{"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":"2025-10-20T18:49:25","modified_gmt":"2025-10-20T09:49:25","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 \uc911\uc694\ud55c \ud2b9\uc131 \uc911 \ud558\ub098\ub294 \ucf64\ud329\ud2b8\uc131\uc774\ub2e4. \uc774\uac83\uc740 \uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc720\ud55c\uc131\uacfc \uad00\ub828\ub41c \uc911\uc694\ud55c \uc131\uc9c8\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>\\(\\varSigma\\)\uac00 \uac00\uc0b0\uc778 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\uc758 \ubb38\uc7a5\uc758 \ubaa8\uc784\uc774\uace0 \\(\\varSigma\\)\uc758 \uc784\uc758\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc774 \ubaa8\ub378\uc744 \uac00\uc9c0\uba74 \\(\\varSigma\\) \uc790\uccb4\ub3c4 \ubaa8\ub378\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\uc644\uc804\uc131 \uc815\ub9ac\uc5d0 \uc758\ud558\uba74, \\(\\varSigma\\)\uac00 \ubaa8\ub378\uc744 \uac00\uc9c8 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\varSigma\\)\uac00 \ubb34\ubaa8\uc21c\uc778 \uac83\uc774\ub2e4. \ub9cc\uc57d \\(\\varSigma\\)\uac00 \ubaa8\uc21c\uc744 \ud3ec\ud568\ud55c\ub2e4\uba74, \uadf8 \ubaa8\uc21c\uc740 \uc720\ud55c\ud55c \uc99d\uba85\uc5d0 \uc758\ud574 \ub3c4\ucd9c\ub418\ubbc0\ub85c \\(\\varSigma\\)\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc73c\ub85c\ubd80\ud130 \ubaa8\uc21c\uc774 \ub3c4\ucd9c\ub41c\ub2e4. \uadf8\ub7ec\ub098 \uac00\uc815\uc5d0 \uc758\ud574 \ubaa8\ub4e0 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc740 \ubaa8\ub378\uc744 \uac00\uc9c0\ubbc0\ub85c \ubaa8\uc21c\uc744 \ud3ec\ud568\ud560 \uc218 \uc5c6\ub2e4. \ub530\ub77c\uc11c \\(\\varSigma\\)\ub294 \ubb34\ubaa8\uc21c\uc774\uace0, \uc644\uc804\uc131 \uc815\ub9ac\uc5d0 \uc758\ud574 \ubaa8\ub378\uc744 \uac00\uc9c4\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ucf64\ud329\ud2b8\uc131 \uc815\ub9ac\ub294 \uc218\ud559\uc758 \ub2e4\uc591\ud55c \uc0c1\ud669\uc5d0\uc11c \ud65c\uc6a9\ub41c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc720\ud55c \ud3c9\uba74\uc9c0\ub3c4\uc758 \uc0ac\uc0c9 \uc815\ub9ac\ub85c\ubd80\ud130 \uc784\uc758\uc758 \ud3c9\uba74\uc9c0\ub3c4(\ubb34\ud55c \ud3ec\ud568)\uc5d0 \ub300\ud55c \uc0ac\uc0c9 \uc815\ub9ac\ub97c \uc99d\uba85\ud558\ub294 \ub370 \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \ub610 \ub2e4\ub978 \uc911\uc694\ud55c \ud2b9\uc131\uc740 \ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 17.2. (\ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac)<\/span><\/p>\n<p>\\(\\varSigma\\)\uac00 \uac00\uc0b0\uc778 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\uc758 \ubb38\uc7a5\uc758 \ubaa8\uc784\uc774\uace0 \ubaa8\ub378\uc744 \uac00\uc9c0\uba74 \\(\\varSigma\\)\ub294 \uc720\ud55c\uc774\uac70\ub098 \uac00\uc0b0\uc778 \ubaa8\ub378\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\\(\\varSigma\\)\uac00 \ubb34\ubaa8\uc21c\uc774\ub77c\uba74, \uc644\uc804\uc131 \uc815\ub9ac\uc758 \uc99d\uba85\uc5d0\uc11c \uc0ac\uc6a9\ud55c \ubc29\ubc95\uc73c\ub85c \\(\\varSigma\\)\uc758 \ubaa8\ub378\uc744 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ol class=\"parenthesis\" style=\"margin-left: 3em;\">\n<li>\\(\\varSigma\\)\uc5d0 \uac00\uc0b0 \uac1c\uc758 \uc0c8\ub85c\uc6b4 \uc0c1\uc218\uae30\ud638\ub97c \ucd94\uac00\ud558\uc5ec \uc644\uc804\ud55c \uc9d1\ud569 \\(T\\)\ub97c \uad6c\uc131\ud55c\ub2e4.<\/li>\n<li>\uc5b8\uc5b4\uc758 \ub2eb\ud78c\ud56d\ub4e4\ub85c \uad6c\uc131\ub41c \uad6c\uc870 \\(N\\)\uc744 \uad6c\uc131\ud55c\ub2e4.<\/li>\n<li>\\(N\\)\uc5d0\uc11c \uc801\uc808\ud55c \ub3d9\uce58\uad00\uacc4\ub97c \uc815\uc758\ud558\uace0 \uc0c1\uc9d1\ud569\uc744 \ucde8\ud558\uc5ec \ucd5c\uc885 \uad6c\uc870\ub97c \ub9cc\ub4e0\ub2e4.<\/li>\n<\/ol>\n<p>\uc774\ub807\uac8c \uad6c\uc131\ub41c \ubaa8\ub378\uc740 \uac00\uc0b0\uc774\uba70 \\(\\varSigma\\)\uc758 \ubaa8\ub378\uc774 \ub41c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc989 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\ub85c \ud45c\ud604 \uac00\ub2a5\ud55c \uc774\ub860\uc774 \ubb34\ud55c \ubaa8\ub378\uc744 \uac00\uc9c4\ub2e4\uba74, \uadf8 \uc774\ub860\uc740 \uac00\uc0b0 \ubb34\ud55c \ubaa8\ub378\ub3c4 \uac00\uc9c4\ub2e4.<\/p>\n<p>\ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac\uc758 \ud655\uc7a5\uc73c\ub85c \ub2e4\uc74c \uc815\ub9ac\ub3c4 \uc54c\ub824\uc838 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 17.3. (\uc704\ubc29\ud5a5 \ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac)<\/span><\/p>\n<p>\\(\\varSigma\\)\uac00 \ubb38\uc7a5\uc758 \uc9d1\ud569\uc774\uace0 \ubb34\ud55c \ubaa8\ub378\uc744 \uac00\uc9c4\ub2e4\uba74, \\(\\varSigma\\)\ub294 \uc784\uc758\uc758 \ubb34\ud55c \uae30\uc218\ubcf4\ub2e4 \ud070 \ubaa8\ub378\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\ub4e4\uc740 \uc77c\uacc4\ub17c\ub9ac\uc758 \ub3c5\ud2b9\ud55c \uc131\uc9c8\uc744 \ubcf4\uc5ec\uc900\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc2e4\uc218\uccb4 \\(\\mathbb{R}\\)\ub9cc\uc744 \uc720\uc77c\ud55c \ubaa8\ub378\ub85c \uac00\uc9c0\ub294 \uc77c\uacc4\ub17c\ub9ac \uacf5\ub9ac \uccb4\uacc4\ub294 \uc874\uc7ac\ud560 \uc218 \uc5c6\ub2e4. \uc65c\ub0d0\ud558\uba74 \ub8b0\ubca4\ud558\uc784-\uc2a4\ucf5c\ub818 \uc815\ub9ac\uc5d0 \uc758\ud574 \uadf8\ub7ec\ud55c \uacf5\ub9ac \uccb4\uacc4\ub294 \uac00\uc0b0 \ubaa8\ub378\ub3c4 \uac00\uc838\uc57c \ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-logic\/\">\uc9d1\ud569\uacfc \uc218\ub9ac\ub17c\ub9ac \uccab\uac78\uc74c \ubaa9\ucc28 \ubcf4\uae30<\/a><\/p>\n<p><span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch01-naive-logic\/\">\uba85\uc81c\uc640 \ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch02-sets\">\uc9d1\ud569\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch03-algebra-of-classes\">\ub2e4\uc591\ud55c \uc9d1\ud569\uc758 \uc5f0\uc0b0<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch04-relations-and-functions\">\uad00\uacc4\uc640 \ud568\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">\uc790\uc5f0\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">\uc9d1\ud569\uc758 \uae30\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch08-ordinal-numbers\">\uc9d1\ud569\uc758 \uc11c\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">\uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch10-axiom-of-choice\">\uc120\ud0dd \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch11-formal-logic\">\ud615\uc2dd\ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch12-propositional-logic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch13-soundness-completeness-proplogic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac74\uc804\uc131\uacfc \uc644\uc804\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch15-semantics-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch16-inference-rule-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucd94\ub860\uaddc\uce59<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch17-compactness-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch18-peano-arithmetics\">\ud398\uc544\ub178 \uc0b0\uc220<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch19-incompleteness-theorem\">\ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/a><\/span>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \uc911\uc694\ud55c \ud2b9\uc131 \uc911 \ud558\ub098\ub294 \ucf64\ud329\ud2b8\uc131\uc774\ub2e4. \uc774\uac83\uc740 \uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc720\ud55c\uc131\uacfc \uad00\ub828\ub41c \uc911\uc694\ud55c \uc131\uc9c8\uc774\ub2e4. \uc815\ub9ac 17.1. (\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131) \\(\\varSigma\\)\uac00 \uac00\uc0b0\uc778 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\uc758 \ubb38\uc7a5\uc758 \ubaa8\uc784\uc774\uace0 \\(\\varSigma\\)\uc758 \uc784\uc758\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc774 \ubaa8\ub378\uc744 \uac00\uc9c0\uba74 \\(\\varSigma\\) \uc790\uccb4\ub3c4 \ubaa8\ub378\uc744 \uac00\uc9c4\ub2e4. \uc99d\uba85 \uac1c\uc694 \uc644\uc804\uc131 \uc815\ub9ac\uc5d0 \uc758\ud558\uba74, \\(\\varSigma\\)\uac00 \ubaa8\ub378\uc744 \uac00\uc9c8 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\varSigma\\)\uac00 \ubb34\ubaa8\uc21c\uc778 \uac83\uc774\ub2e4. \ub9cc\uc57d \\(\\varSigma\\)\uac00 \ubaa8\uc21c\uc744 \ud3ec\ud568\ud55c\ub2e4\uba74, \uadf8 \ubaa8\uc21c\uc740 \uc720\ud55c\ud55c \uc99d\uba85\uc5d0 \uc758\ud574 \ub3c4\ucd9c\ub418\ubbc0\ub85c \\(\\varSigma\\)\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc73c\ub85c\ubd80\ud130 \ubaa8\uc21c\uc774 \ub3c4\ucd9c\ub41c\ub2e4. \uadf8\ub7ec\ub098 \uac00\uc815\uc5d0 \uc758\ud574 \ubaa8\ub4e0 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc740 \ubaa8\ub378\uc744 \uac00\uc9c0\ubbc0\ub85c&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":5,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9282\/revisions"}],"predecessor-version":[{"id":9412,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9282\/revisions\/9412"}],"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}]}}