{"id":9282,"date":"2025-10-17T20:23:48","date_gmt":"2025-10-17T11:23:48","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9282"},"modified":"2026-09-27T12:33:36","modified_gmt":"2026-09-27T03:33:36","slug":"ch17-compactness-first-order-logic","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch17-compactness-first-order-logic\/","title":{"rendered":"\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>17. \uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131<\/h2>\n\n --><\/p>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131\uc740 \ubb34\ud55c\ud55c \ubb38\uc7a5 \uc9d1\ud569\uc758 \ub9cc\uc871 \uac00\ub2a5\uc131\uc744 \uadf8 \uc720\ud55c\ud55c \ubd80\ubd84\ub4e4\ub9cc\uc73c\ub85c \ud310\uc815\ud560 \uc218 \uc788\ub2e4\ub294 \uc815\ub9ac\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 17.1. (\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131)<\/span><\/p>\n<p>\\(\\mathcal L\\)\uc774 \uac00\uc0b0 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\uc774\uace0 \\(\\varSigma\\)\uac00 \\(\\mathcal L\\)-\ubb38\uc7a5\ub4e4\uc758 \uc9d1\ud569\uc774\uba74 \ub2e4\uc74c\uc740 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\varSigma\\)\ub294 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<li>\\(\\varSigma\\)\uc758 \ubaa8\ub4e0 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc740 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n(1)\uc5d0\uc11c (2)\ub294 \uc790\uba85\ud558\ub2e4. \ubc18\ub300\ub85c \ubaa8\ub4e0 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc774 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \\(\\varSigma\\)\uac00 \ubb34\ubaa8\uc21c\uc774 \uc544\ub2c8\uba74 \uc5b4\ub5a4 \ubb38\uc7a5 \\(\\phi\\)\uc5d0 \ub300\ud574<br \/>\n\\[<br \/>\n\\varSigma\\vdash\\phi<br \/>\n\\quad\\text{\uc640}\\quad<br \/>\n\\varSigma\\vdash\\neg\\phi<br \/>\n\\]<br \/>\n\uc778 \uc720\ud55c\ud55c \ub450 \uc99d\uba85\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub450 \uc99d\uba85\uc5d0\uc11c \uc2e4\uc81c\ub85c \uc0ac\uc6a9\ub41c \uac00\uc815\ub4e4\uc744 \ubaa8\ub450 \ubaa8\uc73c\uba74 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569 \\(\\varSigma_0\\subseteq\\varSigma\\)\ub97c \uc5bb\uace0, \\(\\varSigma_0\\)\ub3c4 \ubaa8\uc21c\uc801\uc774\ub2e4. <a href=\"\/blog\/invitation-to-mathematical-logic\/ch16-inference-rule-first-order-logic\">\uac74\uc804\uc131 \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \\(\\varSigma_0\\)\ub294 \ub9cc\uc871 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc544 \uac00\uc815\uacfc \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\varSigma\\)\ub294 \ubb34\ubaa8\uc21c\uc774\uace0, <a href=\"\/blog\/invitation-to-mathematical-logic\/ch16-inference-rule-first-order-logic\">\uc644\uc804\uc131 \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ucf64\ud329\ud2b8\uc131\uc758 \ub300\ud45c\uc801\uc778 \uc751\uc6a9\uc740 \u201c\uc784\uc758\ub85c \ud070 \uc720\ud55c \ubaa8\ud615\u201d\uc5d0\uc11c \uc2e4\uc81c \ubb34\ud55c \ubaa8\ud615\uc744 \ub9cc\ub4dc\ub294 \uac83\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 17.2. (\ubb34\ud55c \ubaa8\ud615\uc758 \uc874\uc7ac)<\/span><\/p>\n<p>\\(\\mathcal L\\)\uc774 \uac00\uc0b0 \uc5b8\uc5b4\uc774\uace0 \\(T\\)\uac00 \\(\\mathcal L\\)-\ubb38\uc7a5\ub4e4\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud574 \uc6d0\uc18c\uac00 \uc801\uc5b4\ub3c4 \\(n\\)\uac1c\uc778 \\(T\\)\uc758 \ubaa8\ud615\uc774 \uc874\uc7ac\ud558\uba74 \\(T\\)\ub294 \ubb34\ud55c \ubaa8\ud615\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(\\sigma_n\\)\uc744 \u201c\uc11c\ub85c \ub2e4\ub978 \uc6d0\uc18c\uac00 \uc801\uc5b4\ub3c4 \\(n\\)\uac1c \uc874\uc7ac\ud55c\ub2e4\u201d\ub294 \ubb38\uc7a5<br \/>\n\\[<br \/>\n(\\exists x_1)\\cdots(\\exists x_n)<br \/>\n\\bigwedge_{1\\le i&lt;j\\le n}x_i\\ne x_j<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \uc9d1\ud569<br \/>\n\\[<br \/>\nT^*=T\\cup\\{\\sigma_n\\mid n\\ge1\\}<br \/>\n\\]<br \/>\n\uc744 \uc0dd\uac01\ud55c\ub2e4.<\/p>\n<p>\\(T^*\\)\uc758 \uc784\uc758\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569 \\(\\Delta\\)\ub97c \uc7a1\uc790. \\(\\Delta\\)\uc5d0 \\(\\sigma_n\\)\uc774 \ud558\ub098\ub77c\ub3c4 \ub098\ud0c0\ub098\uba74 \uadf8 \ucca8\uc790\ub4e4\uc758 \ucd5c\ub313\uac12\uc744 \\(N\\)\uc774\ub77c \ud558\uc790. \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \uc6d0\uc18c\uac00 \uc801\uc5b4\ub3c4 \\(N\\)\uac1c\uc778 \\(T\\)\uc758 \ubaa8\ud615\uc774 \uc874\uc7ac\ud558\uace0, \uc774 \ubaa8\ud615\uc740 \\(\\Delta\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(\\Delta\\)\uc5d0 \\(\\sigma_n\\)\uc774 \ud558\ub098\ub3c4 \ub098\ud0c0\ub098\uc9c0 \uc54a\uc73c\uba74 \uc6d0\uc18c\uac00 \uc801\uc5b4\ub3c4 \ud558\ub098\uc778 \\(T\\)\uc758 \ubaa8\ud615\uc744 \ud0dd\ud558\uba74 \ub41c\ub2e4. \ub530\ub77c\uc11c \\(T^*\\)\uc758 \ubaa8\ub4e0 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc740 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\ucf64\ud329\ud2b8\uc131 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(T^*\\)\ub294 \ubaa8\ud615\uc744 \uac16\ub294\ub2e4. \uc774 \ubaa8\ud615\uc740 \ubaa8\ub4e0 \\(\\sigma_n\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c \ubb34\ud55c\ud558\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 17.1.<\/span><br \/>\n\uc815\ub9ac 17.2\uc5d0\uc11c \uc0ac\uc6a9\ud55c \ubb38\uc7a5 \\(\\sigma_n\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\sigma_2,\\sigma_3,\\sigma_4\\)\ub97c \\(\\bigwedge\\) \uae30\ud638\ub97c \uc4f0\uc9c0 \uc54a\uace0 \ud480\uc5b4 \uc4f0\uc2dc\uc624.<\/li>\n<li>\ubaa8\ub4e0 \\(n\\ge1\\)\uc5d0 \ub300\ud558\uc5ec \\(\\sigma_{n+1}\\models\\sigma_n\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\ud55c \uad6c\uc870\uac00 \ubaa8\ub4e0 \\(\\sigma_n\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \uadf8 \uc601\uc5ed\uc774 \ubb34\ud55c\uc784\uc744 \uc815\uc758\uc5d0\uc11c \uc9c1\uc811 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 17.2.<\/span><br \/>\n\\(T_{\\mathrm{grp}}\\)\ub97c <a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\">14\uc7a5<\/a>\uc5d0\uc11c \uc801\uc740 \uad70 \uacf5\ub9ac\ub4e4\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(n\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac16\ub294 \uc21c\ud658\uad70\uc774 \uc874\uc7ac\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc0ac\uc6a9\ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc815\ub9ac 17.2\uc758 \uac00\uc815\uc774 \\(T_{\\mathrm{grp}}\\)\uc5d0 \ub300\ud574 \uc131\ub9bd\ud568\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\ucf64\ud329\ud2b8\uc131\uc73c\ub85c\ubd80\ud130 \\(T_{\\mathrm{grp}}\\)\uac00 \ubb34\ud55c \ubaa8\ud615\uc744 \uac00\uc9d0\uc744 \uacb0\ub860\ub0b4\ub9ac\uc2dc\uc624.<\/li>\n<li>\uc774 \ub17c\uc99d\uc5d0\uc11c \uac01 \\(\\sigma_n\\)\uc774 \ud558\ub294 \uc5ed\ud560\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 17.3.<\/span><br \/>\n\uc5b8\uc5b4 \\(\\mathcal L=\\{0,S,&lt;\\}\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\mathcal N=(\\mathbb N,0,S,&lt;)<br \/>\n\\]<br \/>\n\uc744 \uc0dd\uac01\ud558\uace0<br \/>\n\\[<br \/>\n\\overline n=S^n(0)<br \/>\n\\]<br \/>\n\uc73c\ub85c \uc4f0\uc790. \uc0c8 \uc0c1\uc218\uae30\ud638 \\(c\\)\ub97c \ucd94\uac00\ud55c \uc5b8\uc5b4\uc5d0\uc11c<br \/>\n\\[<br \/>\nT=\\operatorname{Th}(\\mathcal N)\\cup\\{\\overline n&lt;c\\mid n\\in\\mathbb N\\}<br \/>\n\\]<br \/>\n\ub85c \ub450\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(T\\)\uc758 \uc784\uc758\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc774 \ub9cc\uc871 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\ub54c \\(c\\)\ub97c \\(\\mathcal N\\)\uc758 \uc5b4\ub5a4 \uc6d0\uc18c\ub85c \ud574\uc11d\ud558\uba74 \ub418\ub294\uc9c0 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\ucf64\ud329\ud2b8\uc131 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uc5ec \\(T\\)\uc758 \ubaa8\ud615 \\(\\mathcal M\\)\uc774 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\mathcal M\\)\uc5d0\uc11c \\(c^{\\mathcal M}\\)\uc774 \ubaa8\ub4e0 \\(\\overline n^{\\mathcal M}\\)\ubcf4\ub2e4 \ud070 \uc6d0\uc18c\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc65c \uc774 \ubaa8\ud615\uc774 \uc6d0\ub798\uc758 \\(\\mathcal N\\)\uacfc \ub3d9\ud615\uc77c \uc218 \uc5c6\ub294\uac00?<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 17.4.<\/span><br \/>\n\ub4f1\ud638\ub9cc \uc788\ub294 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\ub97c \uc0dd\uac01\ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\n\\[<br \/>\nT_{\\infty}=\\{\\sigma_n\\mid n\\ge1\\}<br \/>\n\\]<br \/>\n\uc758 \ubaa8\ud615\uc774 \uc815\ud655\ud788 \ubb34\ud55c\ud55c \uad6c\uc870\ub4e4\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.\n<\/li>\n<li>\uc720\ud55c\ud55c \uad6c\uc870\ub4e4\ub9cc\uc744, \uadf8\ub9ac\uace0 \ubaa8\ub4e0 \uc720\ud55c\ud55c \uad6c\uc870\ub97c \uc815\ud655\ud788 \ubaa8\ud615\uc73c\ub85c \uac16\ub294 \ubb38\uc7a5 \uc9d1\ud569 \\(T_{\\mathrm{fin}}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(T_{\\mathrm{fin}}\\cup T_{\\infty}\\)\uc758 \ubaa8\ub4e0 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc774 \ub9cc\uc871 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\ucf64\ud329\ud2b8\uc131 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec (2)\uc758 \uac00\uc815\uc774 \ubaa8\uc21c\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \ub530\ub77c\uc11c \u201c\uc601\uc5ed\uc774 \uc720\ud55c\ud558\ub2e4\u201d\ub294 \uc131\uc9c8\uc740 \uc77c\uacc4\ub17c\ub9ac\uc758 \ubb38\uc7a5 \uc9d1\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uacf5\ub9ac\ud654\ud560 \uc218 \uc5c6\uc74c\uc744 \uacb0\ub860\ub0b4\ub9ac\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \ub610 \ub2e4\ub978 \uc911\uc694\ud55c \uc131\uc9c8\uc740 \ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac\uc774\ub2e4. \uba3c\uc800 \uac00\uc0b0 \uc5b8\uc5b4\uc5d0\uc11c \ud544\uc694\ud55c \ud615\ud0dc\ub97c \ubcf4\uc790.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 17.3. (\uac00\uc0b0 \ubaa8\ud615 \uc815\ub9ac)<\/span><\/p>\n<p>\uac00\uc0b0 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\uc758 \ub9cc\uc871 \uac00\ub2a5\ud55c \ubb38\uc7a5 \uc9d1\ud569\uc740 \uc6d0\uc18c\uac00 \uc720\ud55c \uac1c\uc774\uac70\ub098 \uac00\uc0b0\ubb34\ud55c \uac1c\uc778 \ubaa8\ud615\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n<a href=\"\/blog\/invitation-to-mathematical-logic\/ch16-inference-rule-first-order-logic\">\uc644\uc804\uc131 \uc815\ub9ac<\/a>\uc758 \ud56d \ubaa8\ud615 \uad6c\uc131\uc5d0\uc11c \ud655\uc7a5\ub41c \uc5b8\uc5b4\ub294 \uac00\uc0b0\uc774\uace0, \uc720\ud55c \ubb38\uc790\uc5f4\uc778 \ub2eb\ud78c\ud56d\ub3c4 \uac00\uc0b0 \uac1c\ubfd0\uc774\ub2e4. \ud56d \ubaa8\ud615\uc758 \uc601\uc5ed\uc740 \ub2eb\ud78c\ud56d\ub4e4\uc758 \ub3d9\uce58\ub958\ub85c \uc774\ub8e8\uc5b4\uc9c0\ubbc0\ub85c \ub9ce\uc544\uc57c \uac00\uc0b0\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 17.4. (\ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac: \uac00\uc0b0 \uc5b8\uc5b4\uc758 \uc774\ub860\ud615)<\/span><\/p>\n<p>\uac00\uc0b0 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\uc758 \ubb38\uc7a5 \uc9d1\ud569 \\(T\\)\uac00 \ubb34\ud55c \ubaa8\ud615\uc744 \uac00\uc9c0\uba74 \\(T\\)\ub294 \uac00\uc0b0\ubb34\ud55c \ubaa8\ud615\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uc704 \uc99d\uba85\uc5d0\uc11c \uc0ac\uc6a9\ud55c \ubb38\uc7a5 \\(\\sigma_n\\)\ub4e4\uc744 \ubaa8\ub450 \ucd94\uac00\ud55c<br \/>\n\\[<br \/>\nT^*=T\\cup\\{\\sigma_n\\mid n\\ge1\\}<br \/>\n\\]<br \/>\n\uc744 \uc0dd\uac01\ud55c\ub2e4. \uc8fc\uc5b4\uc9c4 \ubb34\ud55c \ubaa8\ud615\uc774 \\(T^*\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c \\(T^*\\)\ub294 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4. \uc815\ub9ac 17.3\uc5d0 \uc758\ud574 \\(T^*\\)\ub294 \ub9ce\uc544\uc57c \uac00\uc0b0\uc778 \ubaa8\ud615\uc744 \uac00\uc9c0\uba70, \ubaa8\ub4e0 \\(\\sigma_n\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c \uc720\ud55c\ud560 \uc218 \uc5c6\ub2e4. \ub530\ub77c\uc11c \uac00\uc0b0\ubb34\ud55c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc77c\ubc18\uc801\uc778 \ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac\ub294 \uc5b8\uc5b4\uc758 \ud06c\uae30\uac00 \uc784\uc758\uc758 \uae30\uc218\uc77c \ub54c\ub3c4 \uc131\ub9bd\ud55c\ub2e4. \ud2b9\ud788 \uc120\ud0dd\uacf5\ub9ac\uc640 \uc784\uc758 \ud06c\uae30\uc758 \uc5b8\uc5b4\uc5d0 \ub300\ud55c \ucf64\ud329\ud2b8\uc131 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uba74 \ub2e4\uc74c \uc704\ubc29\ud5a5 \ud615\ud0dc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 17.5. (\uc704\ubc29\ud5a5 \ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac)<\/span><\/p>\n<p>\uc77c\uacc4\ub17c\ub9ac \uc774\ub860 \\(T\\)\uac00 \ubb34\ud55c \ubaa8\ud615\uc744 \uac00\uc9c0\uba74, \ucda9\ubd84\ud788 \ud070 \uc784\uc758\uc758 \ubb34\ud55c \uae30\uc218 \\(\\kappa\\)\uc5d0 \ub300\ud574 \\(T\\)\ub294 \uae30\uc218\uac00 \uc801\uc5b4\ub3c4 \\(\\kappa\\)\uc778 \ubaa8\ud615\uc744 \uac00\uc9c4\ub2e4. \ub354 \uc815\ubc00\ud558\uac8c\ub294 \ud45c\uc900\uc801\uc778 \uc77c\ubc18\ud615\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\kappa\\ge\\max\\{|\\mathcal L|,\\aleph_0\\}<br \/>\n\\]<br \/>\n\uc774\uba74 \uae30\uc218\uac00 \\(\\kappa\\)\uc778 \ubaa8\ud615\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n<p>\uc704 \uc815\ub9ac\uc758 \ud45c\uc900 \uc99d\uba85\uc5d0\uc11c\ub294 \\(\\kappa\\)\uac1c\uc758 \uc0c8 \uc0c1\uc218\uae30\ud638 \\(c_\\alpha\\)\ub97c \ucd94\uac00\ud558\uace0<br \/>\n\\[<br \/>\nc_\\alpha\\ne c_\\beta<br \/>\n\\]<br \/>\n\ub77c\ub294 \ubb38\uc7a5\ub4e4\uc744 \ub123\uc740 \ub4a4 \uc77c\ubc18\ud615 \ucf64\ud329\ud2b8\uc131 \uc815\ub9ac\ub97c \uc801\uc6a9\ud55c\ub2e4. \uc815\ud655\ud788 \uae30\uc218 \\(\\kappa\\)\uc778 \ubaa8\ud615\uc744 \uc5bb\ub294 \ub9c8\uc9c0\ub9c9 \ub2e8\uacc4\uc5d0\ub294 \uc77c\ubc18\ud615 \uc544\ub798\ubc29\ud5a5 \ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac\ub97c \ud568\uaed8 \uc0ac\uc6a9\ud55c\ub2e4. \uc774 \uc77c\ubc18\ud615\ub4e4\uc758 \uc644\uc804\ud55c \uc99d\uba85\uc740 \uc774 \ucc45\uc758 \ubc94\uc704\ub97c \ub118\uc73c\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uc0ac\uc6a9\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\uc774 \uacb0\uacfc\ub4e4\uc740 \uc77c\uacc4\ub17c\ub9ac\uc758 \ud45c\ud604\ub825\uc5d0 \ubcf8\uc9c8\uc801\uc778 \ud55c\uacc4\uac00 \uc788\uc74c\uc744 \ubcf4\uc5ec\uc900\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \uac00\uc0b0 \uc5b8\uc5b4\ub85c \\(\\mathbb R\\)\uc744 \uae30\uc220\ud558\ub294 \uc77c\uacc4\ub17c\ub9ac \uc774\ub860\uc774 \ubb34\ud55c \ubaa8\ud615\uc744 \uac00\uc9c4\ub2e4\uba74 \uac00\uc0b0\ubb34\ud55c \ubaa8\ud615\ub3c4 \uac00\uc9c0\ubbc0\ub85c, \\(\\mathbb R\\)\uacfc \ub3d9\ud615\uc778 \uad6c\uc870\ub9cc\uc744 \uc720\uc77c\ud55c \ubaa8\ud615\uc73c\ub85c \uac16\ub294 \uc77c\uacc4\ub17c\ub9ac \uc774\ub860\uc740 \uc874\uc7ac\ud560 \uc218 \uc5c6\ub2e4. \uc774 \ud604\uc0c1\uc740 \uc77c\uacc4\ub17c\ub9ac\uc758 \uc57d\uc810\uc778 \ub3d9\uc2dc\uc5d0, \ub9e4\uc6b0 \uac15\ud55c \uc77c\ubc18 \ubaa8\ud615\uc774\ub860\uc744 \uac00\ub2a5\ud558\uac8c \ud558\ub294 \ud2b9\uc9d5\uc774\uae30\ub3c4 \ud558\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 17.5.<\/span><br \/>\n\uc720\ud55c\ud55c \uc5b8\uc5b4<br \/>\n\\[<br \/>\n\\mathcal L=\\{0,1,+,\\cdot,&lt;\\}<br \/>\n\\]<br \/>\n\uc5d0\uc11c \uc2e4\uc218\uc758 \uc21c\uc11c\uccb4 \uad6c\uc870<br \/>\n\\[<br \/>\n\\mathcal R=(\\mathbb R,0,1,+,\\cdot,&lt;)<br \/>\n\\]<br \/>\n\ub97c \uc0dd\uac01\ud558\uace0 \\(T=\\operatorname{Th}(\\mathcal R)\\)\ub85c \ub450\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\ubaa8\ub4e0 \\(n\\ge1\\)\uc5d0 \ub300\ud574 \\(\\sigma_n\\in T\\)\uc784\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\uac00\uc0b0 \ubaa8\ud615 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(T\\)\uac00 \uac00\uc0b0\ubb34\ud55c \ubaa8\ud615 \\(\\mathcal M\\)\uc744 \uac00\uc9d0\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\mathcal M\\)\uacfc \\(\\mathcal R\\)\uc774 \ub3d9\ud615\uc77c \uc218 \uc5c6\ub294 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\uadf8\ub7fc\uc5d0\ub3c4 \\(\\mathcal M\\models T\\)\uc774\ubbc0\ub85c \\(\\mathcal M\\)\uc740 \\(\\mathcal R\\)\uc5d0\uc11c \ucc38\uc778 \ubaa8\ub4e0 \\(\\mathcal L\\)-\ubb38\uc7a5\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774 \uc0ac\uc2e4\uc774 \uac00\uc0b0 \uc5b8\uc5b4\uc758 \uc77c\uacc4\ub17c\ub9ac \uc774\ub860\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(\\mathcal R\\)\uc744 \ub3d9\ud615\uae4c\uc9c0 \uc720\uc77c\ud558\uac8c \uaddc\uc815\ud560 \uc218 \uc5c6\ub2e4\ub294 \uacb0\ub860\uacfc \uc5b4\ub5bb\uac8c \uc5f0\uacb0\ub418\ub294\uc9c0 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-logic\/\">\uc9d1\ud569\uacfc \uc218\ub9ac\ub17c\ub9ac \uccab\uac78\uc74c \ubaa9\ucc28 \ubcf4\uae30<\/a><\/p>\n<p><span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch01-naive-logic\/\">\uba85\uc81c\uc640 \ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch02-sets\">\uc9d1\ud569\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch03-algebra-of-classes\">\ub2e4\uc591\ud55c \uc9d1\ud569\uc758 \uc5f0\uc0b0<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch04-relations-and-functions\">\uad00\uacc4\uc640 \ud568\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">\uc790\uc5f0\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">\uc9d1\ud569\uc758 \uae30\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch08-ordinal-numbers\">\uc9d1\ud569\uc758 \uc11c\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">\uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch10-axiom-of-choice\">\uc120\ud0dd \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch11-formal-logic\">\ud615\uc2dd\ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch12-propositional-logic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch13-soundness-completeness-proplogic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac74\uc804\uc131\uacfc \uc644\uc804\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch15-semantics-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch16-inference-rule-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucd94\ub860\uaddc\uce59<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch17-compactness-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch18-peano-arithmetics\">\ud398\uc544\ub178 \uc0b0\uc220<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch19-incompleteness-theorem\">\ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/a><\/span>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131\uc740 \ubb34\ud55c\ud55c \ubb38\uc7a5 \uc9d1\ud569\uc758 \ub9cc\uc871 \uac00\ub2a5\uc131\uc744 \uadf8 \uc720\ud55c\ud55c \ubd80\ubd84\ub4e4\ub9cc\uc73c\ub85c \ud310\uc815\ud560 \uc218 \uc788\ub2e4\ub294 \uc815\ub9ac\uc774\ub2e4. \uc815\ub9ac 17.1. (\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131) \\(\\mathcal L\\)\uc774 \uac00\uc0b0 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\uc774\uace0 \\(\\varSigma\\)\uac00 \\(\\mathcal L\\)-\ubb38\uc7a5\ub4e4\uc758 \uc9d1\ud569\uc774\uba74 \ub2e4\uc74c\uc740 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4. \\(\\varSigma\\)\ub294 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4. \\(\\varSigma\\)\uc758 \ubaa8\ub4e0 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc740 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4. \uc99d\uba85 (1)\uc5d0\uc11c (2)\ub294 \uc790\uba85\ud558\ub2e4. \ubc18\ub300\ub85c \ubaa8\ub4e0 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc774 \ub9cc\uc871 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \\(\\varSigma\\)\uac00 \ubb34\ubaa8\uc21c\uc774 \uc544\ub2c8\uba74 \uc5b4\ub5a4 \ubb38\uc7a5 \\(\\phi\\)\uc5d0 \ub300\ud574 \\( \\varSigma\\vdash\\phi \\,\\text{\uc640}\\, \\varSigma\\vdash\\neg\\phi \\) \uc778 \uc720\ud55c\ud55c \ub450 \uc99d\uba85\uc774 \uc874\uc7ac\ud55c\ub2e4.&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":117,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9282","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9282","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=9282"}],"version-history":[{"count":7,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9282\/revisions"}],"predecessor-version":[{"id":10097,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9282\/revisions\/10097"}],"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=9282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}