{"id":9264,"date":"2025-10-17T20:11:52","date_gmt":"2025-10-17T11:11:52","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9264"},"modified":"2025-10-20T18:48:54","modified_gmt":"2025-10-20T09:48:54","slug":"ch09-axiomatic-set-theory","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\/","title":{"rendered":"\uc9d1\ud569\ub860\uc758 \uacf5\ub9ac"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p>19\uc138\uae30 \ub9d0 \uce78\ud1a0\uc5b4\uc5d0 \uc758\ud574 \ucc3d\uc2dc\ub41c \uc9c1\uad00\uc801 \uc9d1\ud569\ub860\uc740 \uc218\ud559\uc5d0 \ud601\uba85\uc801 \ubcc0\ud654\ub97c \uac00\uc838\uc654\uc9c0\ub9cc, \ub3d9\uc2dc\uc5d0 \ub7ec\uc140\uc758 \uc5ed\uc124\uacfc \uac19\uc740 \ubaa8\uc21c\ub3c4 \ub4dc\ub7ec\ub0ac\ub2e4. \uc774\ub7ec\ud55c \ubb38\uc81c\ub97c \ud574\uacb0\ud558\uae30 \uc704\ud574 20\uc138\uae30 \ucd08\uc5d0\ub294 \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc774 \uac1c\ubc1c\ub418\uc5c8\ub2e4. \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \uc874\uc7ac\uc640 \uc131\uc9c8\uc744 \uc5c4\ubc00\ud55c \uacf5\ub9ac \uccb4\uacc4 \uc704\uc5d0 \uc138\uc6cc \ubaa8\uc21c\uc744 \ud53c\ud558\uace0 \uc218\ud559\uc758 \uae30\ucd08\ub97c \ud655\uace0\ud788 \ud55c\ub2e4.<\/p>\n<p>ZFC \uacf5\ub9ac\uacc4\ub294 \ud604\ub300 \uc218\ud559\uc758 \uae30\ucd08\uac00 \ub418\ub294 \uacf5\ub9ac \uccb4\uacc4\uc774\ub2e4. \uc774 \uacf5\ub9ac\ub97c \ud1b5\ud574 \uc6b0\ub9ac\uac00 \uc9c1\uad00\uc801\uc73c\ub85c \uc0ac\uc6a9\ud574\uc628 \uc9d1\ud569\uc758 \uc131\uc9c8\uc774 \uc5c4\ubc00\ud558\uac8c \uc99d\uba85\ub418\uba70, \uc5b4\ub5a4 \ub300\uc0c1\uc774 \uc9d1\ud569\uc774 \ub420 \uc218 \uc788\uace0 \uc5b4\ub5a4 \uac83\uc740 \ub420 \uc218 \uc5c6\ub294\uc9c0\uac00 \uba85\ud655\ud574\uc9c4\ub2e4. \ud2b9\ud788 \uc120\ud0dd \uacf5\ub9ac\ub294 \uc9c1\uad00\uc801\uc73c\ub85c\ub294 \ub2f9\uc5f0\ud574 \ubcf4\uc774\uc9c0\ub9cc \ub180\ub77c\uc6b4 \uacb0\uacfc\ub4e4\uc744 \uac00\uc838\uc624\ub294 \ub3c5\ud2b9\ud55c \uacf5\ub9ac\uc774\ub2e4.<\/p>\n<p>\uc774 \ubd80\uc5d0\uc11c\ub294 ZFC \uacf5\ub9ac\uacc4\ub97c \uad6c\uc131\ud558\uace0 \uc788\ub294 \uacf5\ub9ac\ub97c \uc0b4\ud3b4\ubcf4\uace0, \uc774\ub4e4\uc774 \uc5b4\ub5bb\uac8c \uc9d1\ud569\ub860\uc758 \uae30\ubcf8 \uac1c\ub150\uacfc \uc774\ub860\uc744 \ub4b7\ubc1b\uce68\ud558\ub294\uc9c0 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<p><!-- \n\n<h2>9. \uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/h2>\n\n --><\/p>\n<p>19\uc138\uae30 \ub9d0 \uce78\ud1a0\uc5b4\uac00 \ucc3d\uc2dc\ud55c \uc18c\ubc15\ud55c \uc9d1\ud569\ub860\uc740 \uc9c1\uad00\uc801 \uac1c\ub150\uc744 \ubc14\ud0d5\uc73c\ub85c \uad6c\uc131\ub418\uc5c8\uc73c\uba70, \ub7ec\uc140\uc758 \uc5ed\uc124\uacfc \uac19\uc740 \ubaa8\uc21c\uc744 \ud3ec\ud568\ud558\uace0 \uc788\uc5c8\ub2e4. \uc774\ub7ec\ud55c \ubb38\uc81c\ub97c \ud574\uacb0\ud558\uae30 \uc704\ud574 20\uc138\uae30 \ucd08\uc5d0 <span class=\"defined\">\uacf5\ub9ac\uc801 \uc9d1\ud569\ub860<\/span>(axiomatic set theory)\uc774 \uac1c\ubc1c\ub418\uc5c8\ub2e4. \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \uc218\ud559\uc801 \ub300\uc0c1\uc744 \uc5c4\ubc00\ud55c \uacf5\ub9ac \uccb4\uacc4 \uc704\uc5d0 \uad6c\ucd95\ud55c\ub2e4.<\/p>\n<h3>1. \ud074\ub798\uc2a4\uc640 \uc9d1\ud569<\/h3>\n<p>\uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \uac00\uc9c0 \uae30\ubcf8 \uac1c\ub150\uc73c\ub85c\ubd80\ud130 \ucd9c\ubc1c\ud55c\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\ud074\ub798\uc2a4<\/span>(class): \uc5b4\ub5a4 \uc131\uc9c8\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub300\uc0c1\ub4e4\uc758 \ubaa8\uc784<\/li>\n<li><span class=\"defined\">\uc18d\ud55c\ub2e4<\/span>(membership relation): \uae30\ud638 \\(\\in\\)\uc73c\ub85c \ub098\ud0c0\ub0b4\ub294 \uad00\uacc4<\/li>\n<\/ul>\n<p>\ubaa8\ub4e0 \uc218\ud559\uc801 \ub300\uc0c1\uc740 \ud074\ub798\uc2a4\uc774\uba70, \ud074\ub798\uc2a4\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ub450 \uc885\ub958\ub85c \ub098\ub25c\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\uc9d1\ud569<\/span>(set): \ub2e4\ub978 \ud074\ub798\uc2a4\uc758 \uc6d0\uc18c\uac00 \ub420 \uc218 \uc788\ub294 \ud074\ub798\uc2a4<\/li>\n<li><span class=\"defined\">\uace0\uc720\ud074\ub798\uc2a4<\/span>(proper class): \ub2e4\ub978 \ud074\ub798\uc2a4\uc758 \uc6d0\uc18c\uac00 \ub420 \uc218 \uc5c6\ub294 \ud074\ub798\uc2a4<\/li>\n<\/ul>\n<p>\uc989, \ud074\ub798\uc2a4 \\(A\\)\uac00 \uc9d1\ud569\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc5b4\ub5a4 \ud074\ub798\uc2a4 \\(B\\)\uc5d0 \ub300\ud574 \\(A \\in B\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<p>\ub7ec\uc140\uc758 \uc5ed\uc124\uc744 \ud53c\ud558\uae30 \uc704\ud574, &#8216;\ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4&#8217;\ub098 &#8216;\uc790\uae30 \uc790\uc2e0\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4&#8217; \uac19\uc740 \uac83\ub4e4\uc740 \uace0\uc720\ud074\ub798\uc2a4\uac00 \ub41c\ub2e4. \uace0\uc720\ud074\ub798\uc2a4\ub294 \ub108\ubb34 &#8216;\ud06c\uae30&#8217; \ub54c\ubb38\uc5d0 \ub2e4\ub978 \ud074\ub798\uc2a4\uc758 \uc6d0\uc18c\uac00 \ub420 \uc218 \uc5c6\ub2e4.<\/p>\n<h3>2. ZFC \uacf5\ub9ac\uacc4<\/h3>\n<p><span class=\"defined\">\uccb4\ub974\uba5c\ub85c-\ud504\ub81d\ucf08 \uacf5\ub9ac\uacc4<\/span>(Zermelo-Fraenkel axioms)\uc5d0 \uc120\ud0dd\uacf5\ub9ac(Axiom of Choice)\ub97c \ucd94\uac00\ud55c \uac83\uc744 <span class=\"defined\">ZFC<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. ZFC\ub294 \ud604\ub300 \uc218\ud559\uc758 \ud45c\uc900\uc801\uc778 \uae30\ucd08\uac00 \ub418\ub294 \uacf5\ub9ac \uccb4\uacc4\uc774\ub2e4.<\/p>\n<p>ZFC \uacf5\ub9ac\uacc4\ub294 \ub2e8\ubc88\uc5d0 \uc644\uc131\ub41c \uac83\uc774 \uc544\ub2c8\ub77c \uc810\uc9c4\uc801\uc73c\ub85c \ubc1c\uc804\ud574\uc628 \uacb0\uacfc\uc774\ub2e4. 1908\ub144 \uccb4\ub974\uba5c\ub85c\uac00 \ub7ec\uc140\uc758 \uc5ed\uc124\uc744 \ud53c\ud558\uae30 \uc704\ud574 \ucd5c\ucd08\uc758 \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860(Z)\uc744 \uc81c\uc2dc\ud588\uace0, 1922\ub144 \ud504\ub81d\ucf08\uacfc \uc2a4\ucf5c\ub818\uc774 \ub3c5\ub9bd\uc801\uc73c\ub85c \uce58\ud658 \uacf5\ub9ac\uc758 \ud544\uc694\uc131\uc744 \uc778\uc2dd\ud588\ub2e4. 1925\ub144 \ud3f0 \ub178\uc774\ub9cc\uc774 \uc815\uce59\uc131 \uacf5\ub9ac\ub97c \uc81c\uc548\ud568\uc73c\ub85c\uc368 \ud604\uc7ac \uc6b0\ub9ac\uac00 \uc544\ub294 ZF\uac00 \uc644\uc131\ub418\uc5c8\ub2e4. ZFC\ub294 \uc5ec\uae30\uc5d0 \uc120\ud0dd \uacf5\ub9ac\ub97c \ud3ec\ud568\ud55c \uccb4\uacc4\uc774\ub2e4.<\/p>\n<p>\ud55c\ud3b8, 1925\ub144 \ud3f0 \ub178\uc774\ub9cc\uc740 \uc9d1\ud569\ub860\uc758 \uc5ed\uc124\uc744 \ud574\uacb0\ud558\ub294 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \ud074\ub798\uc2a4\uc640 \uace0\uc720\ud074\ub798\uc2a4\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud588\ub2e4. \uc774\uac83\uc740 ZFC\uc640\ub294 \ubcc4\uac1c\uc758 \uc811\uadfc\uc73c\ub85c, \ub098\uc911\uc5d0 \ubca0\ub974\ub098\uc774\uc2a4(1937)\uc640 \uad34\ub378(1940)\uc5d0 \uc758\ud574 \ubc1c\uc804\ub418\uc5b4 NBG(von Neumann-Bernays-G\u00f6del) \uacf5\ub9ac\uacc4\uac00 \ub418\uc5c8\ub2e4. \ud604\uc7ac \uc9d1\ud569\ub860\uc744 \ub17c\ud560 \ub54c \ud074\ub798\uc2a4 \uac1c\ub150\uc744 \ud568\uaed8 \uc5b8\uae09\ud558\ub294 \uac83\uc740 \uad50\uc721\uc801 \ud3b8\uc758\ub97c \uc704\ud55c \uac83\uc774\uba70, \uc5ed\uc0ac\uc801\uc73c\ub85c\ub294 ZFC\uac00 \uba3c\uc800 \ud655\ub9bd\ub418\uace0 \ud074\ub798\uc2a4 \uac1c\ub150\uc774 \ub098\uc911\uc5d0 \ucd94\uac00\ub41c \uac83\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c ZFC\uc758 \uacf5\ub9ac\ub4e4\uc744 \ucc28\ub840\ub85c \uc0b4\ud3b4\ubcf4\uc790. \uc774 \uacf5\ub9ac\ub4e4\uc740 \ubaa8\ub450 \uc77c\uacc4\ub17c\ub9ac\uc758 \uc5b8\uc5b4\ub85c \ud45c\ud604\ud560 \uc218 \uc788\uc9c0\ub9cc, \uc5ec\uae30\uc11c\ub294 \uc9c1\uad00\uc801\uc778 \uc124\uba85\uacfc \ud568\uaed8 \uc81c\uc2dc\ud55c\ub2e4.<\/p>\n<h4>(1) \uc678\uc5f0 \uacf5\ub9ac (Axiom of Extension)<\/h4>\n<p>\ub450 \ud074\ub798\uc2a4\uac00 \uac19\uc740 \uc6d0\uc18c\ub97c \uac00\uc9c0\uba74 \ub450 \ud074\ub798\uc2a4\ub294 \uac19\ub2e4.<br \/>\n\\[\\forall A \\, \\forall B \\left( A = B \\leftrightarrow \\forall x (x \\in A \\leftrightarrow x \\in B) \\right)\\]<br \/>\n\uc774 \uacf5\ub9ac\ub294 \ud074\ub798\uc2a4\uac00 \uadf8 \uc6d0\uc18c\ub4e4\uc5d0 \uc758\ud574 \uc644\uc804\ud788 \uacb0\uc815\ub428\uc744 \ubcf4\uc7a5\ud55c\ub2e4.<\/p>\n<h4>(2) \uacf5\uc9d1\ud569 \uacf5\ub9ac (Empty Set Axiom)<\/h4>\n<p>\uc5b4\ub5a4 \uc6d0\uc18c\ub3c4 \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[\\exists A\\, \\forall x (x \\notin A)\\]<br \/>\n\uc678\uc5f0 \uacf5\ub9ac\uc5d0 \uc758\ud574 \uc774\ub7ec\ud55c \uc9d1\ud569 \\(A\\)\ub294 \uc720\uc77c\ud558\ub2e4. \uc774 \uc9d1\ud569\uc744 <span class=\"defined\">\uacf5\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(\\varnothing\\)\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<h4>(3) \uc9dd \uacf5\ub9ac (Pairing Axiom)<\/h4>\n<p>\uc784\uc758\uc758 \ub450 \uc9d1\ud569 \\(x\\), \\(y\\)\uc5d0 \ub300\ud574, \uc815\ud655\ud788 \\(x\\)\uc640 \\(y\\)\ub9cc\uc744 \uc6d0\uc18c\ub85c \uac16\ub294 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[\\forall x\\, \\forall y\\, \\exists A\\, \\forall z (z \\in A \\leftrightarrow z = x \\vee z = y)\\]<br \/>\n\uc774 \uc9d1\ud569\uc744 \\(\\{x,\\, y\\}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud2b9\ud788 \\(x = y\\)\uc77c \ub54c \\(\\{x,\\, x\\} = \\{x\\}\\)\ub294 \ub2e8\uc6d0\uc18c \uc9d1\ud569\uc774 \ub41c\ub2e4.<\/p>\n<h4>(4) \ud569\uc9d1\ud569 \uacf5\ub9ac (Union Axiom)<\/h4>\n<p>\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud574, \\(A\\)\uc758 \uc6d0\uc18c\ub4e4\uc758 \uc6d0\uc18c\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[\\forall A\\, \\exists B\\, \\forall x (x \\in B \\leftrightarrow \\exists y (y \\in A \\wedge x \\in y))\\]<br \/>\n\uc774 \uc9d1\ud569\uc744 \\(\\bigcup A\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(A = \\{\\{1,\\, 2\\},\\, \\{2,\\, 3\\}\\}\\)\uc774\uba74 \\(\\bigcup A = \\{1,\\, 2,\\, 3\\}\\)\uc774\ub2e4.<\/p>\n<h4>(5) \uba71\uc9d1\ud569 \uacf5\ub9ac (Power Set Axiom)<\/h4>\n<p>\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud574, \\(A\\)\uc758 \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[\\forall A\\, \\exists B\\, \\forall x (x \\in B \\leftrightarrow x \\subseteq A)\\]<br \/>\n\uc774 \uc9d1\ud569\uc744 \\(A\\)\uc758 <span class=\"defined\">\uba71\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70, \\(\\mathcal{P}(A)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(\\mathcal{P}(\\{1,\\, 2\\}) = \\{\\varnothing,\\, \\{1\\},\\, \\{2\\},\\, \\{1,\\, 2\\}\\}\\)\uc774\ub2e4.<\/p>\n<h4>(6) \ubb34\ud55c \uacf5\ub9ac (Axiom of Infinity)<\/h4>\n<p>\ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(\\varnothing \\in A\\);<\/li>\n<li>\\(x \\in A\\)\uc774\uba74 \\(x \\cup \\{x\\} \\in A\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \uacf5\ub9ac\ub294 \uc790\uc5f0\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc774 \uc874\uc7ac\ud568\uc744 \ubcf4\uc7a5\ud55c\ub2e4. \uc2e4\uc81c\ub85c \uc774 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac00\uc7a5 \uc791\uc740 \uc9d1\ud569\uc774 \uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\omega = \\{0,\\, 1,\\, 2,\\, \\ldots\\}\\)\uc774\ub2e4.<\/p>\n<h4>(7) \ubd84\ub958 \uacf5\ub9ac (Axiom Schema of Separation)<\/h4>\n<p>\uc9d1\ud569 \\(A\\)\uc640 \uc131\uc9c8 \\(\\phi(x)\\)\uc5d0 \ub300\ud574, \\(A\\)\uc758 \uc6d0\uc18c \uc911 \\(\\phi(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\ub4e4\ub9cc\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[\\forall A\\, \\exists B\\, \\forall x (x \\in B \\leftrightarrow x \\in A \\wedge \\phi(x))\\]<br \/>\n\uc774 \uc9d1\ud569\uc744 \\(\\{x \\in A \\mid \\phi(x)\\}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc911\uc694\ud55c \uc810\uc740 \uc774\ubbf8 \uc874\uc7ac\ud558\ub294 \uc9d1\ud569 \\(A\\)\ub85c\ubd80\ud130 \ubd80\ubd84\uc9d1\ud569\uc744 \ub9cc\ub4e0\ub2e4\ub294 \uac83\uc774\ub2e4.<\/p>\n<h4>(8) \uce58\ud658 \uacf5\ub9ac (Axiom Schema of Replacement)<\/h4>\n<p>\uc9d1\ud569 \\(A\\)\uc640 \ud568\uc218\uc801 \uc131\uc9c8 \\(\\psi(x,\\, y)\\)\uc5d0 \ub300\ud574, \\(A\\)\uc758 \uc6d0\uc18c\ub4e4\uc758 \uc0c1(image)\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569 \\(B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[\\text{If}\\,\\,\\, \\forall x \\in A\\, \\exists! y \\, \\psi(x,\\, y), \\,\\,\\, \\text{then} \\,\\,\\, \\exists B\\, \\forall y (y \\in B \\leftrightarrow \\exists x \\in A \\, \\psi(x,\\, y))\\]<br \/>\n\uc774 \uacf5\ub9ac\ub294 \ud568\uc218\uc758 \uce58\uc5ed\uc774 \uc9d1\ud569\uc784\uc744 \ubcf4\uc7a5\ud55c\ub2e4.<\/p>\n<h4>(9) \uc815\uce59\uc131 \uacf5\ub9ac (Axiom of Foundation)<\/h4>\n<p>\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubaa8\ub4e0 \uc9d1\ud569\uc740 \uc790\uc2e0\uacfc \uc11c\ub85c\uc18c\uc778 \uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4.<br \/>\n\\[\\forall A (A \\neq \\varnothing \\rightarrow \\exists x \\in A (x \\cap A = \\varnothing))\\]<br \/>\n\uc774 \uacf5\ub9ac\ub294 \\(x \\in x\\)\ub098 \ubb34\ud55c\ud558\uac15\uc5f4 \\(x_0 \\ni x_1 \\ni x_2 \\ni \\cdots\\) \uac19\uc740 \uac1c\uccb4\uac00 \uc874\uc7ac\ud560 \uc218 \uc5c6\ub3c4\ub85d \ud574\uc900\ub2e4.<\/p>\n<h4>(10) \uc120\ud0dd \uacf5\ub9ac (Axiom of Choice)<\/h4>\n<p>\uacf5\uc9d1\ud569\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc871\uc5d0 \ub300\ud574, \uac01 \uc9d1\ud569\uc5d0\uc11c \uc6d0\uc18c\ub97c \ud558\ub098\uc529 \uc120\ud0dd\ud558\ub294 \ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[\\forall F \\left( \\varnothing \\notin F \\rightarrow \\exists f : F \\to \\bigcup F \\, \\forall A \\in F (f(A) \\in A) \\right)\\]<br \/>\n\uc774 \uacf5\ub9ac\ub294 \uc55e\uc758 \ub2e4\ub978 \uacf5\ub9ac\ub4e4\uacfc \ub3c5\ub9bd\uc801\uc774\ub2e4. \uc989 ZF\uc640 \ubaa8\uc21c\ub418\uc9c0\ub3c4 \uc54a\uace0 ZF\ub85c\ubd80\ud130 \uc99d\uba85\ub418\uc9c0\ub3c4 \uc54a\ub294\ub2e4.<\/p>\n<h3>3. \uc9d1\ud569 \uad6c\uc131\uc758 \uc608<\/h3>\n<p>ZFC \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc2e4\uc81c\ub85c \uc9d1\ud569\uc744 \uad6c\uc131\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<h4>(1) \uc21c\uc11c\uc30d\uc758 \uad6c\uc131<\/h4>\n<p>\ucfe0\ub77c\ud1a0\ud504\uc2a4\ud0a4(Kuratowski) \uc815\uc758\uc5d0 \uc758\ud558\uba74 \uc21c\uc11c\uc30d\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc9d1\ud569\uc73c\ub85c \uc815\uc758\ub41c\ub2e4.<br \/>\n\\[(a,\\, b) = \\{\\{a\\},\\, \\{a,\\, b\\}\\}.\\]<br \/>\n\uc774\uac83\uc740 \uc9dd \uacf5\ub9ac\ub97c \ub450 \ubc88 \uc801\uc6a9\ud558\uc5ec \uc5bb\uc744 \uc218 \uc788\ub2e4. \uc774 \uc815\uc758\uac00 \uc21c\uc11c\uc30d\uc758 \uae30\ubcf8 \uc131\uc9c8<br \/>\n\\[(a,\\, b) = (c,\\, d) \\quad\\Longleftrightarrow\\quad a = c \\,\\wedge\\, b = d\\]<br \/>\n\ub97c \ub9cc\uc871\ud568\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 9.1.<\/span><br \/>\n\uc21c\uc11c\uc30d\uc758 \uc131\uc9c8\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 9.2.<\/span><br \/>\n\uc9d1\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc131\ubd84\uc774 \\(3\\)\uac1c \uc774\uc0c1\uc778 \uc21c\uc11c\uc30d\uc744 \uc815\uc758\ud558\ub294 \ubc29\ubc95\uc744 \ub17c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h4>(2) \ub370\uce74\ub974\ud2b8 \uacf1\uc758 \uad6c\uc131<\/h4>\n<p>\ub450 \uc9d1\ud569 \\(A\\), \\(B\\)\uc758 \ub370\uce74\ub974\ud2b8 \uacf1\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[A \\times B = \\{(a,\\, b) \\mid a \\in A \\wedge b \\in B\\}.\\]<br \/>\n\uc774\uac83\uc740 \uba71\uc9d1\ud569 \uacf5\ub9ac\uc640 \ubd84\ub958 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \uba3c\uc800 \\((a,\\, b) \\subseteq \\mathcal{P}(A \\cup B)\\)\uc784\uc744 \ubcf4\uc774\uace0, \ubd84\ub958 \uacf5\ub9ac\ub97c \uc801\uc6a9\ud55c\ub2e4.<\/p>\n<h4>(3) \ud568\uc218\uc758 \uad6c\uc131<\/h4>\n<p>\ud568\uc218 \\(f: A \\to B\\)\ub294 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc9d1\ud569\uc774\ub2e4.<\/p>\n<ul>\n<li>\\(f \\subseteq A \\times B\\)<\/li>\n<li>\\(\\forall a \\in A\\, \\exists! b \\in B\\, ((a,\\, b) \\in f)\\)<\/li>\n<\/ul>\n<p>\ud568\uc218\ub3c4 \uacb0\uad6d \uc21c\uc11c\uc30d\ub4e4\uc758 \uc9d1\ud569\uc774\ubbc0\ub85c ZFC\uc5d0\uc11c \uc9d1\ud569\uc73c\ub85c \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<h3>4. \ub7ec\uc140\uc758 \uc5ed\uc124<\/h3>\n<p>ZFC\uc5d0\uc11c\ub294 \ub7ec\uc140\uc758 \uc5ed\uc124\uc774 \ubc1c\uc0dd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc65c\ub0d0\ud558\uba74, \uc790\uae30 \uc790\uc2e0\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4<br \/>\n\\[\\mathcal{R} = \\{x \\mid x \\notin x\\}\\]<br \/>\n\ub294 \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c \uace0\uc720\ud074\ub798\uc2a4\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(\\mathcal{R}\\)\uc774 \uc9d1\ud569\uc774\ub77c\uace0 \uac00\uc815\ud558\uba74<\/p>\n<ul>\n<li>\\(\\mathcal{R} \\in \\mathcal{R}\\)\uc774\uba74 \\(\\mathcal{R} \\notin \\mathcal{R}\\)\uc774\uc5b4\uc57c \ud558\uace0,<\/li>\n<li>\\(\\mathcal{R} \\notin \\mathcal{R}\\)\uc774\uba74 \\(\\mathcal{R} \\in \\mathcal{R}\\)\uc774\uc5b4\uc57c \ud55c\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(\\mathcal{R}\\)\uc740 \uc9d1\ud569\uc774 \ub420 \uc218 \uc5c6\ub2e4. \uc2e4\uc81c\ub85c \uc815\uce59\uc131 \uacf5\ub9ac\uc5d0 \uc758\ud574 \\(x \\in x\\)\uc778 \uc9d1\ud569 \\(x\\)\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \\(\\mathcal{R}\\)\uc740 \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4\uc640 \uac19\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 9.3.<\/span><br \/>\n\uc678\uc5f0 \uacf5\ub9ac\uc640 \uacf5\uc9d1\ud569 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uacf5\uc9d1\ud569\uc774 \uc720\uc77c\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624. \uc989, \uc6d0\uc18c\ub97c \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc774 \ub450 \uac1c \uc788\ub2e4\uba74 \uadf8\ub4e4\uc740 \uac19\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 9.4.<\/span><br \/>\n\uc9dd \uacf5\ub9ac\uc640 \ud569\uc9d1\ud569 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc138 \uc9d1\ud569 \\(a\\), \\(b\\), \\(c\\)\uc5d0 \ub300\ud574 \\(\\{a,\\, b,\\, c\\}\\)\ub97c \uad6c\uc131\ud560 \uc218 \uc788\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 9.5.<\/span><br \/>\n\uc9d1\ud569 \\(A = \\{0,\\, 1,\\, 2\\}\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218\uc801 \uc131\uc9c8 \\(\\psi(x,\\, y)\\)\ub97c &#8220;\\(y = 2x\\)&#8221;\ub77c\uace0 \uc815\uc758\ud560 \ub54c, \uce58\ud658 \uacf5\ub9ac\uc5d0 \uc758\ud574 \uc874\uc7ac\ud558\ub294 \uc9d1\ud569 \\(B\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 9.6.<\/span><br \/>\n\\(A = \\{a,\\, b\\}\\), \\(B = \\{1,\\, 2,\\, 3\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \uba71\uc9d1\ud569 \uacf5\ub9ac\uc640 \ubd84\ub958 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(A \\times B\\)\uac00 \uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 9.7.<\/span><br \/>\n\ub2e4\uc74c \ud074\ub798\uc2a4\uac00 \uc9d1\ud569\uc778\uc9c0 \uace0\uc720\ud074\ub798\uc2a4\uc778\uc9c0 \ud310\ub2e8\ud558\uace0 \uadf8 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4<\/li>\n<li>\ubaa8\ub4e0 \ub2e8\uc6d0\uc18c \uc9d1\ud569\uc758 \ud074\ub798\uc2a4<\/li>\n<li>\ubaa8\ub4e0 \uc720\ud55c\uc9d1\ud569\uc758 \ud074\ub798\uc2a4<\/li>\n<li>\uc790\uae30 \uc790\uc2e0\uc744 \uc6d0\uc18c\ub85c \uac16\ub294 \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 9.8.<\/span><br \/>\nZFC \uacf5\ub9ac\uacc4\uc5d0\uc11c \ub2e4\uc74c \uc9d1\ud569\uc774 \uc874\uc7ac\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\{1,\\, 2,\\, 3\\}\\)<\/li>\n<li>\\(\\{\\varnothing,\\, \\{\\varnothing\\},\\, \\{\\{\\varnothing\\}\\}\\}\\)<\/li>\n<li>\ub450 \uc9d1\ud569 \\(A\\), \\(B\\)\uc758 \uad50\uc9d1\ud569 \\(A \\cap B\\)<\/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>19\uc138\uae30 \ub9d0 \uce78\ud1a0\uc5b4\uc5d0 \uc758\ud574 \ucc3d\uc2dc\ub41c \uc9c1\uad00\uc801 \uc9d1\ud569\ub860\uc740 \uc218\ud559\uc5d0 \ud601\uba85\uc801 \ubcc0\ud654\ub97c \uac00\uc838\uc654\uc9c0\ub9cc, \ub3d9\uc2dc\uc5d0 \ub7ec\uc140\uc758 \uc5ed\uc124\uacfc \uac19\uc740 \ubaa8\uc21c\ub3c4 \ub4dc\ub7ec\ub0ac\ub2e4. \uc774\ub7ec\ud55c \ubb38\uc81c\ub97c \ud574\uacb0\ud558\uae30 \uc704\ud574 20\uc138\uae30 \ucd08\uc5d0\ub294 \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc774 \uac1c\ubc1c\ub418\uc5c8\ub2e4. \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \uc874\uc7ac\uc640 \uc131\uc9c8\uc744 \uc5c4\ubc00\ud55c \uacf5\ub9ac \uccb4\uacc4 \uc704\uc5d0 \uc138\uc6cc \ubaa8\uc21c\uc744 \ud53c\ud558\uace0 \uc218\ud559\uc758 \uae30\ucd08\ub97c \ud655\uace0\ud788 \ud55c\ub2e4. ZFC \uacf5\ub9ac\uacc4\ub294 \ud604\ub300 \uc218\ud559\uc758 \uae30\ucd08\uac00 \ub418\ub294 \uacf5\ub9ac \uccb4\uacc4\uc774\ub2e4. \uc774 \uacf5\ub9ac\ub97c \ud1b5\ud574 \uc6b0\ub9ac\uac00 \uc9c1\uad00\uc801\uc73c\ub85c \uc0ac\uc6a9\ud574\uc628 \uc9d1\ud569\uc758 \uc131\uc9c8\uc774 \uc5c4\ubc00\ud558\uac8c \uc99d\uba85\ub418\uba70, \uc5b4\ub5a4 \ub300\uc0c1\uc774 \uc9d1\ud569\uc774 \ub420&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":109,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9264","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9264","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=9264"}],"version-history":[{"count":4,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9264\/revisions"}],"predecessor-version":[{"id":9404,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9264\/revisions\/9404"}],"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=9264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}