{"id":9293,"date":"2025-10-17T20:26:04","date_gmt":"2025-10-17T11:26:04","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9293"},"modified":"2025-10-21T12:26:43","modified_gmt":"2025-10-21T03:26:43","slug":"ch19-incompleteness-theorem","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch19-incompleteness-theorem\/","title":{"rendered":"\ubd88\uc644\uc804\uc131 \uc815\ub9ac"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>19. \ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/h2>\n\n --><\/p>\n<p>\ud398\uc544\ub178 \uc0b0\uc220\uc774 \uc644\uc804\uc131\uc744 \uac00\uc9c8\uae4c? \uadf8\ub807\uc9c0 \uc54a\ub2e4\ub294 \uac83\uc774 1930\ub144 \uad34\ub378\uc5d0 \uc758\ud558\uc5ec \ubc1d\ud600\uc84c\ub2e4. \uc989 \\(\\omega\\)\uc5d0\uc11c \ucc38\uc774\uc9c0\ub9cc \ud398\uc544\ub178\uc758 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud560 \uc218 \uc5c6\ub294 \ubb38\uc7a5\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uad34\ub378\uc758 \ubd88\uc644\uc804\uc131 \uc815\ub9ac\ub294 20\uc138\uae30\uc5d0 \uc190\uaf3d\ud788\ub294 \uc704\ub300\ud55c \uc9c0\uc801 \uc5c5\uc801 \uc911 \ud558\ub098\uc774\ub2e4. \uc5ec\uae30\uc11c\ub294 \ubd88\uc644\uc804\uc131 \uc815\ub9ac\uc758 \ud575\uc2ec\uc801\uc778 \uc544\uc774\ub514\uc5b4\uc640 \uc99d\uba85\uc758 \uac1c\uc694\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<h3>1. \uad34\ub378 \uc218\ub9e4\uae40<\/h3>\n<p>\uad34\ub378\uc758 \uc544\uc774\ub514\uc5b4\ub294 \uac01 \ub17c\ub9ac\uc2dd\uc744 \\(0\\) \uc774\uc0c1\uc758 \uc815\uc218\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4. \uc774\uac83\uc744 <span class=\"defined\">\uad34\ub378 \uc218\ub9e4\uae40<\/span>(G\u00f6del numbering)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc2ed\uc9c4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b8 \uc815\uc218\ub294 \uc22b\uc790 \uae30\ud638\ub4e4\ub85c \uad6c\uc131\ub41c \ubb38\uc790\uc5f4\uc774\uba70, \ub17c\ub9ac\uc2dd \uc5ed\uc2dc \uc5ec\ub7ec \uae30\ud638\ub4e4\ub85c \uad6c\uc131\ub41c \ubb38\uc790\uc5f4\uc774\ubbc0\ub85c \ub17c\ub9ac\uc2dd\uc744 \uc790\uc5f0\uc218\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \uac83\uc774 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\uac04\ub2e8\ud55c \uc608\ub85c, \uc5b8\uc5b4\ub97c \uad6c\uc131\ud558\ub294 \uae30\ud638\uc5d0 \ub2e4\uc74c \ud45c\uc640 \uac19\uc774 \uc22b\uc790\ub97c \ubc30\uc815\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div><a href=\"\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table03.png\" data-rel=\"penci-gallery-image-content\" ><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table03.png\" alt=\"\" width=\"427\" height=\"62\" class=\"aligncenter size-full wp-image-9457\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table03.png 1068w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table03-300x44.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table03-1024x149.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table03-768x111.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table03-585x85.png 585w\" sizes=\"(max-width: 427px) 100vw, 427px\" \/><\/a><\/div>\n<p><!-- width=\"1068\" height=\"155\" --><\/p>\n<p>\uc5ec\uae30\uc5d0 \ub450 \uac1c\uc758 \uae30\ud638\uac00 \ub354 \ud544\uc694\ud558\ub2e4. \uc989 &#8216;\ubcc0\uc218 \ud45c\uc2dc \uae30\ud638&#8217;\uc640 &#8216;\ub17c\ub9ac\uc2dd \ud45c\uc2dc \uae30\ud638&#8217;\uac00 \ud544\uc694\ud558\ub2e4. \uc774\ub4e4 \ub450 \uae30\ud638\ub97c \uac01\uac01 \\(A,\\) \\(B\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uc774\ub85c\uc368 \ub17c\ub9ac\uc2dd\uc744 \uc815\uc218\ub85c \ub098\ud0c0\ub0b4\ub294 \ub370\uc5d0\ub294 10\uc9c4\ubc95\uc774 \uc544\ub2c8\ub77c 12\uc9c4\ubc95\uc744 \uc0ac\uc6a9\ud574\uc57c \ud55c\ub2e4.<\/p>\n<p>\uc774 \ubc29\uc2dd\uc73c\ub85c \ubaa8\ub4e0 \uae30\ud638\uc5d0 \uc218\ub97c \ud560\ub2f9\ud558\uace0, \ubcc0\uc218\ub3c4 \uc801\uc808\ud788 \ucc98\ub9ac\ud55c \ud6c4, \ub17c\ub9ac\uc2dd\uc744 \uc774 \uc22b\uc790\ub4e4\uc758 \ub098\uc5f4\ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4. \uc774\ub807\uac8c \uc5bb\uc740 \uc815\uc218\ub97c \ud574\ub2f9 \ub17c\ub9ac\uc2dd\uc758 <span class=\"defined\">\uad34\ub378 \uc218<\/span>(G\u00f6del number)\ub77c\uace0 \ubd80\ub978\ub2e4.[\uad34\ub378 \uc218\ub9e4\uae40\uc740 \uc774\ub7ec\ud55c \ubc29\ubc95 \uc678\uc5d0\ub3c4 \uc18c\uc218\uc758 \uac70\ub4ed\uc81c\uacf1\uc744 \uc0ac\uc6a9\ud558\ub294 \ubc29\ubc95\ub3c4 \uc788\ub2e4.]<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4, \ud398\uc544\ub178 \uacf5\ub9ac \uc911 \ud558\ub098\uc778 \ub17c\ub9ac\uc2dd \\((\\forall x_0)(\\lnot (s(x_0) = 0))\\)\uc744 \uad34\ub378 \uc218\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[43A0A541474A0A56055\\]<br \/>\n\uc774\ub7ec\ud55c \ubc29\uc2dd\uc73c\ub85c \ub17c\ub9ac\uc2dd\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc99d\uba85 \uc804\uccb4\ub3c4 \uad34\ub378 \uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \ub17c\ub9ac\uc2dd \\(\\phi\\)\uc758 \uad34\ub378 \uc218\ub97c \\(G(\\phi)\\)\ub85c \ub098\ud0c0\ub0b4\uc790.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 19.1.<\/span><br \/>\n\uc2dd \\(2+2=4\\)\ub97c \uc0b0\uc220\uc758 \uc77c\ucc28\ub17c\ub9ac \uc5b8\uc5b4\ub85c \ub098\ud0c0\ub0b4\uace0, \uc774 \uc2dd\uc758 \uad34\ub378\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>2. \uc790\uae30\ucc38\uc870\uc758 \ud615\uc2dd\ud654<\/h3>\n<p>\uad34\ub378\uc740 \ud615\uc2dd\uacc4\uc5d0\uc11c \uc790\uae30\ucc38\uc870\ub97c \ud615\uc2dd\ud654\ud568\uc73c\ub85c\uc368 \ubd88\uc644\uc804\uc131 \uc815\ub9ac\ub97c \uc99d\uba85\ud55c\ub2e4. \uc989 &#8220;\uc774 \ubb38\uc7a5\uc740 \uc99d\uba85\ud560 \uc218 \uc5c6\ub2e4&#8221;\ub77c\ub294 \ubb38\uc7a5\uc744 \uc218\ud559 \ub0b4\uc5d0\uc11c \uad6c\uc131\ud558\uc600\ub2e4. \uc774 \ubb38\uc7a5\uc744 \\(\\zeta\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\zeta\\)\uac00 \uc99d\uba85 \uac00\ub2a5\ud558\ub2e4\uba74, \\(\\zeta\\)\uc758 \ub0b4\uc6a9(&#8220;\\(\\zeta\\)\ub294 \uc99d\uba85 \ubd88\uac00\ub2a5\ud558\ub2e4&#8221;)\uc774 \uac70\uc9d3\uc774 \ub418\ubbc0\ub85c \uccb4\uacc4\uac00 \uac74\uc804\ud558\uc9c0 \uc54a\ub2e4. \ubc18\ub300\ub85c \\(\\zeta\\)\uac00 \uc99d\uba85 \ubd88\uac00\ub2a5\ud558\ub2e4\uba74, \\(\\zeta\\)\uc758 \ub0b4\uc6a9\uc774 \ucc38\uc774 \ub418\uc9c0\ub9cc \uc99d\uba85\ud560 \uc218 \uc5c6\ub294 \ucc38\uc778 \ubb38\uc7a5\uc774 \uc874\uc7ac\ud558\uac8c \ub41c\ub2e4.<\/p>\n<p>\uc790\uae30 \ucc38\uc870\ub97c \ud615\uc2dd\ud654\ud558\ub824\uba74 &#8220;\uc99d\uba85 \uac00\ub2a5\ud558\ub2e4&#8221;\ub77c\ub294 \uc0c1\uc704\uc218\ud559\uc801 \uc9c4\uc220\uc744 \ud615\uc2dd\uc801\uc73c\ub85c \ud45c\ud604\ud574\uc57c \ud55c\ub2e4. \uc774\uac83\uc744 \uc704\ud574 \ub2e4\uc74c\uacfc \uac19\uc740 \ub17c\ub9ac\uc2dd\uc758 \uc874\uc7ac\uc131\uc744 \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<p class=\"marginbottomhalf\">(a) \ub450 \uac1c\uc758 \ubcc0\uc218 \\(x_0,\\, x_1\\)\uc744 \uac00\uc9c4 \uc77c\uacc4\ub17c\ub9ac \uc0b0\uc220\uc758 \ub17c\ub9ac\uc2dd \\(\\pi\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<ul>\n<li>\\(\\omega \\models \\pi [n\/x_0,\\, m\/x_1]\\) (\\(m = G(\\phi)\\)\uc774\uace0 \\(n\\)\uc774 \ud398\uc544\ub178 \uc0b0\uc220\uc5d0\uc11c \\(\\phi\\)\uc758 \uc99d\uba85\uc758 \uad34\ub378 \uc218\uc77c \ub54c)<\/li>\n<li>\\(\\omega \\models (\\lnot \\pi [n\/x_0,\\, m\/x_1])\\) (\uadf8 \uc678\uc758 \uacbd\uc6b0)<\/li>\n<\/ul>\n<p class=\"marginbottomhalf\">(b) \ub450 \uac1c\uc758 \ubcc0\uc218 \\(x_0,\\, x_1\\)\uc744 \uac00\uc9c4 \uc77c\uacc4\ub17c\ub9ac \uc0b0\uc220\uc758 \ub17c\ub9ac\uc2dd \\(\\tau\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<ul>\n<li>\\(\\omega \\models \\tau [m\/x_0,\\, n\/x_1]\\) (\ubcc0\uc218 \\(x_0\\)\uac00 \uc790\uc720\ubcc0\uc218\ub85c\uc11c \ub098\ud0c0\ub098\ub294 \uc801\ub2f9\ud55c \ub17c\ub9ac\uc2dd \\(\\phi\\)\uc640 \ud398\uc544\ub178 \uc0b0\uc220\uc5d0\uc11c\uc758 \\(\\phi [m\/x_0]\\)\uc758 \uc99d\uba85\uc758 \uad34\ub378 \uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(m=G(\\phi)\\)\uc77c \ub54c)<\/li>\n<li>\\(\\omega \\models (\\lnot \\tau [m\/x_0,\\, n\/x_1])\\) (\uadf8 \uc678\uc758 \uacbd\uc6b0)<\/li>\n<\/ul>\n<p>(a)\uc5d0\uc11c \ub17c\ub9ac\uc2dd \\(\\pi[n\/x_0,\\, m\/x_1]\\)\ub294 \uc9c1\uad00\uc801\uc73c\ub85c &#8220;\\(n\\)\uc740 \uad34\ub378 \uc218\uac00 \\(m\\)\uc778 \ub17c\ub9ac\uc2dd\uc758 \uc99d\uba85\uc758 \uad34\ub378 \uc218\uc774\ub2e4&#8221;\ub97c \uc758\ubbf8\ud55c\ub2e4. (b)\uc5d0\uc11c \ub17c\ub9ac\uc2dd \\(\\tau\\)\ub294 \ub300\uac01\ud654(diagonalization) \uacfc\uc815\uc744 \ud45c\ud604\ud55c\ub2e4. \uc989, \uc5b4\ub5a4 \ub17c\ub9ac\uc2dd\uc5d0 \uadf8 \uc790\uc2e0\uc758 \uad34\ub378 \uc218\ub97c \ub300\uc785\ud558\ub294 \uacfc\uc815\uc744 \uc0b0\uc220\ud654\ud55c \uac83\uc774\ub2e4.<\/p>\n<p>\uc774\uc640 \uac19\uc740 \ub17c\ub9ac\uc2dd \\(\\pi\\)\uc640 \\(\\tau\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\ub294 \uac83\uc740 \uad34\ub378 \uc218\ub9e4\uae40\uc744 \ud1b5\ud574 \ub17c\ub9ac\uc2dd\uc758 \uad6c\ubb38\uc801 \uc131\uc9c8\uc744 \uc0b0\uc220\uc801 \uc131\uc9c8\ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc989, &#8220;\uc774 \ubb38\uc7a5\uc740 \uc99d\uba85\ud560 \uc218 \uc5c6\ub2e4&#8221;\uc640 \uac19\uc740 \uc0c1\uc704\uc218\ud559\uc801 \uc9c4\uc220\uc744 \uc77c\uacc4\ub17c\ub9ac \ub0b4\uc5d0\uc11c \ud615\uc2dd\ud654\ud560 \uc218 \uc788\uac8c \ub41c\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 19.2.<\/span><br \/>\n\uad34\ub378\uc758 &#8216;\ub300\uac01\ud654 \ubcf4\uc870\uc815\ub9ac&#8217;\uc5d0 \ub300\ud558\uc5ec \uc870\uc0ac\ud574 \ubcf4\uc790.<\/p>\n<\/div>\n<h3>3. \uad34\ub378\uc758 \ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/h3>\n<p>\uc774\uc81c \uad34\ub378\uc758 \ubd88\uc644\uc804\uc131 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ud575\uc2ec \uc544\uc774\ub514\uc5b4\ub97c \ubc14\ud0d5\uc73c\ub85c, \\(p\\)\uac00 \ub17c\ub9ac\uc2dd \\((\\forall x_1)(\\lnot \\pi)\\)\uc758 \uad34\ub378 \uc218\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(\\pi\\)\ub294 \uc790\uc720\ubcc0\uc218 \\(x_0\\)\uc640 \\(x_1\\)\uc744 \uac00\uc9c0\uace0 \uc788\ub2e4. \uc774 \ub17c\ub9ac\uc2dd\uc5d0\uc11c \\(x_0\\)\ub97c \\(p\\)\ub85c \uce58\ud658\ud55c \ub17c\ub9ac\uc2dd\uc744 \\(\\zeta\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(\\zeta\\)\ub294 \uc790\uae30 \uc790\uc2e0\uc758 \uc99d\uba85 \ubd88\uac00\ub2a5\uc131\uc744 \uc8fc\uc7a5\ud558\ub294 \ubb38\uc7a5\uc774\ub2e4. \uc989, \\(\\zeta\\)\uac00 \ucc38\uc774\ub77c\uba74 \\(\\zeta\\)\ub294 \uc99d\uba85 \ubd88\uac00\ub2a5\ud558\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 19.1. (\uad34\ub378\uc758 \ubd88\uc644\uc804\uc131 \uc815\ub9ac)<\/span><\/p>\n<p>\ud398\uc544\ub178 \uc0b0\uc220\uc774 \ubb34\ubaa8\uc21c\uc774\uba74 \ud398\uc544\ub178 \uc0b0\uc220\uc5d0\uc11c \uc99d\uba85\ub3c4 \ubd88\uac00\ub2a5\ud558\uace0 \ubc18\uc99d\ub3c4 \ubd88\uac00\ub2a5\ud55c \ubb38\uc7a5\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\uc774 \uc815\ub9ac\uc758 \uc99d\uba85\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc9c4\ud589\ub41c\ub2e4.<\/p>\n<ol class=\"parenthesis\" style=\"margin-left: 3em;\">\n<li>\ud398\uc544\ub178 \uc0b0\uc220\uc774 \ubb34\ubaa8\uc21c\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790.<\/li>\n<li>\\(\\zeta\\)\uac00 \uc99d\uba85 \uac00\ub2a5\ud558\ub2e4\uace0 \uac00\uc815\ud558\uace0, \uadf8 \uc99d\uba85\uc758 \uad34\ub378 \uc218\ub97c \\(q\\)\ub77c\uace0 \ud558\uc790.<\/li>\n<li>\uadf8\ub7ec\uba74 \\(\\pi[p\/x_0,\\, q\/x_1]\\)\uc774 \uc99d\uba85 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<li>\uadf8\ub7ec\ub098 \\(\\zeta\\)\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \\((\\forall x_1)(\\lnot \\pi[p\/x_0])\\)\ub3c4 \uc99d\uba85 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<li>\uc774\ub85c\ubd80\ud130 \\((\\lnot \\pi[p\/x_0,\\, q\/x_1])\\)\uc774 \uc99d\uba85 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<li>\uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c, \\(\\zeta\\)\ub294 \uc99d\uba85 \ubd88\uac00\ub2a5\ud558\ub2e4.<\/li>\n<li>\uc720\uc0ac\ud55c \ubc29\ubc95\uc73c\ub85c \\((\\lnot \\zeta)\\)\ub3c4 \uc99d\uba85 \ubd88\uac00\ub2a5\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<\/li>\n<\/ol>\n<p>\ub530\ub77c\uc11c \\(\\zeta\\)\ub294 \ud398\uc544\ub178 \uc0b0\uc220\uc5d0\uc11c \uc99d\uba85\ub3c4 \ubd88\uac00\ub2a5\ud558\uace0 \ubc18\uc99d\ub3c4 \ubd88\uac00\ub2a5\ud55c \ubb38\uc7a5\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ubd88\uc644\uc804\uc131 \uc815\ub9ac\ub294 \uc218\ud559\uc758 \ubd88\uc644\uc804\uc131\uc744 \uc758\ubbf8\ud558\ub294 \uac83\uc774 \uc544\ub2c8\uba70, \ud398\uc544\ub178 \uc0b0\uc220\uc744 \ud3ec\ud568\ud55c \uacf5\ub9ac\uacc4\uc758 \ud2b9\uc131\uc744 \ub098\ud0c0\ub0bc \ubfd0\uc774\ub2e4. \uc624\ud788\ub824, \ubd88\uc644\uc804\uc131 \uc815\ub9ac\ub294 \ub2e8\uc77c \ud615\uc2dd\uacc4\ub97c \ud1b5\ud574 &#8216;\uc218\ud559\uc758 \uc138\uacc4&#8217;\ub97c \uc644\uc131\ud560 \uc218 \uc5c6\ub2e4\ub294 \uc810\uc744 \uc2dc\uc0ac\ud558\ubbc0\ub85c, \uc218\ud559\uc758 \uc138\uacc4\uc5d0\uc11c \uc778\uac04\uc758 \uc720\uc758\ubbf8\ud55c \uc5ed\ud560\uc5d0 \ub300\ud55c \uae0d\uc815\uc801\uc778 \uce21\uba74\uc744 \ub4dc\ub7ec\ub0b8\ub2e4\uace0 \ud560 \uc218 \uc788\ub2e4.<\/p>\n<h3>4. \uc81c 2 \ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/h3>\n<p>\uad34\ub378\uc758 \uc81c 2 \ubd88\uc644\uc804\uc131 \uc815\ub9ac\ub294 \ud398\uc544\ub178 \uc0b0\uc220\uc774 \uc790\uc2e0\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \uc99d\uba85\ud560 \uc218 \uc5c6\ub2e4\ub294 \uac83\uc744 \ubcf4\uc5ec\uc900\ub2e4. [\uc774 \uc815\ub9ac\uc640 \uad6c\ubd84\ud558\uae30 \uc704\ud558\uc5ec \uc815\ub9ac 19.1\uc744 &#8216;\uc81c 1 \ubd88\uc644\uc804\uc131 \uc815\ub9ac&#8217;\ub77c\uace0\ub3c4 \ubd80\ub978\ub2e4.]<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 19.2. (\uc81c 2 \ubd88\uc644\uc804\uc131 \uc815\ub9ac)<\/span><\/p>\n<p>\ub9cc\uc57d \ud398\uc544\ub178 \uc0b0\uc220 \\(\\mathrm{PA}\\)\uac00 \ubb34\ubaa8\uc21c\uc774\uba74, \\(\\mathrm{PA}\\)\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \ub098\ud0c0\ub0b4\ub294 \ub17c\ub9ac\uc2dd\uc740 \uc99d\uba85 \ubd88\uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>PA\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \ub098\ud0c0\ub0b4\ub294 \ub17c\ub9ac\uc2dd\uc744 \\(\\gamma\\)\ub77c\uace0 \ud558\uc790. \ubd88\uc644\uc804\uc131 \uc815\ub9ac(19.1)\uc5d0 \uc758\ud558\uc5ec \uc99d\uba85\uacfc \ubc18\uc99d\uc774 \ubd88\uac00\ub2a5\ud55c \ub17c\ub9ac\uc2dd \\(\\zeta\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\gamma\\)\uac00 PA\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \ub098\ud0c0\ub0b4\ub294\ub370, \\(\\zeta\\)\uac00 \uc99d\uba85\uacfc \ubc18\uc99d\uc774 \ubd88\uac00\ub2a5\ud55c \ub17c\ub9ac\uc2dd\uc774\ubbc0\ub85c \\(\\mathrm{PA}\\cup\\left\\{ \\gamma \\right\\} \\vdash \\zeta\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \ucd94\ub860 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\mathrm{PA}\\vdash (\\gamma \\rightarrow \\zeta)\\)\ub97c \uc5bb\ub294\ub2e4. \ub9cc\uc57d PA\ub85c\ubd80\ud130 \\(\\gamma\\)\uac00 \ucd94\ub860 \uac00\ub2a5\ud558\ub2e4\uba74 MP\uc5d0 \uc758\ud558\uc5ec \\(\\gamma\\)\uc758 \uc99d\uba85\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(\\gamma\\)\ub294 \uc99d\uba85 \ubd88\uac00\ub2a5\ud558\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc790\uc5f0\uc218\uacc4\ub294 \uc9d1\ud569\ub860\uc744 \uc0ac\uc6a9\ud558\uc5ec \ud398\uc544\ub178 \uacf5\ub9ac\uacc4\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub3c4\ub85d \uad6c\uc131\ub420 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc9d1\ud569\ub860\uc758 ZF \uacf5\ub9ac\uacc4\ub294 \ud398\uc544\ub178 \uacf5\ub9ac\uacc4\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \uc99d\uba85\ud558\uae30\uc5d0 \ucda9\ubd84\ud558\ub2e4. \uc989 \ud398\uc544\ub178 \uc0b0\uc220\uc740 \uc2a4\uc2a4\ub85c\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \uc99d\uba85\ud560 \uc218 \uc5c6\uc9c0\ub9cc \uadf8\ubcf4\ub2e4 \ub354 \ud070 \uccb4\uacc4\uc5d0\uc11c \ud398\uc544\ub178 \uc0b0\uc220\uc758 \ubb34\ubaa8\uc21c\uc131\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uc774 \uacbd\uc6b0\uc5d0\ub3c4 \ub354 \ud070 \uccb4\uacc4\uc758 \ubb34\ubaa8\uc21c\uc131\uacfc \uad00\ub828\ub41c \ubb38\uc81c\ub294 \uc5ec\uc804\ud788 \ub0a8\uc544 \uc788\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>\ud398\uc544\ub178 \uc0b0\uc220\uc774 \uc644\uc804\uc131\uc744 \uac00\uc9c8\uae4c? \uadf8\ub807\uc9c0 \uc54a\ub2e4\ub294 \uac83\uc774 1930\ub144 \uad34\ub378\uc5d0 \uc758\ud558\uc5ec \ubc1d\ud600\uc84c\ub2e4. \uc989 \\(\\omega\\)\uc5d0\uc11c \ucc38\uc774\uc9c0\ub9cc \ud398\uc544\ub178\uc758 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud560 \uc218 \uc5c6\ub294 \ubb38\uc7a5\uc774 \uc874\uc7ac\ud55c\ub2e4. \uad34\ub378\uc758 \ubd88\uc644\uc804\uc131 \uc815\ub9ac\ub294 20\uc138\uae30\uc5d0 \uc190\uaf3d\ud788\ub294 \uc704\ub300\ud55c \uc9c0\uc801 \uc5c5\uc801 \uc911 \ud558\ub098\uc774\ub2e4. \uc5ec\uae30\uc11c\ub294 \ubd88\uc644\uc804\uc131 \uc815\ub9ac\uc758 \ud575\uc2ec\uc801\uc778 \uc544\uc774\ub514\uc5b4\uc640 \uc99d\uba85\uc758 \uac1c\uc694\ub97c \uc0b4\ud3b4\ubcf4\uc790. 1. \uad34\ub378 \uc218\ub9e4\uae40 \uad34\ub378\uc758 \uc544\uc774\ub514\uc5b4\ub294 \uac01 \ub17c\ub9ac\uc2dd\uc744 \\(0\\) \uc774\uc0c1\uc758 \uc815\uc218\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4. \uc774\uac83\uc744 \uad34\ub378 \uc218\ub9e4\uae40(G\u00f6del numbering)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc2ed\uc9c4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b8 \uc815\uc218\ub294 \uc22b\uc790 \uae30\ud638\ub4e4\ub85c \uad6c\uc131\ub41c \ubb38\uc790\uc5f4\uc774\uba70,&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":119,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9293","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9293","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=9293"}],"version-history":[{"count":10,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9293\/revisions"}],"predecessor-version":[{"id":9603,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9293\/revisions\/9603"}],"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=9293"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}