{"id":9258,"date":"2025-10-17T20:02:36","date_gmt":"2025-10-17T11:02:36","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9258"},"modified":"2026-09-27T11:56:08","modified_gmt":"2026-09-27T02:56:08","slug":"ch07-cardinal-numbers","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\/","title":{"rendered":"\uc9d1\ud569\uc758 \uae30\uc218"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>7. \uc9d1\ud569\uc758 \uae30\uc218<\/h2>\n\n --><\/p>\n<p><a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">5\uc7a5<\/a>\uc5d0\uc11c \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \uc77c\ub300\uc77c\ub300\uc751\uc744 \ud1b5\ud574 \ube44\uad50\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\uc218(cardinal number)\ub97c \ub3c4\uc785\ud558\uace0, \uae30\uc218\uc758 \ub300\uc18c\uad00\uacc4\uc640 \uc5f0\uc0b0\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uae30\uc218\ub294 \uc720\ud55c\uc9d1\ud569\uc758 \uc6d0\uc18c \uc218\ub97c \ubb34\ud55c\uc9d1\ud569\uae4c\uc9c0 \ud655\uc7a5\ud55c \uac1c\ub150\uc774\ub2e4.<\/p>\n<h3>1. \uc9d1\ud569\uc758 \ub300\ub4f1\uacfc \uae30\uc218<\/h3>\n<p><a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">5\uc7a5<\/a>\uc5d0\uc11c \uc0ac\uc6a9\ud55c \ud45c\uae30\ub97c \ub2e4\uc2dc \uc4f0\uba74, \ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\) \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud560 \ub54c<br \/>\n\\[A\\sim B\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub300\ub4f1\uc740 \ub3d9\uce58\uad00\uacc4\uc774\uba70, \ubb38\uc81c 4.16\uc5d0\uc11c \uc774\ub97c \ud655\uc778\ud558\uc600\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 7.1. (\uae30\uc218)<\/span><\/p>\n<p>\uc9d1\ud569 \\(A\\)\uc758 <span class=\"defined\">\uae30\uc218<\/span>(cardinal number) \ub610\ub294 <span class=\"defined\">\ub18d\ub3c4<\/span>(cardinality)\ub97c \\(|A|\\) \ub610\ub294 \\(\\operatorname{card}(A)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uae30\uc218\ub294 \uc9d1\ud569\uc758 \ub300\ub4f1\ub958\ub97c \ub098\ud0c0\ub0b4\ub294 \ucd94\uc0c1\uc801\uc778 \ud06c\uae30 \ud45c\uc9c0\uc774\uba70,<br \/>\n\\[|A|=|B|\\quad\\Longleftrightarrow\\quad A\\sim B\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\ub3c4\ub85d \uc774\ud574\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc5ec\uae30\uc11c \u201c\ub300\ub4f1\ub958\u201d\ub77c\ub294 \ub9d0\uc740 \ubaa8\ub4e0 \ub300\ub4f1\ud55c \uc9d1\ud569\ub4e4\uc744 \ud558\ub098\uc758 \uc9d1\ud569\uc73c\ub85c \ubaa8\uc740\ub2e4\ub294 \ub73b\uc774 \uc544\ub2c8\ub2e4. \ubaa8\ub4e0 \uc9d1\ud569\uc744 \ub300\uc0c1\uc73c\ub85c \ud558\ub294 \ub300\ub4f1\ub958\ub294 \uc77c\ubc18\uc801\uc73c\ub85c \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c \uace0\uc720\ub958\uac00 \ub418\ubbc0\ub85c, \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc704\uc758 \ub3d9\uce58\ub9cc\uc744 \uae30\uc218\uc5d0 \uad00\ud55c \uae30\ubcf8 \uc6d0\ub9ac\ub85c \uc0ac\uc6a9\ud55c\ub2e4. \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \uae30\uc218\ub97c \uc2e4\uc81c \uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ud558\ub294 \ubc29\ubc95\uc744 \ubcc4\ub3c4\ub85c \ub9c8\ub828\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc720\ud55c\uc9d1\ud569\uc758 \uacbd\uc6b0 \uae30\uc218\ub294 \uc790\uc5f0\uc218\uc640 \uc77c\uce58\ud55c\ub2e4. \uc608\ub97c \ub4e4\uba74<br \/>\n\\[|\\varnothing|=0,\\quad |\\{a\\}|=1,\\quad |\\{a,b\\}|=2\\quad(a\\ne b)\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc77c \ub54c\ub294 \\(|A|=n(A)\\)\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \ubb34\ud55c\uc9d1\ud569\uc758 \uae30\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<ul>\n<li>\\(|\\mathbb N|=\\aleph_0\\)\uc774\ub2e4. \\(\\aleph_0\\)\ub294 \u2018\uc54c\ub808\ud504 \uc601\u2019 \ub610\ub294 \u2018\uc54c\ub808\ud504 \ub110\u2019\uc774\ub77c\uace0 \uc77d\ub294\ub2e4.<\/li>\n<li>\\(|\\mathbb R|=\\mathfrak c\\)\uc774\ub2e4. \ubb38\uc81c 5.14\uc5d0 \uc758\ud574<br \/>\n\\[\\mathfrak c=|\\mathcal P(\\mathbb N)|=2^{\\aleph_0}\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uba74 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">5\uc7a5<\/a>\uc5d0\uc11c \ub2e4\ub8ec \uc5f0\uc18d\uccb4 \uac00\uc124\uc740, \uc11c\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\aleph_1\\)\uc744 \uc815\uc758\ud55c \ub4a4\uc5d0\ub294 \\(2^{\\aleph_0}=\\aleph_1\\)\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<p>\ud55c\ud3b8 \uc5f0\uc18d\uccb4 \uac00\uc124\uc744 \uc77c\ubc18\ud654\ud558\uc5ec, \ubaa8\ub4e0 \uc54c\ub808\ud504 \uae30\uc218\uc5d0 \ub300\ud558\uc5ec \uba71\uc9d1\ud569 \uc5f0\uc0b0\uc774 \ubc14\ub85c \ub2e4\uc74c \uc54c\ub808\ud504 \uae30\uc218\uac00 \ub41c\ub2e4\ub294 \uc8fc\uc7a5\uc744 <span class=\"defined\">\uc77c\ubc18 \uc5f0\uc18d\uccb4 \uac00\uc124<\/span>(Generalized Continuum Hypothesis, GCH)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<h3>2. \uae30\uc218\uc758 \ub300\uc18c\uad00\uacc4<\/h3>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 7.2. (\uae30\uc218\uc758 \ub300\uc18c\uad00\uacc4)<\/span><\/p>\n<p>\ub450 \uae30\uc218 \\(\\kappa=|A|\\), \\(\\lambda=|B|\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ud568\uc218\uac00 \uc874\uc7ac\ud558\uba74<br \/>\n\\[\\kappa\\le\\lambda\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud55c\ub2e4. \ub610\ud55c \\(\\kappa\\le\\lambda\\)\uc774\uace0 \\(\\kappa\\ne\\lambda\\)\uc774\uba74 \\(\\kappa<\\lambda\\)\ub77c\uace0 \uc4f4\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc815\uc758\ub294 \ub300\ud45c \uc9d1\ud569\uc758 \uc120\ud0dd\uc5d0 \uc758\uc874\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc2e4\uc81c\ub85c \\(A\\sim A&#8217;\\), \\(B\\sim B&#8217;\\)\uc774\uace0 \\(f\\colon A\\to B\\)\uac00 \uc77c\ub300\uc77c\ud568\uc218\uc774\uba74, \uc77c\ub300\uc77c\ub300\uc751 \\(u\\colon A&#8217;\\to A\\), \\(v\\colon B\\to B&#8217;\\)\ub97c \ud0dd\ud558\uc5ec \\(v\\circ f\\circ u\\colon A&#8217;\\to B&#8217;\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 7.3. (\uae30\uc218\uc758 \ub300\uc18c\uad00\uacc4)<\/span><\/p>\n<p>\uae30\uc218 \\(\\kappa\\), \\(\\lambda\\), \\(\\mu\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\kappa\\le\\kappa\\)\uc774\ub2e4.<\/li>\n<li>\\(\\kappa\\le\\lambda\\)\uc774\uace0 \\(\\lambda\\le\\kappa\\)\uc774\uba74 \\(\\kappa=\\lambda\\)\uc774\ub2e4.<\/li>\n<li>\\(\\kappa\\le\\lambda\\)\uc774\uace0 \\(\\lambda\\le\\mu\\)\uc774\uba74 \\(\\kappa\\le\\mu\\)\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\ub294 \ud56d\ub4f1\ud568\uc218, \uc14b\uc9f8\ub294 \uc77c\ub300\uc77c\ud568\uc218\uc758 \ud569\uc131\uc744 \uc0ac\uc6a9\ud558\uba74 \ub41c\ub2e4. \ub458\uc9f8\ub294 \uc815\ub9ac 5.1\uc758 \uce78\ud1a0\uc5b4-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac\uc5d0 \uc758\ud574 \ub450 \ub300\ud45c \uc9d1\ud569\uc774 \ub300\ub4f1\ud558\ubbc0\ub85c \uae30\uc218\uc758 \uc815\uc758\uc5d0\uc11c \ub530\ub978\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.1.<\/span><br \/>\n\ub2e4\uc74c \uc138 \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[<br \/>\nA=\\{1,\\,2,\\,3\\},\\quad B=\\{a,\\,b,\\,c\\},\\quad C=\\{0,\\,1,\\,2,\\,3\\}.<br \/>\n\\]<br \/>\n\ub2e4\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A\\)\uc5d0\uc11c \\(B\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc744 \ud558\ub098 \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(A\\)\uc5d0\uc11c \\(C\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ud568\uc218\ub97c \ud558\ub098 \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(|A|=|B|\\), \\(|A|<|C|\\), \\(|C|\\le|B|\\) \uac00\uc6b4\ub370 \ucc38\uc778 \uac83\uc744 \ubaa8\ub450 \uace0\ub974\uace0 \uadf8 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\\(|A|\\le|B|\\)\uc640 \\(|B|\\le|A|\\)\uac00 \ub3d9\uc2dc\uc5d0 \uc131\ub9bd\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc5d0\uc11c \uc5b4\ub5a4 \uacb0\ub860\uc744 \uc5bb\uc744 \uc218 \uc788\ub294\uc9c0 \uc815\ub9ac 7.3\uc744 \uc774\uc6a9\ud558\uc5ec \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\uc784\uc758\uc758 \ub450 \uae30\uc218 \\(\\kappa\\), \\(\\lambda\\)\uc5d0 \ub300\ud558\uc5ec \ud56d\uc0c1 \\(\\kappa\\le\\lambda\\) \ub610\ub294 \\(\\lambda\\le\\kappa\\)\uac00 \uc131\ub9bd\ud55c\ub2e4\ub294 \uba85\uc81c\ub97c <span class=\"defined\">\uae30\uc218\uc758 \ube44\uad50\uac00\ub2a5\uc131<\/span>(cardinal comparability)\uc774\ub77c\uace0 \ud55c\ub2e4. \uc774 \uba85\uc81c\ub294 \uc120\ud0dd\uacf5\ub9ac\uc640 \ub3d9\uce58\uc774\uba70, \ub530\ub77c\uc11c \uc120\ud0dd\uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc9c0 \uc54a\ub294 ZF\uc5d0\uc11c\ub294 \uc784\uc758\uc758 \ub450 \uae30\uc218\ub97c \ud56d\uc0c1 \ube44\uad50\ud560 \uc218 \uc788\ub2e4\uace0 \uac00\uc815\ud574\uc11c\ub294 \uc548 \ub41c\ub2e4.<\/p>\n<h3>3. \uae30\uc218\uc758 \ub367\uc148<\/h3>\n<p>\ub450 \uc9d1\ud569 \\(A\\), \\(B\\)\uc758 <span class=\"defined\">\uc11c\ub85c\uc18c \ud569<\/span>\uc744<br \/>\n\\[A\\sqcup B=(A\\times\\{0\\})\\cup(B\\times\\{1\\})\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b4\uc790. \ub450 \ubd80\ubd84\uc740 \ud56d\uc0c1 \uc11c\ub85c\uc18c\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 7.4. (\uae30\uc218\uc758 \ub367\uc148)<\/span><\/p>\n<p>\ub450 \uae30\uc218 \\(\\kappa=|A|\\), \\(\\lambda=|B|\\)\uc758 <span class=\"defined\">\uae30\uc218\uc758 \ud569<\/span>\uc744<br \/>\n\\[\\kappa+\\lambda=|A\\sqcup B|\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\\(A\\sim A&#8217;\\), \\(B\\sim B&#8217;\\)\uc774\uba74 \uac01 \uc77c\ub300\uc77c\ub300\uc751\uc744 \ub450 \ud0dc\uadf8 \uc704\uc5d0\uc11c \ub530\ub85c \uc801\uc6a9\ud558\uc5ec \\(A\\sqcup B\\sim A&#8217;\\sqcup B&#8217;\\)\ub97c \uc5bb\uc73c\ubbc0\ub85c \uc774 \uc815\uc758\ub294 \ub300\ud45c \uc9d1\ud569\uc758 \uc120\ud0dd\uacfc \ubb34\uad00\ud558\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 7.5. (\uae30\uc218 \ub367\uc148\uc758 \uae30\ubcf8 \ubc95\uce59)<\/span><\/p>\n<p>\uae30\uc218 \\(\\kappa\\), \\(\\lambda\\), \\(\\mu\\), \\(\\kappa&#8217;\\), \\(\\lambda&#8217;\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\kappa+\\lambda=\\lambda+\\kappa\\).<\/li>\n<li>\\((\\kappa+\\lambda)+\\mu=\\kappa+(\\lambda+\\mu)\\).<\/li>\n<li>\\(\\kappa+0=\\kappa\\).<\/li>\n<li>\\(\\kappa\\le\\kappa&#8217;\\)\uc774\uace0 \\(\\lambda\\le\\lambda&#8217;\\)\uc774\uba74 \\(\\kappa+\\lambda\\le\\kappa&#8217;+\\lambda&#8217;\\).<\/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 \uc11c\ub85c\uc18c \ud569\uc758 \ud0dc\uadf8\ub97c \ubc14\uafb8\uac70\ub098 \ub2e4\uc2dc \ubb36\ub294 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc77c\ub300\uc77c\ub300\uc751\uc73c\ub85c \uc99d\uba85\ud55c\ub2e4. \uc14b\uc9f8\ub294 \\(A\\sqcup\\varnothing\\sim A\\)\uc5d0\uc11c \ub530\ub978\ub2e4. \ub137\uc9f8\ub294 \uc77c\ub300\uc77c\ud568\uc218 \\(f\\colon A\\to A&#8217;\\), \\(g\\colon B\\to B&#8217;\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \uac01 \ud0dc\uadf8\uc5d0\uc11c \\(f\\), \\(g\\)\ub97c \uc801\uc6a9\ud558\uba74 \\(A\\sqcup B\\to A&#8217;\\sqcup B&#8217;\\)\uc778 \uc77c\ub300\uc77c\ud568\uc218\ub97c \uc5bb\ub294\ub2e4\ub294 \uc0ac\uc2e4\uc5d0\uc11c \ub530\ub978\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.2.<\/span><br \/>\n\\(A=\\{1,\\,2\\}\\), \\(B=\\{2,\\,3\\}\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc11c\ub85c\uc18c \ud569 \\(A\\sqcup B\\)\ub97c \uc6d0\uc18c\ub97c \ubaa8\ub450 \uc368\uc11c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n<li>\\(|A|+|B|\\)\uc640 \\(|A\\cup B|\\)\ub97c \uac01\uac01 \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>(2)\uc758 \ub450 \uac12\uc774 \ub2e4\ub978 \uc774\uc720\ub97c \uc11c\ub85c\uc18c \ud569\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\uc77c\ubc18\uc801\uc73c\ub85c \\(A\\cap B=\\varnothing\\)\uc774\uba74 \\(|A|+|B|=|A\\cup B|\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.3.<\/span><br \/>\n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(n\\)\uc774 \uc720\ud55c\uae30\uc218\uc77c \ub54c \\(\\aleph_0+n=\\aleph_0\\)\uc774\ub2e4.<\/li>\n<li>\\(\\aleph_0+\\aleph_0=\\aleph_0\\)\uc774\ub2e4.<\/li>\n<li>\\(\\mathfrak c+\\aleph_0=\\mathfrak c\\)\uc774\ub2e4.<\/li>\n<li>\\(\\mathfrak c+\\mathfrak c=\\mathfrak c\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<h3>4. \uae30\uc218\uc758 \uacf1\uc148<\/h3>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 7.6. (\uae30\uc218\uc758 \uacf1\uc148)<\/span><\/p>\n<p>\ub450 \uae30\uc218 \\(\\kappa=|A|\\), \\(\\lambda=|B|\\)\uc758 <span class=\"defined\">\uae30\uc218\uc758 \uacf1<\/span>\uc744<br \/>\n\\[\\kappa\\lambda=|A\\times B|\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\\(A\\sim A&#8217;\\), \\(B\\sim B&#8217;\\)\uc774\uba74 \ub450 \uc77c\ub300\uc77c\ub300\uc751\uc744 \uc131\ubd84\ubcc4\ub85c \uc801\uc6a9\ud558\uc5ec \\(A\\times B\\sim A&#8217;\\times B&#8217;\\)\ub97c \uc5bb\uc73c\ubbc0\ub85c \uc774 \uc815\uc758\ub3c4 \ub300\ud45c \uc9d1\ud569\uc758 \uc120\ud0dd\uacfc \ubb34\uad00\ud558\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 7.7. (\uae30\uc218 \uacf1\uc148\uc758 \uae30\ubcf8 \ubc95\uce59)<\/span><\/p>\n<p>\uae30\uc218 \\(\\kappa\\), \\(\\lambda\\), \\(\\mu\\), \\(\\kappa&#8217;\\), \\(\\lambda&#8217;\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\kappa\\lambda=\\lambda\\kappa\\).<\/li>\n<li>\\((\\kappa\\lambda)\\mu=\\kappa(\\lambda\\mu)\\).<\/li>\n<li>\\(\\kappa\\cdot1=\\kappa\\).<\/li>\n<li>\\(\\kappa\\cdot0=0\\).<\/li>\n<li>\\(\\kappa(\\lambda+\\mu)=\\kappa\\lambda+\\kappa\\mu\\).<\/li>\n<li>\\(\\kappa\\le\\kappa&#8217;\\)\uc774\uace0 \\(\\lambda\\le\\lambda&#8217;\\)\uc774\uba74 \\(\\kappa\\lambda\\le\\kappa&#8217;\\lambda&#8217;\\).<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab\uc9f8\ub294 \\((a,b)\\mapsto(b,a)\\), \ub458\uc9f8\ub294 \\(((a,b),c)\\mapsto(a,(b,c))\\)\uac00 \uc8fc\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc5d0\uc11c \ub530\ub978\ub2e4. \uc14b\uc9f8\uc640 \ub137\uc9f8\ub294 \\(A\\times\\{0\\}\\sim A\\), \\(A\\times\\varnothing=\\varnothing\\)\uc5d0\uc11c \ub530\ub978\ub2e4. \ub2e4\uc12f\uc9f8\ub294<br \/>\n\\[A\\times(B\\sqcup C)\\sim(A\\times B)\\sqcup(A\\times C)\\]<br \/>\n\uc778 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc77c\ub300\uc77c\ub300\uc751\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \ub9c8\uc9c0\ub9c9 \uc131\uc9c8\uc740 \ub450 \uc77c\ub300\uc77c\ud568\uc218\uc758 \ub370\uce74\ub974\ud2b8 \uacf1\uc744 \ucde8\ud558\uba74 \ub41c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.4.<\/span><br \/>\n\\(A=\\{a,\\,b,\\,c\\}\\), \\(B=\\{0,\\,1\\}\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A\\times B\\)\uc758 \uc6d0\uc18c\ub97c \ubaa8\ub450 \ub098\uc5f4\ud558\uc2dc\uc624.<\/li>\n<li>\\(|A|\\cdot|B|\\)\ub97c \uad6c\ud558\uace0, (1)\uc5d0\uc11c \uad6c\ud55c \\(|A\\times B|\\)\uc640 \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<li>\\((a,b)\\mapsto(b,a)\\)\ub85c \uc815\uc758\ub41c \ub300\uc751\uc774 \\(A\\times B\\)\uc640 \\(B\\times A\\) \uc0ac\uc774\uc758 \uc77c\ub300\uc77c\ub300\uc751\uc784\uc744 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<li>\\(C=\\{x,\\,y\\}\\)\uc77c \ub54c \\(|A\\times(B\\sqcup C)|\\)\uc640 \\(|(A\\times B)\\sqcup(A\\times C)|\\)\ub97c \uac01\uac01 \uad6c\ud558\uc5ec \ubd84\ubc30\ubc95\uce59\uc744 \uc218\ub85c \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>5. \uae30\uc218\uc758 \uac70\ub4ed\uc81c\uacf1<\/h3>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 7.8. (\uae30\uc218\uc758 \uac70\ub4ed\uc81c\uacf1)<\/span><\/p>\n<p>\ub450 \uae30\uc218 \\(\\kappa=|A|\\), \\(\\lambda=|B|\\)\uc5d0 \ub300\ud558\uc5ec <span class=\"defined\">\uae30\uc218\uc758 \uac70\ub4ed\uc81c\uacf1<\/span>\uc744<br \/>\n\\[\\kappa^\\lambda=|A^B|\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(A^B\\)\ub294 \\(B\\)\uc5d0\uc11c \\(A\\)\ub85c \uac00\ub294 \ubaa8\ub4e0 \ud568\uc218\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<\/p>\n<\/div>\n<p>\\(A\\sim A&#8217;\\), \\(B\\sim B&#8217;\\)\uc774\uba74 \uc77c\ub300\uc77c\ub300\uc751 \\(u\\colon A\\to A&#8217;\\), \\(v\\colon B&#8217;\\to B\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[f\\longmapsto u\\circ f\\circ v\\]<br \/>\n\ub294 \\(A^B\\)\uc5d0\uc11c \\((A&#8217;)^{B&#8217;}\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\ubbc0\ub85c \uc774 \uc815\uc758\ub3c4 \uc798 \uc815\uc758\ub41c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 7.9. (\uae30\uc218 \uac70\ub4ed\uc81c\uacf1\uc758 \uae30\ubcf8 \ubc95\uce59)<\/span><\/p>\n<p>\uae30\uc218 \\(\\kappa\\), \\(\\lambda\\), \\(\\mu\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\kappa^{\\lambda+\\mu}=\\kappa^\\lambda\\kappa^\\mu\\).<\/li>\n<li>\\((\\kappa\\lambda)^\\mu=\\kappa^\\mu\\lambda^\\mu\\).<\/li>\n<li>\\((\\kappa^\\lambda)^\\mu=\\kappa^{\\lambda\\mu}\\).<\/li>\n<li>\\(\\kappa^0=1\\).<\/li>\n<li>\\(\\kappa^1=\\kappa\\).<\/li>\n<li>\\(1^\\lambda=1\\).<\/li>\n<li>\\(0^\\lambda=0\\)\uc774\ub2e4. \ub2e8, \\(\\lambda\\ne0\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\ud2b9\ud788 \\(0^0=1\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab\uc9f8\ub294 \uc11c\ub85c\uc18c \ud569 \\(B\\sqcup C\\) \uc704\uc758 \ud568\uc218\uac00 \\(B\\)\uc640 \\(C\\) \uc704\uc758 \ub450 \ud568\uc218\uc640 \uc815\ud655\ud788 \ub300\uc751\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc5d0\uc11c \ub530\ub978\ub2e4. \ub458\uc9f8\ub294 \ud568\uc218 \\(f\\colon C\\to A\\times B\\)\ub97c \ub450 \uc131\ubd84\ud568\uc218\ub85c \ub098\ub204\uba74 \ub41c\ub2e4. \uc14b\uc9f8\ub294 \ud568\uc218 \\(C\\to A^B\\)\uc640 \ud568\uc218 \\(B\\times C\\to A\\) \uc0ac\uc774\uc758 \uc77c\ub300\uc77c\ub300\uc751<br \/>\n\\[F\\longmapsto\\bigl((b,\\,c)\\mapsto F(c)(b)\\bigr)\\]<br \/>\n\uc5d0\uc11c \ub530\ub978\ub2e4. \ub098\uba38\uc9c0\ub294 \uacf5\uc9d1\ud569\uc5d0\uc11c \uc784\uc758\uc758 \uc9d1\ud569\uc73c\ub85c \uac00\ub294 \ud568\uc218\uac00 \uc815\ud655\ud788 \ud558\ub098\uc774\uace0, \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc5d0\uc11c \uacf5\uc9d1\ud569\uc73c\ub85c \uac00\ub294 \ud568\uc218\ub294 \uc5c6\ub2e4\ub294 \uc0ac\uc2e4\uc5d0 \uc758\ud558\uc5ec \ubc14\ub85c \uc720\ub3c4\ub41c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.5.<\/span><br \/>\n\\(A=\\{0,\\,1\\}\\), \\(B=\\{a,\\,b,\\,c\\}\\)\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(f\\colon B\\to A\\)\ub97c<br \/>\n\\[<br \/>\n(f(a),\\,f(b),\\,f(c))<br \/>\n\\]<br \/>\n\uc640 \uac19\uc740 \uc21c\uc11c\uc0bc\uc911\ud56d\uc73c\ub85c \ub098\ud0c0\ub0b4\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A^B\\)\uc758 \uc6d0\uc18c\ub97c \ubaa8\ub450 \ub098\uc5f4\ud558\uc2dc\uc624.<\/li>\n<li>\\(|A^B|\\)\ub97c \uad6c\ud558\uace0 \\(2^3\\)\uacfc \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<li>\\(|A^\\varnothing|\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(|\\varnothing^B|\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\\(2=|\\{0,1\\}|\\)\ub85c \ub450\uba74 \uc784\uc758\uc758 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \ubd80\ubd84\uc9d1\ud569 \\(S\\subseteq A\\)\uc640 \ud2b9\uc131\ud568\uc218<br \/>\n\\[<br \/>\n\\chi_S\\colon A\\to\\{0,\\,1\\},\\quad<br \/>\n\\chi_S(a)=<br \/>\n\\begin{cases}<br \/>\n1 &#038; (a\\in S),\\\\[5pt]<br \/>\n0 &#038; (a\\notin S)<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uac00 \uc77c\ub300\uc77c\ub85c \ub300\uc751\ud55c\ub2e4. \ub530\ub77c\uc11c \\(|A|=\\kappa\\)\uc774\uba74<br \/>\n\\[2^\\kappa=|\\mathcal P(A)|\\]<br \/>\n\uc774\ub2e4. \uc815\ub9ac 5.3\uc758 \uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac\uc5d0 \uc758\ud574 \ud56d\uc0c1 \\(\\kappa<2^\\kappa\\)\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.6.<\/span><br \/>\n\\(A=\\{a,b,c\\}\\)\ub77c \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\mathcal P(A)\\)\uc758 \uc6d0\uc18c\ub97c \ubaa8\ub450 \ub098\uc5f4\ud558\uace0 \\(|\\mathcal P(A)|\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(S=\\{a,c\\}\\)\uc5d0 \ub300\uc751\ud558\ub294 \ud2b9\uc131\ud568\uc218 \\(\\chi_S\\colon A\\to\\{0,1\\}\\)\uc758 \uac12\uc744 \ubaa8\ub450 \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(f\\colon A\\to\\{0,1\\}\\)\uac00<br \/>\n\\[<br \/>\nf(a)=0,\\quad f(b)=1,\\quad f(c)=1<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(f=\\chi_T\\)\uac00 \ub418\uac8c \ud558\ub294 \ubd80\ubd84\uc9d1\ud569 \\(T\\subseteq A\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\uc704 \uacc4\uc0b0\uc744 \uc774\uc6a9\ud558\uc5ec \\(|\\mathcal P(A)|=2^{|A|}\\)\ub97c \uc774 \uacbd\uc6b0\uc5d0 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.7.<\/span><br \/>\n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc720\ud55c\uae30\uc218 \\(m\\), \\(n\\)\uc5d0 \ub300\ud558\uc5ec \uae30\uc218\uc758 \ub367\uc148, \uacf1\uc148, \uac70\ub4ed\uc81c\uacf1\uc774 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">6\uc7a5<\/a>\uc5d0\uc11c \uc815\uc758\ud55c \uc790\uc5f0\uc218\uc758 \uc5f0\uc0b0\uacfc \uc77c\uce58\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(2^{\\aleph_0}=\\mathfrak c\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\aleph_0^{\\aleph_0}=\\mathfrak c\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\mathfrak c^{\\aleph_0}=\\mathfrak c\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>6. \ubb34\ud55c\uae30\uc218\uc758 \ud2b9\ubcc4\ud55c \uc131\uc9c8<\/h3>\n<p>\uc774 \uc808\uc758 \uc77c\ubc18 \uba85\uc81c\uc5d0\uc11c\ub294 \uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud55c\ub2e4. \uc120\ud0dd\uacf5\ub9ac \uc544\ub798\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \ubb34\ud55c\uae30\uc218 \\(\\kappa\\)\uc640 \ubaa8\ub4e0 \uc591\uc758 \uc720\ud55c\uae30\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\kappa+\\kappa=\\kappa,\\quad<br \/>\n\\kappa\\cdot\\kappa=\\kappa,\\quad<br \/>\n\\kappa+n=\\kappa,\\quad<br \/>\n\\kappa\\cdot n=\\kappa<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub354 \uc77c\ubc18\uc801\uc73c\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 7.10. (\ubb34\ud55c\uae30\uc218\uc758 \ud569\uacfc \uacf1)<\/span><\/p>\n<p>\uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uc790. \\(\\kappa\\), \\(\\lambda\\)\uac00 \ubb34\ud55c\uae30\uc218\uc774\uba74<br \/>\n\\[\\kappa+\\lambda=\\kappa\\cdot\\lambda=\\max\\{\\kappa,\\lambda\\}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\uc758 \uc77c\ubc18\uc801\uc778 \uc99d\uba85\uc5d0\ub294 \uc120\ud0dd\uacf5\ub9ac\uc640 \ubb34\ud55c\uc9d1\ud569\uc758 \uc815\ub82c\uac00\ub2a5\uc131\uc744 \uc0ac\uc6a9\ud558\ub294 \ub17c\uc758\uac00 \ud544\uc694\ud558\ubbc0\ub85c <a href=\"\/blog\/invitation-to-mathematical-logic\/ch10-axiom-of-choice\">10\uc7a5<\/a>\uc5d0\uc11c \ub2e4\uc2dc \ub2e4\ub8ec\ub2e4. \ub2e4\ub9cc \\(\\aleph_0+\\aleph_0=\\aleph_0\\)\uacfc \\(\\aleph_0\\cdot\\aleph_0=\\aleph_0\\) \uac19\uc740 \ud2b9\uc815\ud55c \ub4f1\uc2dd\uc740 \uc120\ud0dd\uacf5\ub9ac \uc5c6\uc774\ub3c4 \uba85\uc2dc\uc801\uc778 \uc77c\ub300\uc77c\ub300\uc751\uc73c\ub85c \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.8.<\/span><br \/>\n\uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uace0 \uc815\ub9ac 7.10\uc744 \uc774\uc6a9\ud558\uc5ec \ub2e4\uc74c\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\aleph_0\\mathfrak c\\).<\/li>\n<li>\\(\\mathfrak c\\cdot\\mathfrak c\\).<\/li>\n<li>\\((\\mathfrak c+\\aleph_0)\\aleph_0\\).<\/li>\n<li>\\(|\\mathbb N\\times\\mathbb R|\\).<\/li>\n<li>\\(|\\mathcal P(\\mathbb N)\\times\\mathbb R|\\).<\/li>\n<\/ol>\n<\/div>\n<h3>7. \uae30\uc218 \uc5f0\uc0b0\uc758 \uc608<\/h3>\n<p>\uae30\uc218\uc758 \uc5f0\uc0b0\uc740 \ud568\uc218\ub4e4\uc758 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \uc138\ub294 \ub370 \uc720\uc6a9\ud558\ub2e4. \uba3c\uc800 \\(\\mathbb R\\)\uc5d0\uc11c \\(\\mathbb R\\)\ub85c \uac00\ub294 \ubaa8\ub4e0 \ud568\uc218\ub4e4\uc758 \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790. \ubd80\ubd84\uc9d1\ud569 \\(S\\subseteq\\mathbb R\\)\ub9c8\ub2e4 \ud2b9\uc131\ud568\uc218 \\(\\chi_S\\colon\\mathbb R\\to\\{0,1\\}\\subseteq\\mathbb R\\)\ub97c \ub300\uc751\uc2dc\ud0a4\uba74<br \/>\n\\[2^{\\mathfrak c}\\le|\\mathbb R^{\\mathbb R}|\\]<br \/>\n\uc774\ub2e4. \ubc18\ub300\ub85c \ud568\uc218\ub294 \uadf8 \uadf8\ub798\ud504\ub85c \uacb0\uc815\ub418\ubbc0\ub85c<br \/>\n\\[\\mathbb R^{\\mathbb R}\\preccurlyeq\\mathcal P(\\mathbb R\\times\\mathbb R)\\]<br \/>\n\uc774\uace0, \uc815\ub9ac 5.2\uc5d0 \uc758\ud574 \\(\\mathbb R\\times\\mathbb R\\sim\\mathbb R\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[|\\mathbb R^{\\mathbb R}|=2^{\\mathfrak c}.\\]<br \/>\n\uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc774\ub294 \\(\\mathfrak c\\)\ubcf4\ub2e4 \ud070 \uae30\uc218\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \uc5f0\uc18d\ud568\uc218\ub4e4\uc758 \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790. \uc5f0\uc18d\ud568\uc218 \\(f\\colon\\mathbb R\\to\\mathbb R\\)\ub294 \uc870\ubc00\ud55c \ubd80\ubd84\uc9d1\ud569 \\(\\mathbb Q\\)\uc5d0\uc11c\uc758 \uac12\uc5d0 \uc758\ud558\uc5ec \uc644\uc804\ud788 \uacb0\uc815\ub41c\ub2e4. \ub530\ub77c\uc11c \ubb38\uc81c 7.7\uc758 (4)\uc5d0 \uc758\ud574 \uc5f0\uc18d\ud568\uc218\uc758 \uac1c\uc218\ub294 \ub9ce\uc544\uc57c<br \/>\n\\[|\\mathbb R^{\\mathbb Q}|=\\mathfrak c^{\\aleph_0}=\\mathfrak c\\]<br \/>\n\uc774\ub2e4. \ud55c\ud3b8 \uc0c1\uc218\ud568\uc218\ub9cc \ud574\ub3c4 \\(\\mathfrak c\\)\uac1c \uc788\uc73c\ubbc0\ub85c, \uc5f0\uc18d\ud568\uc218\uc758 \uac1c\uc218\ub294 \uc815\ud655\ud788 \\(\\mathfrak c\\)\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.9.<\/span><br \/>\n\ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uac00\uc0b0\uc9d1\ud569 \\(A_n\\)\uacfc \uc77c\ub300\uc77c\ud568\uc218 \\(i_n\\colon A_n\\to\\mathbb N\\)\uc774 \uac01 \\(n\\in\\mathbb N\\)\uc5d0 \ub300\ud558\uc5ec \uc8fc\uc5b4\uc838 \uc788\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(\\displaystyle\\bigcup_{n\\in\\mathbb N}A_n\\)\uc740 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\\(\\mathbb R\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc5f4\ub9b0\uad6c\uac04\ub4e4\uc758 \uac1c\uc218\ub294 \\(\\mathfrak c\\)\uc774\ub2e4.<\/li>\n<li>\\(x<y\\)\uc774\uba74 \\(f(x)\\le f(y)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(f\\colon\\mathbb R\\to\\mathbb R\\)\uc758 \uac1c\uc218\ub294 \\(\\mathfrak c\\)\uc774\ub2e4. \ub2e8\uc870\ud568\uc218\uc758 \ubd88\uc5f0\uc18d\uc810 \uc9d1\ud569\uc774 \uac00\uc0b0\uc9d1\ud569\uc774\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p><!-- TODO: theorem:tarskicardinalsquare\uc758 \ucd5c\uc885 \uc815\ub9ac \ubc88\ud638\ub294 \ud574\ub2f9 \uc7a5 \uac1c\uc815 \uc2dc \ud655\uc778 --><\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 7.10.<\/span><br \/>\n\uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uc790. \ubb34\ud55c\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[|A\\times A|=|A|\\]<br \/>\n\uc784\uc744 \uc815\ub9ac 7.10\uc744 \uc774\uc6a9\ud558\uc5ec \ubcf4\uc774\uc2dc\uc624. (\ucc38\uace0\ub85c \ub4a4\uc5d0\uc11c \ub2e4\ub8f0 \ud0c0\ub974\uc2a4\ud0a4\uc758 \uc815\ub9ac\uc5d0 \ub530\ub974\uba74 \uc774 \uba85\uc81c\uac00 \ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4\ub294 \uc8fc\uc7a5\uc740 ZF\uc5d0\uc11c \uc120\ud0dd\uacf5\ub9ac\uc640 \ub3d9\uce58\uc774\ub2e4.)<\/p>\n<\/div>\n<h3>8. \uae30\uc218\uc640 \uc120\ud0dd\uacf5\ub9ac<\/h3>\n<p>\uae30\uc218\uc758 \ube44\uad50\uac00\ub2a5\uc131, \uc989 \uc784\uc758\uc758 \ub450 \uae30\uc218 \\(\\kappa\\), \\(\\lambda\\)\uc5d0 \ub300\ud558\uc5ec \\(\\kappa\\le\\lambda\\)\uc774\uac70\ub098 \\(\\lambda\\le\\kappa\\)\ub77c\ub294 \uba85\uc81c\ub294 \uc120\ud0dd\uacf5\ub9ac\uc640 \ub3d9\uce58\uc774\ub2e4. \uc774\ub294 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch10-axiom-of-choice\">10\uc7a5<\/a>\uc5d0\uc11c \ub2e4\ub8f0 \uc815\ub82c \uc815\ub9ac\uc640\ub3c4 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<p>\uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uba74 \uc815\ub9ac 7.10\uacfc \uac19\uc774 \ubaa8\ub4e0 \ubb34\ud55c\uae30\uc218\uc758 \ud569\uacfc \uacf1\uc744 \ub9e4\uc6b0 \uac04\ub2e8\ud558\uac8c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uc120\ud0dd\uacf5\ub9ac\uac00 \uc5c6\uc73c\uba74 \uc774\ub7ec\ud55c \uba85\uc81c\ub97c \uc784\uc758\uc758 \ubb34\ud55c\uc9d1\ud569\uc5d0 \uc77c\uad04\uc801\uc73c\ub85c \uc801\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \uc608\ub97c \ub4e4\uc5b4 ZF\ub9cc\uc73c\ub85c\ub294 \u201c\ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(|A\\times A|=|A|\\)\u201d\uc784\uc744 \uc99d\uba85\ud560 \uc218 \uc5c6\ub2e4. \ub610\ud55c \uac01 \\(A_n\\)\uc774 \uac00\uc0b0\uc9d1\ud569\uc774\ub77c\ub294 \uc0ac\uc2e4\ub9cc\uc73c\ub85c \\(\\bigcup_{n\\in\\mathbb N}A_n\\)\uc774 \uac00\uc0b0\uc774\ub77c\uace0 \uacb0\ub860 \ub0b4\ub9ac\ub294 \uba85\uc81c\uc5d0\ub294 \uac00\uc0b0\uc120\ud0dd\uacf5\ub9ac(countable choice)\uc758 \ud55c \ud615\ud0dc\uac00 \uad00\uc5ec\ud55c\ub2e4. \ubb38\uc81c 7.9\uc758 (1)\uc5d0\uc11c\ub294 \uac01 \\(A_n\\)\uc758 \ub2e8\uc0ac\ud568\uc218 \\(i_n\\)\uc744 \ud568\uaed8 \uc8fc\uc5b4 \uc774 \uc120\ud0dd \ubb38\uc81c\ub97c \ud53c\ud558\uc600\ub2e4. \ubc18\uba74<br \/>\n\\[\\aleph_0+\\aleph_0=\\aleph_0,\\quad \\aleph_0\\cdot\\aleph_0=\\aleph_0\\]<br \/>\n\uc740 \uc790\uc5f0\uc218\uc5d0 \ub300\ud55c \uba85\uc2dc\uc801\uc778 \uc77c\ub300\uc77c\ub300\uc751\uc73c\ub85c ZF\uc5d0\uc11c\ub3c4 \uc99d\uba85\ub41c\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>5\uc7a5\uc5d0\uc11c \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \uc77c\ub300\uc77c\ub300\uc751\uc744 \ud1b5\ud574 \ube44\uad50\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\uc218(cardinal number)\ub97c \ub3c4\uc785\ud558\uace0, \uae30\uc218\uc758 \ub300\uc18c\uad00\uacc4\uc640 \uc5f0\uc0b0\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uae30\uc218\ub294 \uc720\ud55c\uc9d1\ud569\uc758 \uc6d0\uc18c \uc218\ub97c \ubb34\ud55c\uc9d1\ud569\uae4c\uc9c0 \ud655\uc7a5\ud55c \uac1c\ub150\uc774\ub2e4. 1. \uc9d1\ud569\uc758 \ub300\ub4f1\uacfc \uae30\uc218 5\uc7a5\uc5d0\uc11c \uc0ac\uc6a9\ud55c \ud45c\uae30\ub97c \ub2e4\uc2dc \uc4f0\uba74, \ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\) \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud560 \ub54c \\(A\\sim B\\) \ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub300\ub4f1\uc740 \ub3d9\uce58\uad00\uacc4\uc774\uba70, \ubb38\uc81c 4.16\uc5d0\uc11c \uc774\ub97c \ud655\uc778\ud558\uc600\ub2e4. \uc815\uc758 7.1. (\uae30\uc218) \uc9d1\ud569 \\(A\\)\uc758 \uae30\uc218(cardinal number) \ub610\ub294 \ub18d\ub3c4(cardinality)\ub97c \\(|A|\\)&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":107,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9258","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9258","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=9258"}],"version-history":[{"count":10,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9258\/revisions"}],"predecessor-version":[{"id":10082,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9258\/revisions\/10082"}],"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=9258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}