{"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":"2025-10-20T18:48:46","modified_gmt":"2025-10-20T09:48:46","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>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 <span class=\"defined\">\uae30\uc218<\/span>(cardinal number)\ub77c\ub294 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uace0, \uae30\uc218\uc758 \uc5f0\uc0b0\uacfc \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uae30\uc218\ub294 \uc6d0\uc18c\uc758 &#8216;\uac1c\uc218&#8217;\ub97c \ucd94\uc0c1\ud654\ud55c \uac1c\ub150\uc73c\ub85c, \uc720\ud55c\uc9d1\ud569\ubfd0\ub9cc \uc544\ub2c8\ub77c \ubb34\ud55c\uc9d1\ud569\uc758 \ud06c\uae30\ub3c4 \ub2e4\ub8f0 \uc218 \uc788\uac8c \ud574\uc900\ub2e4.<\/p>\n<h3>1. \uc9d1\ud569\uc758 \ub300\ub4f1\uacfc \uae30\uc218<\/h3>\n<p>\ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\) \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud560 \ub54c, &#8220;\\(A\\)\uc640 \\(B\\)\uac00 <span class=\"defined\">\ub300\ub4f1<\/span>(equipotent)\ud558\ub2e4&#8221;\ub77c\uace0 \ub9d0\ud558\uba70 \\(A \\sim B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. 5\uc7a5\uc5d0\uc11c \ubcf4\uc558\ub4ef\uc774 \ub300\ub4f1 \uad00\uacc4\ub294 \ub3d9\uce58\uad00\uacc4\uc774\ub2e4. \uc989, \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\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>\uc9d1\ud569 \\(A\\)\uc758 <span class=\"defined\">\uae30\uc218<\/span>(cardinal number) \ub610\ub294 <span class=\"defined\">\ub18d\ub3c4<\/span>(cardinality)\ub294 \\(A\\)\uc640 \ub300\ub4f1\ud55c \ubaa8\ub4e0 \uc9d1\ud569\ub4e4\uc774 \uacf5\ud1b5\uc73c\ub85c \uac16\ub294 &#8216;\ud06c\uae30&#8217;\ub97c \ub098\ud0c0\ub0b4\ub294 \uac1c\ub150\uc774\ub2e4. \uc9d1\ud569 \\(A\\)\uc758 \uae30\uc218\ub97c \\(|A|\\) \ub610\ub294 \\(\\text{card}(A)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\(\\# A\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\uae30\uc218\ub97c \uc5c4\ubc00\ud558\uac8c \ub2e4\ub8e8\uae30 \uc704\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 <span class=\"defined\">\uae30\uc218 \uacf5\ub9ac<\/span>(axiom of cardinality)\ub97c \ub3c4\uc785\ud55c\ub2e4.<\/p>\n<div class=\"box\">\n<p><span class=\"defined\">\uae30\uc218\uc758 \ubaa8\uc784<\/span> \\(CD\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<ul>\n<li>(K1) \uc784\uc758\uc758 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(A \\sim a\\)\uc778 \\(a\\in CD\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(a\\)\ub97c \\(A\\)\uc758 <span class=\"defined\">\uae30\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0, \\(|A|\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>(K2) \ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uac00 \ub300\ub4f1\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(|A| = |B|\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<\/div>\n<p>\uc720\ud55c\uc9d1\ud569\uc758 \uacbd\uc6b0 \uae30\uc218\ub294 \uc790\uc5f0\uc218\uc774\ub2e4. \uc608\ub97c \ub4e4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\\(|\\varnothing| = 0\\)<\/li>\n<li>\\(|\\{a\\}| = 1\\)<\/li>\n<li>\\(|\\{a,\\, b\\}| = 2\\) (\ub2e8, \\(a \\neq b\\))<\/li>\n<\/ul>\n<p>\uadf8\ub7ec\ubbc0\ub85c \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc77c \ub54c\ub294 \\(A\\)\uc758 \uae30\uc218\ub97c \\(n(A)\\)\ub85c \ub098\ud0c0\ub0b4\ub3c4 \ub41c\ub2e4.<\/p>\n<p>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \ubb34\ud55c\uc9d1\ud569\uc758 \uae30\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ud2b9\ubcc4\ud55c \uae30\ud638\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<ul>\n<li>\\(|\\mathbb{N}| = \\aleph_0\\) (&#8216;\uc54c\ub808\ud504 \ub110&#8217; \ub610\ub294 &#8216;\uc54c\ub808\ud504 \uc601&#8217;\uc774\ub77c\uace0 \uc77d\ub294\ub2e4.)<\/li>\n<li>\\(|\\mathbb{R}| = \\mathfrak{c}\\) \ub610\ub294 \\(2^{\\aleph_0}\\) (\uc54c\ud30c\ubcb3 c\uc640 \uac19\uc740 \ubc1c\uc74c\uc73c\ub85c \uc77d\ub294\ub2e4.)<\/li>\n<li>\uc5f0\uc18d\uccb4 \uac00\uc124\uc744 \uac00\uc815\ud558\uba74, \\(| E | = \\aleph_n\\)\uc77c \ub54c \\( | \\mathcal{P} (E) | = \\aleph_{n+1}\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<h3>2. \uae30\uc218\uc758 \ub300\uc18c\uad00\uacc4<\/h3>\n<p>\ub450 \uae30\uc218 \\(\\kappa\\)\uc640 \\(\\lambda\\)\uc5d0 \ub300\ud558\uc5ec, \\(|A| = \\kappa\\)\uc774\uace0 \\(|B| = \\lambda\\)\uc778 \uc9d1\ud569 \\(A\\), \\(B\\)\ub97c \ud0dd\ud588\uc744 \ub54c, \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c\uc758 \uc77c\ub300\uc77c\ud568\uc218\uac00 \uc874\uc7ac\ud558\uba74 \\(\\kappa \\leq \\lambda\\)\ub77c\uace0 \uc815\uc758\ud55c\ub2e4. \uc774 \uc815\uc758\ub294 \\(A\\), \\(B\\)\uc758 \uc120\ud0dd\uacfc \ubb34\uad00\ud558\ub2e4.<\/p>\n<ul>\n<li>\\(\\kappa \\leq \\lambda\\)\uc774\uace0 \\(\\lambda \\leq \\kappa\\)\uc774\uba74 \\(\\kappa = \\lambda\\)\uc774\ub2e4. (\uc288\ub8b0\ub354-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac)<\/li>\n<li>\\(\\kappa \\leq \\lambda\\)\uc774\uace0 \\(\\lambda \\leq \\mu\\)\uc774\uba74 \\(\\kappa \\leq \\mu\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \ub450 \uae30\uc218 \\(\\kappa\\), \\(\\lambda\\)\uc5d0 \ub300\ud558\uc5ec \\(\\kappa \\leq \\lambda\\)\uc774\uac70\ub098 \\(\\lambda \\leq \\kappa\\)\uc774\ub2e4.[\uc120\ud0dd \uacf5\ub9ac\ub97c \uac00\uc815\ud588\uc744 \ub54c \uc131\ub9bd\ud55c\ub2e4.]<\/li>\n<\/ul>\n<p>\ud2b9\ud788 \\(\\kappa < \\lambda\\)\ub294 \\(\\kappa \\leq \\lambda\\)\uc774\uace0 \\(\\kappa \\neq \\lambda\\)\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<h3>3. \uae30\uc218\uc758 \ub367\uc148<\/h3>\n<p>\ub450 \uae30\uc218 \\(\\kappa = |A|\\), \\(\\lambda = |B|\\)\uc5d0 \ub300\ud558\uc5ec, \\(A \\cap B = \\varnothing\\)\ub77c\uace0 \uac00\uc815\ud558\uc5ec\ub3c4 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\ub294\ub2e4.[\ub9cc\uc57d \\(A,\\) \\(B\\)\uac00 \uc11c\ub85c\uc18c\uac00 \uc544\ub2c8\ub77c\uba74 \\(A &#8216; = A\\times \\left\\{0\\right\\},\\) \\(B &#8216; = B\\times \\left\\{ 1\\right\\}\\)\uc640 \uac19\uc774 \uc11c\ub85c\uc18c\uc778 \\(A &#8216;\\) \\(B &#8216;\\)\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(A,\\) \\(B\\)\ub97c \ub300\uccb4\ud558\uba74 \ub41c\ub2e4.] \uc774\ub54c \uae30\uc218\uc758 <span class=\"defined\">\ud569<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\kappa + \\lambda = |A \\cup B|\\]<br \/>\n\uc774 \uc815\uc758\ub294 \\(A\\), \\(B\\)\uc758 \uc120\ud0dd\uacfc \ubb34\uad00\ud558\ub2e4. \uc2e4\uc81c\ub85c \\(A&#8217; \\sim A\\), \\(B&#8217; \\sim B\\)\uc774\uace0 \\(A&#8217; \\cap B&#8217; = \\varnothing\\)\uc774\uba74 \\(A&#8217; \\cup B&#8217; \\sim A \\cup B\\)\uc774\ub2e4.<\/p>\n<p>\uae30\uc218\uc758 \ub367\uc148\uc740 \ub2e4\uc74c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\uad50\ud658\ubc95\uce59: \\(\\kappa + \\lambda = \\lambda + \\kappa\\)\uc774\ub2e4.<\/li>\n<li>\uacb0\ud569\ubc95\uce59: \\((\\kappa + \\lambda) + \\mu = \\kappa + (\\lambda + \\mu)\\)<\/li>\n<li>\ud56d\ub4f1\uc6d0: \\(\\kappa + 0 = \\kappa\\)<\/li>\n<li>\ub2e8\uc870\uc131: \\(\\kappa \\leq \\kappa&#8217;\\)\uc774\uace0 \\(\\lambda \\leq \\lambda&#8217;\\)\uc774\uba74 \\(\\kappa + \\lambda \\leq \\kappa&#8217; + \\lambda&#8217;\\)<\/li>\n<\/ul>\n<p>\uae30\uc218\uc758 \ub367\uc148\uc758 \uc131\uc9c8\uc740 \uae30\uc218\uc758 \uc815\uc758\uc640 \uc77c\ub300\uc77c\ub300\uc751\uc758 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uba74 \uc99d\uba85\ub41c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.1.<\/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\\)<\/li>\n<li>\\(\\aleph_0 + \\aleph_0 = \\aleph_0\\)<\/li>\n<li>\\(\\mathfrak{c} + \\aleph_0 = \\mathfrak{c}\\)<\/li>\n<li>\\(\\mathfrak{c} + \\mathfrak{c} = \\mathfrak{c}\\)<\/li>\n<\/ol>\n<\/div>\n<h3>4. \uae30\uc218\uc758 \uacf1\uc148<\/h3>\n<p>\ub450 \uae30\uc218 \\(\\kappa = |A|\\), \\(\\lambda = |B|\\)\uc5d0 \ub300\ud558\uc5ec, \uae30\uc218\uc758 <span class=\"defined\">\uacf1<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\kappa \\cdot \\lambda = |A \\times B|\\]<br \/>\n\uc5ec\uae30\uc11c \\(A \\times B\\)\ub294 \ub370\uce74\ub974\ud2b8 \uacf1\uc774\ub2e4. \uc774 \uc815\uc758 \uc5ed\uc2dc \\(A\\), \\(B\\)\uc758 \uc120\ud0dd\uacfc \ubb34\uad00\ud558\ub2e4.<\/p>\n<p>\uae30\uc218\uc758 \uacf1\uc148\uc740 \ub2e4\uc74c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\uad50\ud658\ubc95\uce59: \\(\\kappa \\cdot \\lambda = \\lambda \\cdot \\kappa\\)<\/li>\n<li>\uacb0\ud569\ubc95\uce59: \\((\\kappa \\cdot \\lambda) \\cdot \\mu = \\kappa \\cdot (\\lambda \\cdot \\mu)\\)<\/li>\n<li>\ud56d\ub4f1\uc6d0: \\(\\kappa \\cdot 1 = \\kappa\\)<\/li>\n<li>\uc601\uc6d0: \\(\\kappa \\cdot 0 = 0\\)<\/li>\n<li>\ubd84\ubc30\ubc95\uce59: \\(\\kappa \\cdot (\\lambda + \\mu) = \\kappa \\cdot \\lambda + \\kappa \\cdot \\mu\\)<\/li>\n<li>\ub2e8\uc870\uc131: \\(\\kappa \\leq \\kappa&#8217;\\)\uc774\uace0 \\(\\lambda \\leq \\lambda&#8217;\\)\uc774\uba74 \\(\\kappa \\cdot \\lambda \\leq \\kappa&#8217; \\cdot \\lambda&#8217;\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<h3>5. \uae30\uc218\uc758 \uac70\ub4ed\uc81c\uacf1<\/h3>\n<p>\ub450 \uae30\uc218 \\(\\kappa = |A|\\), \\(\\lambda = |B|\\)\uc5d0 \ub300\ud558\uc5ec, \uae30\uc218\uc758 <span class=\"defined\">\uac70\ub4ed\uc81c\uacf1<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\kappa^\\lambda = \\left| A^B \\right|\\]<br \/>\n\uc5ec\uae30\uc11c \\(A^B\\)\ub294 \\(B\\)\uc5d0\uc11c \\(A\\)\ub85c\uc758 \ubaa8\ub4e0 \ud568\uc218\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<\/p>\n<p>\uae30\uc218\uc758 \uac70\ub4ed\uc81c\uacf1\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\\(\\kappa^{\\lambda + \\mu} = \\kappa^\\lambda \\cdot \\kappa^\\mu\\)<\/li>\n<li>\\((\\kappa \\cdot \\lambda)^\\mu = \\kappa^\\mu \\cdot \\lambda^\\mu\\)<\/li>\n<li>\\((\\kappa^\\lambda)^\\mu = \\kappa^{\\lambda \\cdot \\mu}\\)<\/li>\n<li>\\(\\kappa^0 = 1\\) (\ub2e8, \\(\\kappa \\neq 0\\))<\/li>\n<li>\\(\\kappa^1 = \\kappa\\)<\/li>\n<li>\\(1^\\lambda = 1\\)<\/li>\n<li>\\(0^\\lambda = 0\\) (\ub2e8, \\(\\lambda \\neq 0\\))<\/li>\n<\/ul>\n<p>\ud2b9\ud788 \uc911\uc694\ud55c \uc0ac\uc2e4\uc740 \\(|A| = \\kappa\\)\uc77c \ub54c \\(2^{\\kappa} = |\\mathcal{P}(A)|\\)\ub77c\ub294 \uc810\uc774\ub2e4. \uc2e4\uc81c\ub85c \uc9d1\ud569 \\(A\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc740 \ud2b9\uc131\ud568\uc218 \\(\\chi: A \\to \\{0,\\, 1\\}\\)\uacfc \uc77c\ub300\uc77c \ub300\uc751\ub41c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.2.<\/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 \uc5f0\uc0b0\uc774 \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>\ubb34\ud55c\uae30\uc218\ub294 \uc720\ud55c\uae30\uc218\uc640 \ub2e4\ub978 \ud2b9\ubcc4\ud55c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4. \uc608\ub97c \ub4e4\uba74, \\(\\kappa\\)\uac00 \ubb34\ud55c\uae30\uc218\uc774\uace0 \\(n\\)\uc774 \uc720\ud55c\uae30\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(\\kappa + \\kappa = \\kappa\\)<\/li>\n<li>\\(\\kappa \\cdot \\kappa = \\kappa\\)<\/li>\n<li>\\(\\kappa + n = \\kappa\\)<\/li>\n<li>\\(\\kappa \\cdot n = \\kappa\\) (\ub2e8, \\(n\\ne 0\\))<\/li>\n<\/ul>\n<p>\ub610\ud55c, \\(\\kappa\\)\uc640 \\(\\lambda\\)\uac00 \ubb34\ud55c\uae30\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\kappa + \\lambda = \\kappa \\lambda = \\operatorname{max}\\left\\{ \\kappa ,\\, \\lambda \\right\\}.\\]<\/p>\n<h3>7. \uae30\uc218 \uc5f0\uc0b0\uc758 \uc608<\/h3>\n<p>\uae30\uc218\uc758 \uc5f0\uc0b0\uc758 \uc131\uc9c8\uc740 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ub2e4\ub8e8\ub294 \uc5ec\ub7ec \ubb38\uc81c\uc5d0\uc11c \ud65c\uc6a9\ub41c\ub2e4.<\/p>\n<p>\ub300\ud45c\uc801\uc778 \uc608\ub85c\uc11c \uc2e4\ud568\uc218\uc758 \uac1c\uc218\ub97c \uc138\uc5b4 \ubcf4\uc790. \\(\\mathbb{R}\\)\uc5d0\uc11c \\(\\mathbb{R}\\)\ub85c\uc758 \ubaa8\ub4e0 \ud568\uc218\uc758 \uc9d1\ud569\uc758 \uae30\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[|\\mathbb{R}^{\\mathbb{R}}| = \\mathfrak{c}^{\\mathfrak{c}} = (2^{\\aleph_0})^{2^{\\aleph_0}} = 2^{\\aleph_0 \\cdot 2^{\\aleph_0}} = 2^{2^{\\aleph_0}} = 2^{\\mathfrak{c}}\\]<br \/>\n\uc774\uac83\uc740 \\(\\mathfrak{c}\\)\ubcf4\ub2e4 \ud070 \uae30\uc218\uc774\ub2e4. \uc989, \uc2e4\ud568\uc218\uc758 \uac1c\uc218\ub294 \uc2e4\uc218\uc758 \uac1c\uc218\ubcf4\ub2e4 \ubcf8\uc9c8\uc801\uc73c\ub85c \ub354 \ub9ce\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \uc5f0\uc18d\ud568\uc218\uc758 \uac1c\uc218\ub97c \uc138\uc5b4 \ubcf4\uc790. \ub180\ub78d\uac8c\ub3c4 \\(\\mathbb{R}\\)\uc5d0\uc11c \\(\\mathbb{R}\\)\ub85c\uc758 \uc5f0\uc18d\ud568\uc218\uc758 \uac1c\uc218\ub294 \\(\\mathfrak{c}\\)\uc774\ub2e4. \uc989 \uc720\ub9ac\uc218\uc758 \uc870\ubc00\uc131\uacfc \uc5f0\uc18d\ud568\uc218\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \uc5f0\uc18d\ud568\uc218\ub294 \uc720\ub9ac\uc218\uc5d0\uc11c\uc758 \uac12\uc73c\ub85c \uc644\uc804\ud788 \uacb0\uc815\ub418\ubbc0\ub85c, \uc5f0\uc18d\ud568\uc218\uc758 \uac1c\uc218\ub294 \uae30\uaecf\ud574\uc57c \\(\\mathfrak{c}^{\\aleph_0} = \\mathfrak{c}\\)\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 7.3.<\/span><br \/>\n\ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uac00\uc0b0\ubb34\ud55c\uac1c\uc758 \uac00\uc0b0\uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uc740 \uac00\uc0b0\uc774\ub2e4.<\/li>\n<li>\\(\\mathbb{R}\\)\uc758 \uc5f4\ub9b0\uad6c\uac04\ub4e4\uc758 \uac1c\uc218\ub294 \\(\\mathfrak{c}\\)\uc774\ub2e4.<\/li>\n<li>\\(\\mathbb{R}\\)\uc5d0\uc11c \\(\\mathbb{R}\\)\ub85c\uc758 \ub2e8\uc870\uc99d\uac00 \ud568\uc218\uc758 \uac1c\uc218\ub294 \\(\\mathfrak{c}\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 7.4.<\/span><br \/>\n\ubb34\ud55c\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(|A \\times A| = |A|\\)\uc758 \uc99d\uba85\uc744 \ucc3e\uc544 \ubcf4\uc790.<\/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 \\leq \\lambda\\)\uc774\uac70\ub098 \\(\\lambda \\leq \\kappa\\)\ub77c\ub294 \uc0ac\uc2e4\uc740 \uc120\ud0dd\uacf5\ub9ac\uc640 \ub3d9\uce58\uc774\ub2e4. \uc2e4\uc81c\ub85c \uc774\uac83\uc740 8\uc7a5\uc5d0\uc11c \ub2e4\ub8f0 \uc815\ub82c \uc815\ub9ac\uc640\ub3c4 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<p>\ub610\ud55c \ubb34\ud55c\uae30\uc218 \\(\\kappa\\)\uc5d0 \ub300\ud558\uc5ec \\(\\kappa + \\kappa = \\kappa\\)\ub77c\ub294 \uc131\uc9c8\ub3c4 \uc120\ud0dd\uacf5\ub9ac\ub97c \ud544\uc694\ub85c \ud55c\ub2e4. \uc120\ud0dd\uacf5\ub9ac \uc5c6\uc774\ub294 \\(\\aleph_0 + \\aleph_0 = \\aleph_0\\)\uc870\ucc28 \uc99d\uba85\ud560 \uc218 \uc5c6\ub294 \ubaa8\ub378\uc774 \uc874\uc7ac\ud55c\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)\ub77c\ub294 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uace0, \uae30\uc218\uc758 \uc5f0\uc0b0\uacfc \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uae30\uc218\ub294 \uc6d0\uc18c\uc758 &#8216;\uac1c\uc218&#8217;\ub97c \ucd94\uc0c1\ud654\ud55c \uac1c\ub150\uc73c\ub85c, \uc720\ud55c\uc9d1\ud569\ubfd0\ub9cc \uc544\ub2c8\ub77c \ubb34\ud55c\uc9d1\ud569\uc758 \ud06c\uae30\ub3c4 \ub2e4\ub8f0 \uc218 \uc788\uac8c \ud574\uc900\ub2e4. 1. \uc9d1\ud569\uc758 \ub300\ub4f1\uacfc \uae30\uc218 \ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\) \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud560 \ub54c, &#8220;\\(A\\)\uc640 \\(B\\)\uac00 \ub300\ub4f1(equipotent)\ud558\ub2e4&#8221;\ub77c\uace0 \ub9d0\ud558\uba70 \\(A \\sim B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. 5\uc7a5\uc5d0\uc11c \ubcf4\uc558\ub4ef\uc774 \ub300\ub4f1 \uad00\uacc4\ub294 \ub3d9\uce58\uad00\uacc4\uc774\ub2e4. \uc989, \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ubc18\uc0ac\uc801&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":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9258\/revisions"}],"predecessor-version":[{"id":9439,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9258\/revisions\/9439"}],"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}]}}