{"id":9278,"date":"2025-10-17T20:22:48","date_gmt":"2025-10-17T11:22:48","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9278"},"modified":"2026-09-27T12:28:01","modified_gmt":"2026-09-27T03:28:01","slug":"ch15-semantics-first-order-logic","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch15-semantics-first-order-logic\/","title":{"rendered":"\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>15. \uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/h2>\n\n --><\/p>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860\uc740 \uae30\ud638\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc2dd\uc744 \uc815\uc758\ud55c\ub2e4. \uc758\ubbf8\ub860\uc740 \uc774 \uae30\ud638\ub4e4\uc744 \uc2e4\uc81c \uc9d1\ud569, \ud568\uc218, \uad00\uacc4\ub85c \ud574\uc11d\ud558\uace0 \ub17c\ub9ac\uc2dd\uc774 \uc5b8\uc81c \ucc38\uc778\uc9c0\ub97c \uc815\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 15.1. (\\(\\mathcal L\\)-\uad6c\uc870)<\/span><\/p>\n<p><span class=\"defined\">\\(\\mathcal L\\)-\uad6c\uc870<\/span>(\\(\\mathcal L\\)-structure) \\(\\mathcal M\\)\uc740 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569 \\(M\\)\uacfc \ub2e4\uc74c \ud574\uc11d\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\uac01 \\(n\\)\ud56d \ud568\uc218\uae30\ud638 \\(f\\)\uc5d0 \ud568\uc218<br \/>\n\\[<br \/>\nf^{\\mathcal M}\\colon M^n\\to M,<br \/>\n\\]\n<\/li>\n<li>\uac01 \\(n\\)\ud56d \uad00\uacc4\uae30\ud638 \\(R\\)\uc5d0 \uad00\uacc4<br \/>\n\\[<br \/>\nR^{\\mathcal M}\\subseteq M^n,<br \/>\n\\]\n<\/li>\n<li>\uac01 \uc0c1\uc218\uae30\ud638 \\(c\\)\uc5d0 \uc6d0\uc18c<br \/>\n\\[<br \/>\nc^{\\mathcal M}\\in M.<br \/>\n\\]\n<\/li>\n<\/ul>\n<p>\uc5ec\uae30\uc11c \\(M\\)\uc744 \\(\\mathcal M\\)\uc758 <span class=\"defined\">\uc601\uc5ed<\/span>(domain)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub4f1\ud638\ub294 \ud56d\uc0c1 \\(M\\)\uc5d0\uc11c\uc758 \uc2e4\uc81c \ub3d9\uc77c\uc131\uc73c\ub85c \ud574\uc11d\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\ubcc0\uc218\uc5d0 \uc601\uc5ed\uc758 \uc6d0\uc18c\ub97c \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218<br \/>\n\\[<br \/>\ns\\colon\\{x_0,x_1,x_2,\\ldots\\}\\to M<br \/>\n\\]<br \/>\n\uc744 <span class=\"defined\">\ubcc0\uc218 \uac12\ub9e4\uae40<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(a\\in M\\)\uc77c \ub54c \\(s[x\\mapsto a]\\)\ub294 \\(x\\)\uc5d0\ub9cc \\(a\\)\ub97c \ub300\uc751\uc2dc\ud0a4\uace0 \ub2e4\ub978 \ubcc0\uc218\uc5d0\uc11c\ub294 \\(s\\)\uc640 \uac19\uc740 \uac12\ub9e4\uae40\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 15.2. (\ud56d\uc758 \ud574\uc11d)<\/span><\/p>\n<p>\uad6c\uc870 \\(\\mathcal M\\)\uacfc \ubcc0\uc218 \uac12\ub9e4\uae40 \\(s\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \ud56d \\(t\\)\uc758 \uac12\uc744 \\(t^{\\mathcal M}[s]\\)\ub85c \uc4f0\uace0 \ub2e4\uc74c\uacfc \uac19\uc774 \uc7ac\uadc0\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\\[<br \/>\n\\begin{aligned}<br \/>\nx^{\\mathcal M}[s]&#038;=s(x),\\\\[3pt]<br \/>\nc^{\\mathcal M}[s]&#038;=c^{\\mathcal M},\\\\[3pt]<br \/>\nf(t_1,\\ldots,t_n)^{\\mathcal M}[s]<br \/>\n&#038;=f^{\\mathcal M}\\bigl(t_1^{\\mathcal M}[s],\\ldots,t_n^{\\mathcal M}[s]\\bigr).<br \/>\n\\end{aligned}<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 15.1.<\/span><br \/>\n\uc5b8\uc5b4 \\(\\mathcal L=\\{c,f,g,R\\}\\)\uc5d0\uc11c \\(c\\)\ub294 \uc0c1\uc218\uae30\ud638, \\(f\\)\ub294 1\ud56d \ud568\uc218\uae30\ud638, \\(g\\)\ub294 2\ud56d \ud568\uc218\uae30\ud638, \\(R\\)\uc740 2\ud56d \uad00\uacc4\uae30\ud638\ub77c\uace0 \ud558\uc790. \uc601\uc5ed\uc774<br \/>\n\\[<br \/>\nM=\\{0,1,2,3,4\\}<br \/>\n\\]<br \/>\n\uc778 \uad6c\uc870 \\(\\mathcal M\\)\uc744<br \/>\n\\[<br \/>\nc^{\\mathcal M}=1,\\quad<br \/>\nf^{\\mathcal M}(a)=a+1\\pmod 5,\\quad<br \/>\ng^{\\mathcal M}(a,b)=a+b\\pmod 5<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud558\uc790. \uac12\ub9e4\uae40 \\(s\\)\uac00 \\(s(x)=2\\), \\(s(y)=4\\)\ub97c \ub9cc\uc871\uc2dc\ud0ac \ub54c \ub2e4\uc74c \ud56d\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(c\\).<\/li>\n<li>\\(f(x)\\).<\/li>\n<li>\\(g(x,c)\\).<\/li>\n<li>\\(f(g(x,c))\\).<\/li>\n<li>\\(g(f(x),f(y))\\).<\/li>\n<\/ol>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 15.3. (\ub9cc\uc871\uad00\uacc4)<\/span><\/p>\n<p>\uad6c\uc870 \\(\\mathcal M\\), \uac12\ub9e4\uae40 \\(s\\), \ub17c\ub9ac\uc2dd \\(\\phi\\)\uc5d0 \ub300\ud558\uc5ec \u201c\\(\\phi\\)\uac00 \\((\\mathcal M,s)\\)\uc5d0\uc11c \ucc38\uc774\ub2e4\u201d\ub97c<br \/>\n\\[<br \/>\n\\mathcal M,s\\models\\phi<br \/>\n\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc871\uad00\uacc4\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc7ac\uadc0\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\\[<br \/>\n\\begin{aligned}<br \/>\n\\mathcal M,s\\models R(t_1,\\ldots,t_n)<br \/>\n&#038;\\quad\\Longleftrightarrow\\quad<br \/>\n\\bigl(t_1^{\\mathcal M}[s],\\ldots,t_n^{\\mathcal M}[s]\\bigr)\\in R^{\\mathcal M},\\\\[3pt]<br \/>\n\\mathcal M,s\\models t_1=t_2<br \/>\n&#038;\\quad\\Longleftrightarrow\\quad<br \/>\nt_1^{\\mathcal M}[s]=t_2^{\\mathcal M}[s],\\\\[3pt]<br \/>\n\\mathcal M,s\\models\\neg\\phi<br \/>\n&#038;\\quad\\Longleftrightarrow\\quad<br \/>\n\\mathcal M,s\\not\\models\\phi,\\\\[3pt]<br \/>\n\\mathcal M,s\\models\\phi\\to\\psi<br \/>\n&#038;\\quad\\Longleftrightarrow\\quad<br \/>\n\\mathcal M,s\\not\\models\\phi\\text{ \ub610\ub294 }\\mathcal M,s\\models\\psi,\\\\[3pt]<br \/>\n\\mathcal M,s\\models(\\forall x)\\phi<br \/>\n&#038;\\quad\\Longleftrightarrow\\quad<br \/>\n\\text{\ubaa8\ub4e0 }a\\in M\\text{\uc5d0 \ub300\ud558\uc5ec }\\mathcal M,s[x\\mapsto a]\\models\\phi.<br \/>\n\\end{aligned}<br \/>\n\\]<\/p>\n<p>\uc57d\uc5b4\ub85c \uc815\uc758\ud55c \\(\\wedge,\\vee,\\leftrightarrow,\\exists\\)\ub294 \uc774\uc5d0 \ub530\ub77c \ud1b5\uc0c1\uc801\uc778 \uc758\ubbf8\ub97c \uac00\uc9c4\ub2e4. \ud2b9\ud788<br \/>\n\\[<br \/>\n\\mathcal M,s\\models(\\exists x)\\phi<br \/>\n\\quad\\Longleftrightarrow\\quad<br \/>\n\\text{\uc5b4\ub5a4 }a\\in M\\text{\uc5d0 \ub300\ud558\uc5ec }\\mathcal M,s[x\\mapsto a]\\models\\phi.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 15.2.<\/span><br \/>\n\ubb38\uc81c 15.1\uc758 \uad6c\uc870\uc5d0\uc11c<br \/>\n\\[<br \/>\nR^{\\mathcal M}=\\{(a,b)\\in M^2:a&lt;b\\}<br \/>\n\\]<br \/>\n\ub85c \ub450\uace0, \uc5ed\uc2dc \\(s(x)=2\\), \\(s(y)=4\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \uba85\uc81c\uac00 \ucc38\uc778\uc9c0 \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\mathcal M,s\\models R(x,y)\\).<\/li>\n<li>\\(\\mathcal M,s\\models R(f(x),y)\\).<\/li>\n<li>\\(\\mathcal M,s\\models(\\exists z)R(z,f(z))\\).<\/li>\n<li>\\(\\mathcal M,s\\models(\\forall z)R(z,f(z))\\).<\/li>\n<li>\\(\\mathcal M,s\\models(\\forall z)(\\exists w)R(z,w)\\).<\/li>\n<\/ol>\n<p>\uac70\uc9d3\uc778 \uacbd\uc6b0\uc5d0\ub294 \ubc18\ub840\uac00 \ub418\ub294 \uc601\uc5ed\uc758 \uc6d0\uc18c\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 15.3.<\/span><br \/>\n\uc601\uc5ed\uc774 \\(M=\\{0,1,2\\}\\)\uc778 \uad6c\uc870\uc5d0\uc11c 2\ud56d \uad00\uacc4 \\(R\\)\uc744<br \/>\n\\[<br \/>\nR^{\\mathcal M}=\\{(0,0),(1,2),(2,1)\\}<br \/>\n\\]<br \/>\n\ub85c \ud574\uc11d\ud558\uc790. \ub2e4\uc74c \ubb38\uc7a5\uc758 \uc9c4\ub9bf\uac12\uc744 \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\((\\forall x)(\\exists y)R(x,y)\\).<\/li>\n<li>\\((\\exists y)(\\forall x)R(x,y)\\).<\/li>\n<li>\\((\\forall y)(\\exists x)R(x,y)\\).<\/li>\n<li>\\((\\exists x)(\\forall y)R(x,y)\\).<\/li>\n<\/ol>\n<p>\ucc38\uc778 \uc874\uc7ac\uba85\uc81c\uc5d0\uc11c\ub294 \uac01 \uacbd\uc6b0\uc758 \uc99d\uc778\uc744 \uc801\uace0, \ud55c\uc815\uae30\ud638\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\uba74 \uc9c4\ub9bf\uac12\uc774 \ub2ec\ub77c\uc9c8 \uc218 \uc788\ub294 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ub2e4\uc74c \ubcf4\uc870\uc815\ub9ac\ub294 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\">14\uc7a5<\/a>\uc5d0\uc11c \uc815\uc758\ud55c \uce58\ud658\uacfc \uc758\ubbf8\ub860\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud55c\ub2e4. \uc774 \ubcf4\uc870\uc815\ub9ac\ub294 \ub4a4\uc5d0\uc11c \uac74\uc804\uc131 \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \ub54c \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\ubcf4\uc870\uc815\ub9ac 15.4. (\uce58\ud658 \ubcf4\uc870\uc815\ub9ac)<\/span><\/p>\n<p>\ud56d \\(u,t\\), \ubcc0\uc218 \\(x\\), \uad6c\uc870 \\(\\mathcal M\\), \uac12\ub9e4\uae40 \\(s\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n(u[t\/x])^{\\mathcal M}[s]<br \/>\n=u^{\\mathcal M}\\bigl[s[x\\mapsto t^{\\mathcal M}[s]]\\bigr].<br \/>\n\\]<br \/>\n\ub610 \\(t\\)\uac00 \ub17c\ub9ac\uc2dd \\(\\phi\\)\uc5d0\uc11c \\(x\\)\uc5d0 \ub300\ud574 \uc790\uc720\ub86d\uac8c \ub300\uc785 \uac00\ub2a5\ud558\uba74<br \/>\n\\[<br \/>\n\\mathcal M,s\\models\\phi[t\/x]<br \/>\n\\quad\\Longleftrightarrow\\quad<br \/>\n\\mathcal M,s[x\\mapsto t^{\\mathcal M}[s]]\\models\\phi.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab\uc9f8 \uc2dd\uc740 \ud56d\uc758 \uad6c\uc131\uc5d0 \ub300\ud55c \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud55c\ub2e4. \ub458\uc9f8 \uc2dd\uc740 \ub17c\ub9ac\uc2dd\uc758 \uad6c\uc870\uc5d0 \ub300\ud55c \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uba70, \uc804\uce6d \ud55c\uc815\uae30\ud638 \ub2e8\uacc4\uc5d0\uc11c \u201c\uc790\uc720\ub86d\uac8c \ub300\uc785 \uac00\ub2a5\u201d \uc870\uac74\uc774 \ubcc0\uc218 \ud3ec\ud68d\uc774 \uc77c\uc5b4\ub098\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc7a5\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub2e4\uc74c \ubcf4\uc870\uc815\ub9ac\ub294 \ubb38\uc7a5\uc5d0\uc11c \ubcc0\uc218 \uac12\ub9e4\uae40\uc744 \ub530\ub85c \ud45c\uc2dc\ud560 \ud544\uc694\uac00 \uc5c6\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc124\uba85\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\ubcf4\uc870\uc815\ub9ac 15.5. (\uc790\uc720\ubcc0\uc218 \ubcf4\uc870\uc815\ub9ac)<\/span><\/p>\n<p>\ub17c\ub9ac\uc2dd \\(\\phi\\)\uc758 \ubaa8\ub4e0 \uc790\uc720\ubcc0\uc218\uc5d0\uc11c \ub450 \uac12\ub9e4\uae40 \\(s,s&#8217;\\)\uac00 \uac19\uc740 \uac12\uc744 \uac00\uc9c0\uba74<br \/>\n\\[<br \/>\n\\mathcal M,s\\models\\phi<br \/>\n\\quad\\Longleftrightarrow\\quad<br \/>\n\\mathcal M,s&#8217;\\models\\phi.<br \/>\n\\]<br \/>\n\ud2b9\ud788 \\(\\phi\\)\uac00 \ubb38\uc7a5\uc774\uba74 \uadf8 \uc9c4\ub9bf\uac12\uc740 \ubcc0\uc218 \uac12\ub9e4\uae40\uacfc \ubb34\uad00\ud558\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\ud56d\uacfc \ub17c\ub9ac\uc2dd\uc758 \uad6c\uc131\uc5d0 \ub300\ud55c \uad6c\uc870\uc801 \uadc0\ub0a9\ubc95\uc744 \uc801\uc6a9\ud55c\ub2e4. \uc544\ud1b0\ub17c\ub9ac\uc2dd\uc758 \uacbd\uc6b0 \ud56d\uc758 \uac12\uc774 \uadf8 \ud56d\uc5d0 \uc2e4\uc81c\ub85c \ub098\ud0c0\ub098\ub294 \ubcc0\uc218\uc758 \uac12\uc5d0\ub9cc \uc758\uc874\ud55c\ub2e4. \\(\\neg\\)\uc640 \\(\\to\\)\uc758 \uacbd\uc6b0\uc5d0\ub294 \uadc0\ub0a9\uac00\uc815\uc744 \ubc14\ub85c \uc801\uc6a9\ud558\uba74 \ub41c\ub2e4. \\((\\forall x)\\psi\\)\uc758 \uacbd\uc6b0 \ub450 \uac12\ub9e4\uae40\uc744 \uac01\uac01 \\(x\\)\uc5d0\uc11c \uac19\uc740 \\(a\\in M\\)\ub85c \ubc14\uafb8\uba74 \\(\\psi\\)\uc758 \uc790\uc720\ubcc0\uc218\uc5d0\uc11c \uc5ec\uc804\ud788 \uc77c\uce58\ud558\ubbc0\ub85c \uadc0\ub0a9\uac00\uc815\uc744 \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ubb38\uc7a5 \\(\\phi\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(\\mathcal M,s\\models\\phi\\)\uac00 \uc5b4\ub5a4 \ud558\ub098\uc758 \uac12\ub9e4\uae40\uc5d0\uc11c \uc131\ub9bd\ud558\uba74 \ubaa8\ub4e0 \uac12\ub9e4\uae40\uc5d0\uc11c \uc131\ub9bd\ud55c\ub2e4. \uc774\ub54c<br \/>\n\\[<br \/>\n\\mathcal M\\models\\phi<br \/>\n\\]<br \/>\n\ub85c \uc4f4\ub2e4. \ubb38\uc7a5\ub4e4\uc758 \uc9d1\ud569 \\(\\varSigma\\)\uc5d0 \ub300\ud574 \ubaa8\ub4e0 \\(\\sigma\\in\\varSigma\\)\uc5d0 \\(\\mathcal M\\models\\sigma\\)\uc774\uba74 \u201c\\(\\mathcal M\\)\uc774 \\(\\varSigma\\)\uc758 <span class=\"defined\">\ubaa8\ud615<\/span>\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0<br \/>\n\\[<br \/>\n\\mathcal M\\models\\varSigma<br \/>\n\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774\ub7ec\ud55c \ubaa8\ud615\uc774 \uc874\uc7ac\ud558\uba74 \u201c\\(\\varSigma\\)\uac00 <span class=\"defined\">\ub9cc\uc871 \uac00\ub2a5<\/span>\ud558\ub2e4\u201d\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 15.6. (\ub17c\ub9ac\uc801 \uadc0\uacb0\uacfc \uc774\ub860)<\/span><\/p>\n<p>\ubb38\uc7a5\ub4e4\uc758 \uc9d1\ud569 \\(\\varSigma\\)\uc640 \ubb38\uc7a5 \\(\\phi\\)\uc5d0 \ub300\ud558\uc5ec \ubaa8\ub4e0 \\(\\varSigma\\)\uc758 \ubaa8\ud615\uc774 \\(\\phi\\)\ub3c4 \ub9cc\uc871\uc2dc\ud0a4\uba74<br \/>\n\\[<br \/>\n\\varSigma\\models\\phi<br \/>\n\\]<br \/>\n\ub77c\uace0 \uc4f0\uace0 \\(\\phi\\)\ub97c \\(\\varSigma\\)\uc758 <span class=\"defined\">\ub17c\ub9ac\uc801 \uadc0\uacb0<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\varSigma=\\varnothing\\)\uc77c \ub54c<br \/>\n\\[<br \/>\n\\models\\phi<br \/>\n\\]<br \/>\n\ub77c\uace0 \uc4f0\uba70, \uc774 \uacbd\uc6b0 \u201c\\(\\phi\\)\uac00 <span class=\"defined\">\ub17c\ub9ac\uc801\uc73c\ub85c \uc720\ud6a8<\/span>\ud558\ub2e4(logically valid)\u201d\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4.<\/p>\n<p>\ud55c \uad6c\uc870 \\(\\mathcal M\\)\uc5d0\uc11c \ucc38\uc778 \ubaa8\ub4e0 \\(\\mathcal L\\)-\ubb38\uc7a5\uc758 \uc9d1\ud569\uc744 \\(\\mathcal M\\)\uc758 <span class=\"defined\">\uc774\ub860<\/span>(theory)\uc774\ub77c\uace0 \ubd80\ub974\uace0<br \/>\n\\[<br \/>\n\\operatorname{Th}(\\mathcal M)<br \/>\n\\]<br \/>\n\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 15.4.<\/span><br \/>\n\uc5b8\uc5b4\uac00 \ud558\ub098\uc758 2\ud56d \uad00\uacc4\uae30\ud638 \\(&lt;\\)\ub9cc\uc744 \uac16\ub294\ub2e4\uace0 \ud558\uc790. \ub2e4\uc74c \uc138 \uad6c\uc870\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\[<br \/>\n\\mathcal N=(\\mathbb N,&lt;),\\quad<br \/>\n\\mathcal Z=(\\mathbb Z,&lt;),\\quad<br \/>\n\\mathcal F=(\\{0,1,2\\},&lt;).<br \/>\n\\]<br \/>\n\uc5ec\uae30\uc11c \\(&lt;\\)\ub294 \ubaa8\ub450 \ud1b5\uc0c1\uc801\uc778 \uc21c\uc11c\uc774\ub2e4. \ub2e4\uc74c \ubb38\uc7a5 \uac00\uc6b4\ub370 \uac01 \uad6c\uc870\uc758 \uc774\ub860\uc5d0 \uc18d\ud558\ub294 \uac83\uc744 \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n<p>\\[<br \/>\n\\begin{aligned}<br \/>\n\\alpha&#038;=(\\forall x)\\neg(x&lt;x),\\\\[3pt]<br \/>\n\\beta&#038;=(\\forall x)(\\exists y)(x&lt;y),\\\\[3pt]<br \/>\n\\gamma&#038;=(\\exists x)(\\forall y)\\neg(y&lt;x),\\\\[3pt]<br \/>\n\\delta&#038;=(\\forall x)(\\forall y)\\bigl((x&lt;y)\\vee(x=y)\\vee(y&lt;x)\\bigr).<br \/>\n\\end{aligned}<br \/>\n\\]<\/p>\n<p>\uc989 \\(\\alpha,\\beta,\\gamma,\\delta\\) \uac00\uc6b4\ub370 \uc5b4\ub290 \uac83\uc774 \\(\\operatorname{Th}(\\mathcal N)\\), \\(\\operatorname{Th}(\\mathcal Z)\\), \\(\\operatorname{Th}(\\mathcal F)\\)\uc5d0 \uc18d\ud558\ub294\uc9c0 \uac01\uac01 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc608\ub97c \ub4e4\uc5b4 \\(\\mathcal L_{\\mathrm{grp}}\\)-\uad6c\uc870 \\(\\mathcal M\\)\uc774 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\">14\uc7a5<\/a>\uc5d0 \uc801\uc740 \uc138 \uad70 \uacf5\ub9ac\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74, \\(\\mu^{\\mathcal M}\\)\uc744 \uc5f0\uc0b0\uc73c\ub85c \ud558\ub294 \\(M\\)\uc740 \uad70\uc774\uba70 \\(\\epsilon^{\\mathcal M}\\)\uc740 \uadf8 \ud56d\ub4f1\uc6d0, \\(\\iota^{\\mathcal M}\\)\ub294 \uc5ed\uc6d0 \ud568\uc218\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 15.5.<\/span><br \/>\n1\ud56d \uad00\uacc4\uae30\ud638 \\(P,Q\\)\uac00 \uc788\ub294 \uc5b8\uc5b4\ub97c \uc0dd\uac01\ud558\uc790. \ub17c\ub9ac\uc801 \uadc0\uacb0\uacfc \ub17c\ub9ac\uc801 \uc720\ud6a8\uc131\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\{(\\forall x)(P(x)\\to Q(x)),\\;(\\forall x)P(x)\\}\\models(\\forall x)Q(x)\\)\ub97c \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\{(\\exists x)P(x)\\}\\not\\models(\\forall x)P(x)\\)\uc784\uc744 \ub450 \uc6d0\uc18c \uad6c\uc870\ub97c \uc0ac\uc6a9\ud558\uc5ec \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\models(\\forall x)(x=x)\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(\\not\\models(\\exists x)(\\forall y)(x=y)\\)\uc784\uc744 \ubcf4\uc774\ub294 \uad6c\uc870\ub97c \ud558\ub098 \uc81c\uc2dc\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 \uad6c\ubb38\ub860\uc740 \uae30\ud638\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc2dd\uc744 \uc815\uc758\ud55c\ub2e4. \uc758\ubbf8\ub860\uc740 \uc774 \uae30\ud638\ub4e4\uc744 \uc2e4\uc81c \uc9d1\ud569, \ud568\uc218, \uad00\uacc4\ub85c \ud574\uc11d\ud558\uace0 \ub17c\ub9ac\uc2dd\uc774 \uc5b8\uc81c \ucc38\uc778\uc9c0\ub97c \uc815\ud55c\ub2e4. \uc815\uc758 15.1. (\\(\\mathcal L\\)-\uad6c\uc870) \\(\\mathcal L\\)-\uad6c\uc870(\\(\\mathcal L\\)-structure) \\(\\mathcal M\\)\uc740 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569 \\(M\\)\uacfc \ub2e4\uc74c \ud574\uc11d\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4\ub2e4. \uac01 \\(n\\)\ud56d \ud568\uc218\uae30\ud638 \\(f\\)\uc5d0 \ud568\uc218 \\( f^{\\mathcal M}\\colon M^n\\to M, \\) \uac01 \\(n\\)\ud56d \uad00\uacc4\uae30\ud638 \\(R\\)\uc5d0 \uad00\uacc4 \\( R^{\\mathcal M}\\subseteq M^n, \\) \uac01 \uc0c1\uc218\uae30\ud638 \\(c\\)\uc5d0 \uc6d0\uc18c \\( c^{\\mathcal M}\\in M. \\)&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":115,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9278","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9278","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=9278"}],"version-history":[{"count":8,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9278\/revisions"}],"predecessor-version":[{"id":10092,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9278\/revisions\/10092"}],"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=9278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}