\ud569\ub4e4\uc740 \ubaa8\ub450 \ub300\ub4f1\ud558\ub2e4.<\/p>\n<ul>\n<li>\uad6c\uac04 \\([0,\\, 1]\\)<\/li>\n<li>\uc2e4\uc218 \uc9c1\uc120 \\(\\mathbb{R}\\)<\/li>\n<li>\ud3c9\uba74 \\(\\mathbb{R}^2\\)<\/li>\n<li>3\ucc28\uc6d0 \uacf5\uac04\uc5d0 \ub193\uc778 \ubc18\uc9c1\uc120<\/li>\n<li>\\(n\\)\ucc28\uc6d0 \uacf5\uac04 \\(\\mathbb{R}^n\\) (\ub2e8, \\(n\\ge 1\\))<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.5.<\/span><br \/>\n\ub3c4\ud615\uc744 \uc810\uc758 \uc9d1\ud569\uc73c\ub85c \uac04\uc8fc\ud558\uc600\uc744 \ub54c, \uae38\uc774\uac00 \ub2e4\ub978 \ub450 \uc120\ubd84\uc774 \uc11c\ub85c \ub300\ub4f1\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \ub610\ud55c \uae38\uc774\uac00 \uc720\ud55c\uc778 \uc120\ubd84\uacfc \uae38\uc774\uac00 \ubb34\ud55c\uc778 \uc9c1\uc120\uc774 \ub300\ub4f1\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uad6c\uac04 \\([0,\\, 1]\\)\uacfc \\([0,\\, 1]^2\\) \uc0ac\uc774\uc758 \uc77c\ub300\uc77c\ub300\uc751\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \uc810 \\((x,\\, y) \\in [0,\\, 1]^2\\)\uc5d0 \ub300\ud574 \\(x\\)\uc640 \\(y\\)\ub97c \uc18c\uc218 \uc804\uac1c\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\nx &#038;= 0.x_1\\,x_2\\,x_3\\cdots \\\\[6pt]<br \/>\ny &#038;= 0.y_1\\,y_2\\,y_3\\cdots<br \/>\n\\end{aligned}\\]<br \/>\n\ub77c\uace0 \ud558\uba74, \uc18c\uc218\uc810 \uc544\ub798 \uc22b\uc790\ub97c<br \/>\n\\[z = 0.x_1\\,y_1\\,x_2\\,y_2\\,x_3\\,y_3\\cdots\\]<br \/>\n\uc640 \uac19\uc774 \uc870\ud569\ud558\uc5ec \\([0,\\, 1]\\)\uc758 \uc6d0\uc18c\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. [\uc2e4\uc81c \uc99d\uba85\uc5d0\uc11c\ub294 \uc18c\uc218 \uc804\uac1c\uc758 \uc720\uc77c\uc131 \ubb38\uc81c\ub97c \ucc98\ub9ac\ud574\uc57c \ud55c\ub2e4.]<\/p>\n<p>\uc774\uc640 \uac19\uc740 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uba74, \uc790\uc5f0\uc2a4\ub7fd\uac8c \ubcf5\uc18c\uc218 \uc9d1\ud569 \\(\\mathbb{C}\\)\uac00 \uc2e4\uc218 \uc9d1\ud569 \\(\\mathbb{R}\\)\uacfc \ub300\ub4f1\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \uc989 \\(\\mathbb{C} = \\{a + bi \\mid a,\\,b \\in \\mathbb{R}\\}\\)\uc774\ubbc0\ub85c \\(\\mathbb{C}\\)\uc640 \\(\\mathbb{R}^2\\) \uc0ac\uc774\uc5d0 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc77c\ub300\uc77c \ub300\uc751 \\((a,\\, b) \\mapsto a + bi\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(\\mathbb{R}^2 \\sim \\mathbb{R}\\)\uc774\ubbc0\ub85c, \\(\\mathbb{C} \\sim \\mathbb{R}\\)\uc774\ub2e4. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uba74, \uc784\uc758\uc758 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\mathbb{R} \\sim \\mathbb{R}^n\\)\uc784\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n<p>\ud39c\uc73c\ub85c \uc885\uc774\uc5d0 \uc120\uc744 \ud558\ub098 \uadf8\uc73c\uba74, \uadf8 \uc120 \uc704\uc5d0 \uc788\ub294 \uc810\uc758 \uac1c\uc218\uc640 \uc6b0\ub9ac\uc758 \ubab8\uc744 \uc774\ub8e8\uace0 \uc788\ub294 \uc810\uc758 \uac1c\uc218\ub294 \uac19\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.6.<\/span><br \/>\n\ub370\ub370\ud0a8\ud2b8\uac00 \ubb34\ud55c\uc9d1\ud569\uc744 \uc815\uc758\ud55c \ubc29\ubc95\uc744 \uc870\uc0ac\ud558\uace0, \ub370\ub370\ud0a8\ud2b8\uac00 \uc815\uc758\ud55c \ubb34\ud55c\uc9d1\ud569\uc774 \uc774 \uc7a5\uc5d0\uc11c \uc815\uc758\ud55c \ubb34\ud55c\uc9d1\ud569\uc758 \uc815\uc758\uc640 \ub3d9\uce58\uc784\uc744 \ubc1d\ud788\uc2dc\uc624. [\uc5c4\ubc00\ud788 \ub9d0\ud558\uba74, ZF \uacf5\ub9ac\uacc4\uc5d0\uc11c\ub294 \ub3d9\uce58\uac00 \uc544\ub2c8\uba70, ZFC \uacf5\ub9ac\uacc4\uc5d0\uc11c\ub294 \ub3d9\uce58\uc774\ub2e4.]<\/p>\n<\/div>\n<h3>5. \uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac<\/h3>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ube44\uac00\uc0b0\uc9d1\ud569\uc740 \ubaa8\ub450 \\(\\mathbb{R}\\)\uacfc \ub300\ub4f1\ud55c \uac83\uc774\uc5c8\ub2e4. \uadf8\ub807\ub2e4\uba74 \ubaa8\ub4e0 \ube44\uac00\uc0b0\uc9d1\ud569\uc774 \\(\\mathbb{R}\\)\uacfc \ub300\ub4f1\ud560\uae4c? \uadf8\ub807\uc9c0 \uc54a\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.1. (\uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\uc784\uc758\uc758 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec, \\(A\\)\uc640 \uadf8 \uba71\uc9d1\ud569 \\(\\mathcal{P}(A)\\) \uc0ac\uc774\uc5d0\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc989, \\(|\\mathcal{P}(A)| > |A|\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(f: A \\to \\mathcal{P}(A)\\)\uac00 \uc784\uc758\uc758 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \uc9d1\ud569\uc744 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[B = \\{x \\in A \\mid x \\notin f(x)\\}\\]<br \/>\n\uadf8\ub7ec\uba74 \\(B \\in \\mathcal{P}(A)\\)\uc774\uc9c0\ub9cc, \ubaa8\ub4e0 \\(a \\in A\\)\uc5d0 \ub300\ud574 \\(B \\neq f(a)\\)\uc774\ub2e4. \uadf8 \uc774\uc720\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\ub9cc\uc57d \\(a \\in B\\)\uc774\uba74 \uc815\uc758\uc5d0 \uc758\ud574 \\(a \\notin f(a)\\)\uc774\ubbc0\ub85c \\(B \\neq f(a)\\)\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(a \\notin B\\)\uc774\uba74 \uc815\uc758\uc5d0 \uc758\ud574 \\(a \\in f(a)\\)\uc774\ubbc0\ub85c \\(B \\neq f(a)\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ub530\ub77c\uc11c \\(f\\)\ub294 \uc704\ub85c\uc758 \ud568\uc218\uac00 \uc544\ub2c8\uace0, \uc77c\ub300\uc77c\ub300\uc751\ub3c4 \uc544\ub2c8\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ubb34\ud55c\uc9d1\ud569 \ub610\ud55c \ub2e4\uc591\ud55c \uc885\ub958\uac00 \uc788\uc74c\uc744 \ubcf4\uc5ec\uc900\ub2e4.<br \/>\n\\[|\\mathbb{N}| < |\\mathcal{P}(\\mathbb{N})| < |\\mathcal{P}(\\mathcal{P}(\\mathbb{N}))| < \\cdots\\]<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.7.<\/span><br \/>\n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ubb34\ud55c\uc9d1\ud569 \\(A\\)\uc640 \uc720\ud55c\uc9d1\ud569 \\(B\\)\uc5d0 \ub300\ud558\uc5ec \\(A \\cup B \\sim A\\)\uc774\ub2e4.<\/li>\n<li>\uad6c\uac04 \\((0,\\, 1)\\)\uacfc \\((0,\\ \\infty)\\)\ub294 \ub300\ub4f1\ud558\ub2e4.<\/li>\n<li>\ubb34\ub9ac\uc218 \uc9d1\ud569\uc740 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\ub300\uc218\uc801 \uc218\uc758 \uc9d1\ud569\uc740 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.8.<\/span><br \/>\n\uc2e4\uc218 \uc9d1\ud569 \\(\\mathbb{R}\\)\uacfc \uc790\uc5f0\uc218 \uc9d1\ud569\uc758 \uba71\uc9d1\ud569 \\(\\mathcal{P}(\\mathbb{N})\\)\uc774 \ub300\ub4f1\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h3>6. \uc5f0\uc18d\uccb4 \uac00\uc124<\/h3>\n<p>\uce78\ud1a0\uc5b4\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc9c8\ubb38\uc744 \uc81c\uae30\ud588\ub2e4.<\/p>\n<p style=\"text-align: center;\">&#8220;\\(|\\mathbb{N}| < |A| < |\\mathbb{R}|\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc9d1\ud569 \\(A\\)\uac00 \uc874\uc7ac\ud558\ub294\uac00?\"<\/p>\n<p>\uc774 \uc9c8\ubb38\uc5d0 \ub300\ud55c \ubd80\uc815\uc801 \ub2f5\ubcc0, \uc989 &#8220;\uadf8\ub7ec\ud55c \uc9d1\ud569\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4&#8221;\ub77c\ub294 \uc8fc\uc7a5\uc744 <span class=\"defined\">\uc5f0\uc18d\uccb4 \uac00\uc124<\/span>(Continuum Hypothesis, CH)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uad34\ub378\uacfc \ucf54\uc5b8\uc758 \uc5f0\uad6c\uc5d0 \uc758\ud574, \uc5f0\uc18d\uccb4 \uac00\uc124\uc740 ZFC \uacf5\ub9ac\uacc4\uc5d0\uc11c \uc99d\uba85\ud560 \uc218\ub3c4 \uc5c6\uace0 \ubc18\uc99d\ud560 \uc218\ub3c4 \uc5c6\ub2e4\ub294 \uac83\uc774 \ubc1d\ud600\uc84c\ub2e4. \uc989, \uc5f0\uc18d\uccb4 \uac00\uc124\uc740 ZFC\uc640 \ub3c5\ub9bd\uc801\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>\uc9d1\ud569\uc740 \uc6d0\uc18c\uc758 \uac1c\uc218\uc5d0 \ub530\ub77c \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569\uc73c\ub85c \ub098\ub25c\ub2e4. \ubb34\ud55c\uc9d1\ud569\uc740 \ub2e4\uc2dc \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uacfc \ube44\uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc73c\ub85c \ubd84\ub958\ub41c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \ubb34\ud55c\uc758 \uac1c\ub150\uc744 \uc5c4\ubc00\ud558\uac8c \ub2e4\ub8e8\uace0, \uc11c\ub85c \ub2e4\ub978 \ud06c\uae30\uc758 \ubb34\ud55c\uc774 \uc874\uc7ac\ud568\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. 1. \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569\uc758 \uc815\uc758 \uc9d1\ud569 \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569(finite set)\uc774\ub77c\ub294 \uac83\uc740 \\(A\\)\uac00 \uacf5\uc9d1\ud569\uc774\uac70\ub098, \ub610\ub294 \uc5b4\ub5a4 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\)\uc758 \uc6d0\uc18c\ub97c \\(a_1,\\, a_2,\\, \\ldots,\\, a_k\\)\uc640 \uac19\uc774 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4. \uc774\ub54c \\(k\\)\ub97c \uc9d1\ud569 \\(A\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218 \ub610\ub294 \ud06c\uae30\ub77c\uace0 \ud558\uace0 \\(|A| = k\\) \ub610\ub294&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":105,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9253","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9253","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=9253"}],"version-history":[{"count":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9253\/revisions"}],"predecessor-version":[{"id":9435,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9253\/revisions\/9435"}],"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=9253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}