{"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":"2026-09-27T12:05:14","modified_gmt":"2026-09-27T03:05:14","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":"
<\/p>\n

\uc55e \uc7a5\uae4c\uc9c0\ub294 \uc9d1\ud569\uc744 \uc9c1\uad00\uc801\uc73c\ub85c \uc0ac\uc6a9\ud558\uba74\uc11c \uc790\uc5f0\uc218, \uae30\uc218, \uc11c\uc218\ub97c \uad6c\uc131\ud558\uc600\ub2e4. \uadf8\ub7ec\ub098 \u201c\uc5b4\ub5a4 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub300\uc0c1\ub4e4\uc744 \ubaa8\ub450 \ubaa8\uc73c\uba74 \uc9d1\ud569\uc774 \ub41c\ub2e4\u201d\uace0 \ubb34\uc81c\ud55c\uc73c\ub85c \ud5c8\uc6a9\ud558\uba74 \ub7ec\uc140\uc758 \uc5ed\uc124\uacfc \uac19\uc740 \ubaa8\uc21c\uc774 \uc0dd\uae34\ub2e4. \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \uc5b4\ub5a4 \uc9d1\ud569\uc758 \uc874\uc7ac\ub97c \uc778\uc815\ud560 \uac83\uc778\uc9c0 \uacf5\ub9ac\ub85c \uc81c\ud55c\ud558\uc5ec \uc774\ub7ec\ud55c \ubb38\uc81c\ub97c \ud53c\ud55c\ub2e4.<\/p>\n

\uc774 \ubd80\uc5d0\uc11c\ub294 \ud604\ub300 \uc218\ud559\uc758 \ud45c\uc900\uc801\uc778 \uae30\ucd08 \uac00\uc6b4\ub370 \ud558\ub098\uc778 ZFC\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. ZFC\ub294 \uccb4\ub974\uba5c\ub85c-\ud504\ub81d\ucf08 \uc9d1\ud569\ub860(ZF)\uc5d0 \uc120\ud0dd\uacf5\ub9ac\ub97c \ub354\ud55c \uccb4\uacc4\uc774\ub2e4. \ud2b9\ud788 \uc55e \uc7a5\uc5d0\uc11c \uc0ac\uc6a9\ud55c \uc790\uc5f0\uc218 \uc9d1\ud569\uc758 \uc874\uc7ac, \ucd08\ud55c\uc7ac\uadc0, \ucd08\uae30\uc11c\uc218\uc758 \uc874\uc7ac\uac00 \uc5b4\ub5a4 \uc9d1\ud569 \uc874\uc7ac \uacf5\ub9ac\uc640 \uc5f0\uacb0\ub418\ub294\uc9c0 \ud655\uc778\ud55c\ub2e4.<\/p>\n

<\/p>\n

\uacf5\ub9ac\uc801 \uc9d1\ud569\ub860<\/span>(axiomatic set theory)\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \uc874\uc7ac\uc640 \uc131\uc9c8\uc744 \uba87 \uac1c\uc758 \uae30\ubcf8 \uacf5\ub9ac\uc640 \uacf5\ub9ac\uaf34\ub85c \uaddc\uc815\ud55c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 ZFC\ub97c \uae30\uc900\uc73c\ub85c \uc124\uba85\ud55c\ub2e4.<\/p>\n

1. \uc9d1\ud569\uacfc \ud074\ub798\uc2a4<\/h3>\n

ZFC\uc758 \uc77c\uacc4\ub17c\ub9ac \uc5b8\uc5b4\uc5d0\uc11c \ubcc0\uc218\ub294 \ubaa8\ub450 \uc9d1\ud569\uc744 \ub098\ud0c0\ub0b4\uba70, \uae30\ubcf8\uc801\uc778 \ube44\ub17c\ub9ac \uae30\ud638\ub294 \uc6d0\uc18c\uad00\uacc4 \\(\\in\\)\uc774\ub2e4. \ub530\ub77c\uc11c ZFC \uc790\uccb4\uc5d0\ub294 \u201c\ud074\ub798\uc2a4\u201d\ub97c \uc9d1\ud569\uacfc \ub098\ub780\ud55c \uc0c8\ub85c\uc6b4 \uc885\ub958\uc758 \ub300\uc0c1\uc73c\ub85c \ub450\ub294 \ubcc0\uc218\uac00 \uc5c6\ub2e4.<\/p>\n

\uadf8\ub7fc\uc5d0\ub3c4 \uc124\uba85\uc758 \ud3b8\uc758\ub97c \uc704\ud558\uc5ec \ub2e4\uc74c \uc6a9\uc5b4\ub97c \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n

\n

\uc815\uc758 9.1. (\ud074\ub798\uc2a4\uc640 \uace0\uc720\ud074\ub798\uc2a4)<\/span><\/p>\n

\ub17c\ub9ac\uc2dd \\(\\phi(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubaa8\ub4e0 \uc9d1\ud569\uc744
\n\\[\\{x\\mid \\phi(x)\\}\\]
\n\uc640 \uac19\uc774 \uc801\uace0 \ud074\ub798\uc2a4<\/span>(class)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ud45c\uae30\ub294 \uc0c8\ub85c\uc6b4 \uc9d1\ud569\uc758 \uc874\uc7ac\ub97c \uc8fc\uc7a5\ud558\ub294 \uac83\uc774 \uc544\ub2c8\ub77c, \u201c\\(\\phi(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc9d1\ud569 \\(x\\)\u201d\ub77c\ub294 \uc870\uac74\uc744 \uc904\uc5ec \uc4f4 \uac83\uc774\ub2e4. \uc774\ub7ec\ud55c \ud074\ub798\uc2a4\uac00 \uc2e4\uc81c\ub85c \uc5b4\ub5a4 \uc9d1\ud569\uacfc \uac19\uc740 \uacbd\uc6b0\uc5d0\ub294 \uadf8 \ud074\ub798\uc2a4\ub97c \uc9d1\ud569\uc774\ub77c\uace0 \ubd80\ub974\uace0, \uc5b4\ub5a4 \uc9d1\ud569\uacfc\ub3c4 \uac19\uc9c0 \uc54a\uc740 \uacbd\uc6b0\uc5d0\ub294 \uace0\uc720\ud074\ub798\uc2a4<\/span>(proper class)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

\uc608\ub97c \ub4e4\uc5b4 \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4
\n\\[V=\\{x\\mid x=x\\}\\]
\n\ub294 \uace0\uc720\ud074\ub798\uc2a4\uc774\ub2e4. \ub9cc\uc57d \\(V\\)\uac00 \uc9d1\ud569\uc774\ub77c\uba74 \ub4a4\uc5d0\uc11c \ub2e4\ub8f0 \ubd84\ub9ac \uacf5\ub9ac\uaf34\uc744 \\(V\\)\uc5d0 \uc801\uc6a9\ud558\uc5ec \\(\\{x\\in V\\mid x\\notin x\\}\\)\ub97c \uc9d1\ud569\uc73c\ub85c \ub9cc\ub4e4 \uc218 \uc788\uace0, \ub7ec\uc140\uc758 \uc5ed\uc124\uc774 \uc0dd\uae34\ub2e4. \ubaa8\ub4e0 \uc11c\uc218\uc758 \ud074\ub798\uc2a4\ub3c4 8\uc7a5<\/a>\uc5d0\uc11c \ubcf8 \uac83\ucc98\ub7fc \uace0\uc720\ud074\ub798\uc2a4\uc774\ub2e4.<\/p>\n

\ud074\ub798\uc2a4\ub97c \uc2e4\uc81c \ub300\uc0c1\uc73c\ub85c \ub2e4\ub8e8\ub294 NBG \uac19\uc740 \uacf5\ub9ac\uacc4\ub3c4 \uc788\uc9c0\ub9cc, \uc774 \ucc45\uc5d0\uc11c\ub294 ZFC\ub97c \uc0ac\uc6a9\ud558\uace0 \ud074\ub798\uc2a4 \ud45c\uae30\ub294 \uc704\uc640 \uac19\uc740 \uba54\ud0c0\uc5b8\uc5b4\uc801 \uc57d\uc18d\uc73c\ub85c\ub9cc \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n

2. ZFC \uacf5\ub9ac\uacc4<\/h3>\n
\n

\uc815\uc758 9.2. (ZFC)<\/span><\/p>\n

\uccb4\ub974\uba5c\ub85c-\ud504\ub81d\ucf08 \uc9d1\ud569\ub860<\/span>(Zermelo-Fraenkel set theory)\uc5d0 \uc120\ud0dd\uacf5\ub9ac(Axiom of Choice)\ub97c \ucd94\uac00\ud55c \uccb4\uacc4\ub97c ZFC<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

\uc544\ub798\uc758 \uc2dd\uc5d0\uc11c \\(\\subseteq\\), \\(\\varnothing\\), \ud568\uc218 \ub4f1\uc758 \ud45c\uae30\ub294 \ubaa8\ub450 \\(\\in\\)\uacfc \\(=\\)\ub9cc\uc744 \uc0ac\uc6a9\ud558\uc5ec \ud480\uc5b4 \uc4f8 \uc218 \uc788\ub294 \uc57d\uc2dd \ud45c\uae30\uc774\ub2e4.<\/p>\n

\ubd84\ub9ac\uc640 \uce58\ud658\uc740 \uac01\uac01 \ud558\ub098\uc758 \uacf5\ub9ac\uac00 \uc544\ub2c8\ub77c, \ud5c8\uc6a9\ub418\ub294 \ub17c\ub9ac\uc2dd\ub9c8\ub2e4 \ud558\ub098\uc758 \uacf5\ub9ac\ub97c \uac16\ub294 \uacf5\ub9ac\uaf34<\/span>(axiom schema)\uc774\ub2e4. \ub610\ud55c \uc544\ub798\uc5d0\uc11c\ub294 \uad50\uc721\uc801 \ud3b8\uc758\ub97c \uc704\ud558\uc5ec \uacf5\uc9d1\ud569\uc758 \uc874\uc7ac\ub97c \ubcc4\ub3c4\ub85c \uc801\ub294\ub2e4. \uc774\uac83\uc740 \ud1b5\uc0c1\uc801\uc778 ZF\uc758 \ub2e4\ub978 \uacf5\ub9ac\ub4e4\ub85c\ubd80\ud130\ub3c4 \uc720\ub3c4\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \ub3c5\ub9bd\uc801\uc778 \uacf5\ub9ac\ub85c \ubc18\ub4dc\uc2dc \ub123\uc5b4\uc57c \ud558\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4.<\/p>\n

\n

\uacf5\ub9ac 9.3. (\uc678\uc5f0 \uacf5\ub9ac)<\/span><\/p>\n

\ub450 \uc9d1\ud569\uc774 \uac19\uc740 \uc6d0\uc18c\ub97c \uac00\uc9c0\uba74 \ub450 \uc9d1\ud569\uc740 \uac19\ub2e4.
\n\\[\\forall A\\,\\forall B\\,
\n\\bigl(\\forall x\\,(x\\in A\\leftrightarrow x\\in B)\\to A=B\\bigr).\\]<\/p>\n<\/div>\n

\n

\uacf5\ub9ac 9.4. (\uacf5\uc9d1\ud569\uc758 \uc874\uc7ac)<\/span><\/p>\n

\uc6d0\uc18c\ub97c \ud558\ub098\ub3c4 \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.
\n\\[\\exists A\\,\\forall x\\,(x\\notin A).\\]
\n\uc678\uc5f0 \uacf5\ub9ac\uc5d0 \uc758\ud574 \uc774\ub7ec\ud55c \uc9d1\ud569\uc740 \uc720\uc77c\ud558\uba70 \uc774\ub97c \\(\\varnothing\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n

\n

\uacf5\ub9ac 9.5. (\uc9dd \uacf5\ub9ac)<\/span><\/p>\n

\uc784\uc758\uc758 \ub450 \uc9d1\ud569 \\(x\\), \\(y\\)\uc5d0 \ub300\ud558\uc5ec \uc815\ud655\ud788 \\(x\\)\uc640 \\(y\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.
\n\\[\\forall x\\,\\forall y\\,\\exists A\\,\\forall z\\,
\n\\bigl(z\\in A\\leftrightarrow(z=x\\vee z=y)\\bigr).\\]
\n\uc774 \uc9d1\ud569\uc744 \\(\\{x,y\\}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud2b9\ud788 \\(x=y\\)\uc774\uba74 \ub2e8\uc6d0\uc18c \uc9d1\ud569 \\(\\{x\\}\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<\/div>\n

\n

\uacf5\ub9ac 9.6. (\ud569\uc9d1\ud569 \uacf5\ub9ac)<\/span><\/p>\n

\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\)\uc758 \uc6d0\uc18c\ub4e4\uc758 \uc6d0\uc18c\ub97c \ubaa8\ub450 \ubaa8\uc740 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.
\n\\[\\forall A\\,\\exists B\\,\\forall x\\,
\n\\bigl(x\\in B\\leftrightarrow\\exists y\\,(y\\in A\\wedge x\\in y)\\bigr).\\]
\n\uc774 \uc9d1\ud569\uc744 \\(\\bigcup A\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n

\n

\uacf5\ub9ac 9.7. (\uba71\uc9d1\ud569 \uacf5\ub9ac)<\/span><\/p>\n

\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\)\uc758 \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc744 \uc6d0\uc18c\ub85c \uac16\ub294 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4.
\n\\[\\forall A\\,\\exists B\\,\\forall x\\,(x\\in B\\leftrightarrow x\\subseteq A).\\]
\n\uc774 \uc9d1\ud569\uc740 \uc55e\uc5d0\uc11c \uc815\uc758\ud55c \uba71\uc9d1\ud569 \\(\\mathcal P(A)\\)\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\ubb38\uc81c 9.1.<\/span>
\n\uc9d1\ud569 \\(A=\\{0,1\\}\\)\uacfc \uc784\uc758\uc758 \uc9d1\ud569 \\(B\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

    \n
  1. \uc9dd \uacf5\ub9ac\uc640 \ud569\uc9d1\ud569 \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(A\\cup B\\)\uc758 \uc874\uc7ac\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n
  2. \\(\\mathcal P(A)\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n
  3. \\(\\bigcup\\mathcal P(A)\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n
  4. \\(\\mathcal P(A\\cup B)\\)\uc758 \uc874\uc7ac\ub97c \ubcf4\uc7a5\ud558\uae30 \uc704\ud574 \uc5b4\ub5a4 \uacf5\ub9ac\ub4e4\uc744 \ucc28\ub840\ub85c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub294\uc9c0 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
    \n

    \uacf5\ub9ac 9.8. (\ubb34\ud55c \uacf5\ub9ac)<\/span><\/p>\n

    \uadc0\ub0a9\uc801 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc989,
    \n\\[\\exists A\\,
    \n\\bigl(\\varnothing\\in A\\wedge
    \n\\forall x\\in A\\,(x\\cup\\{x\\}\\in A)\\bigr).\\]<\/p>\n<\/div>\n

    \uacf5\ub9ac 9.8\uc740 6\uc7a5<\/a>\uc5d0\uc11c \uac00\uc815\ud588\ub358 \uadc0\ub0a9\uc801 \uc9d1\ud569\uc758 \uc874\uc7ac\ub97c \ubcf4\uc7a5\ud55c\ub2e4. \uadc0\ub0a9\uc801 \uc9d1\ud569 \ud558\ub098\ub97c \\(I\\)\ub77c \ud558\uba74 \uba71\uc9d1\ud569 \uacf5\ub9ac\uc640 \uc544\ub798\uc758 \ubd84\ub9ac \uacf5\ub9ac\uaf34\uc744 \uc774\uc6a9\ud558\uc5ec \\(\\mathcal P(I)\\) \uc548\uc758 \uadc0\ub0a9\uc801 \ubd80\ubd84\uc9d1\ud569\ub4e4\uc744 \ubaa8\uc744 \uc218 \uc788\uace0, \uadf8\ub4e4\uc758 \uad50\uc9d1\ud569\uc73c\ub85c \uac00\uc7a5 \uc791\uc740 \uadc0\ub0a9\uc801 \uc9d1\ud569 \\(\\omega=\\mathbb N\\)\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \ubb38\uc81c 9.2.<\/span>
    \n\\(I\\)\uac00 \ubb34\ud55c \uacf5\ub9ac\uc5d0\uc11c \ubcf4\uc7a5\ud558\ub294 \uadc0\ub0a9\uc801 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

      \n
    1. \\(0=\\varnothing\\in I\\)\uc784\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n
    2. \\(1=\\{0\\}\\), \\(2=\\{0,1\\}\\), \\(3=\\{0,1,2\\}\\)\uac00 \ucc28\ub840\ub85c \\(I\\)\uc5d0 \uc18d\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
    3. \uc77c\ubc18\uc801\uc73c\ub85c \ubaa8\ub4e0 \uc790\uc5f0\uc218 \\(n\\)\uc774 \\(I\\)\uc5d0 \uc18d\ud568\uc744 \ubcf4\uc77c \ub54c \uc5b4\ub5a4 \uc6d0\ub9ac\ub97c \uc0ac\uc6a9\ud558\ub294\uc9c0 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uacf5\ub9ac 9.9. (\ubd84\ub9ac \uacf5\ub9ac\uaf34)<\/span><\/p>\n

      \uc9d1\ud569 \\(A\\)\uc640 \ub17c\ub9ac\uc2dd \\(\\phi(x,\\vec p)\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(A\\)\uc758 \uc6d0\uc18c \uc911 \\(\\phi\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubd80\ubd84\uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc989 \uac01 \ub17c\ub9ac\uc2dd \\(\\phi\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[\\forall A\\,\\exists B\\,\\forall x\\,
      \n\\bigl(x\\in B\\leftrightarrow(x\\in A\\wedge\\phi(x,\\vec p))\\bigr)\\]
      \n\ub77c\ub294 \uacf5\ub9ac\ub97c \ub454\ub2e4. \uc5ec\uae30\uc11c \\(\\vec p\\)\ub294 \ub9e4\uac1c\ubcc0\uc218\ub4e4\uc774\uba70 \\(B\\)\ub294 \\(\\phi\\)\uc5d0 \uc790\uc720\ubcc0\uc218\ub85c \ub098\ud0c0\ub098\uc9c0 \uc54a\ub294\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4.<\/p>\n<\/div>\n

      \ubd84\ub9ac \uacf5\ub9ac\uaf34\uc758 \ud575\uc2ec\uc740 \uc774\ubbf8 \uc874\uc7ac\ud558\ub294 \uc9d1\ud569 \\(A\\) \uc548\uc5d0\uc11c\ub9cc \uc6d0\uc18c\ub97c \uace8\ub77c \ub0b8\ub2e4\ub294 \uc810\uc774\ub2e4. \ub530\ub77c\uc11c \u201c\uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubaa8\ub4e0 \ub300\uc0c1\uc758 \uc9d1\ud569\u201d\uc744 \ubb34\uc81c\ud55c\uc73c\ub85c \ub9cc\ub4dc\ub294 \uc18c\ubc15\ud55c \ubd84\ub958 \uc6d0\ub9ac\ub97c \ud5c8\uc6a9\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

      \n

      \ubb38\uc81c 9.3.<\/span>
      \n\\(A=6=\\{0,1,2,3,4,5\\}\\)\ub77c \ud558\uc790. \ubd84\ub9ac \uacf5\ub9ac\uaf34\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \uc9d1\ud569\uc774 \uc874\uc7ac\ud568\uc744 \uc124\uba85\ud558\uace0, \uac01 \uc9d1\ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

        \n
      1. \\(B=\\{x\\in A\\mid x\\in3\\}\\)<\/li>\n
      2. \\(C=\\{x\\in A\\mid3\\in x\\}\\)<\/li>\n
      3. \\(D=\\{x\\in A\\mid x=0\\vee x=5\\}\\)<\/li>\n<\/ol>\n<\/div>\n
        \n

        \uacf5\ub9ac 9.10. (\uce58\ud658 \uacf5\ub9ac\uaf34)<\/span><\/p>\n

        \ub17c\ub9ac\uc2dd \\(\\psi(x,y,\\vec p)\\)\uac00 \uc9d1\ud569 \\(A\\)\uc758 \uac01 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc720\uc77c\ud55c \\(y\\)\ub97c \uc815\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uadf8 \\(y\\)\ub4e4\uc744 \ubaa8\ub450 \ubaa8\uc740 \uc9d1\ud569\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc989 \uac01 \ub17c\ub9ac\uc2dd \\(\\psi\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[
        \n\\forall A\\,
        \n\\left(
        \n \\forall x\\in A\\,\\exists!y\\,\\psi(x,y,\\vec p)
        \n \\to
        \n \\exists B\\,\\forall y\\,
        \n \\bigl(y\\in B\\leftrightarrow\\exists x\\in A\\,\\psi(x,y,\\vec p)\\bigr)
        \n\\right).
        \n\\]<\/p>\n<\/div>\n

        \uce58\ud658 \uacf5\ub9ac\uaf34\uc740 \uc774\ubbf8 \uc9d1\ud569\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \ud568\uc218\uc758 \uce58\uc5ed\ub9cc\uc744 \ub9cc\ub4e4\uae30 \uc704\ud574 \ud544\uc694\ud55c \uac83\uc774 \uc544\ub2c8\ub2e4. \uc9d1\ud569\uc744 \uc815\uc758 \uac00\ub2a5\ud55c \uaddc\uce59\uc5d0 \ub530\ub77c \ubcc0\ud658\ud588\uc744 \ub54c \uadf8 \uac12\ub4e4\uc758 \ubaa8\uc784\uc774 \ub2e4\uc2dc \uc9d1\ud569\uc784\uc744 \ubcf4\uc7a5\ud558\ub294 \uac83\uc774 \ud575\uc2ec\uc774\ub2e4. \ud2b9\ud788 8\uc7a5<\/a>\uc758 \ucd08\ud55c\uc7ac\uadc0\uc5d0\uc11c \ub2e8\uacc4\ubcc4\ub85c \ub9cc\ub4e4\uc5b4\uc9c0\ub294 \uac12\ub4e4\uc744 \ud558\ub098\uc758 \uc9d1\ud569\uc73c\ub85c \ubaa8\uc73c\ub294 \ub370 \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud55c\ub2e4.<\/p>\n

        \n

        \uacf5\ub9ac 9.11. (\uc815\uce59\uc131 \uacf5\ub9ac)<\/span><\/p>\n

        \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubaa8\ub4e0 \uc9d1\ud569 \\(A\\)\uc5d0\ub294 \\(A\\)\uc640 \uc11c\ub85c\uc18c\uc778 \uc6d0\uc18c\uac00 \uc874\uc7ac\ud55c\ub2e4.
        \n\\[\\forall A\\,
        \n\\bigl(A\\ne\\varnothing\\to\\exists x\\in A\\,(x\\cap A=\\varnothing)\\bigr).\\]<\/p>\n<\/div>\n

        \uc815\uce59\uc131 \uacf5\ub9ac\ub85c\ubd80\ud130 \\(x\\in x\\)\uc778 \uc9d1\ud569\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\uba70, \uc9d1\ud569\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c \ud558\uac15\uc5f4
        \n\\[x_0\\ni x_1\\ni x_2\\ni\\cdots\\]
        \n\ub3c4 \uc874\uc7ac\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n

        \n

        \ubb38\uc81c 9.4.<\/span>
        \n\uc815\uce59\uc131 \uacf5\ub9ac\uc5d0 \uad00\ud558\uc5ec \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

          \n
        1. \\(A=\\{0,1,2\\}\\)\uc5d0\uc11c \\(x\\cap A=\\varnothing\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(x\\in A\\)\ub97c \ubaa8\ub450 \uad6c\ud558\uc2dc\uc624.<\/li>\n
        2. \ub450 \uc9d1\ud569 \\(x\\), \\(y\\)\uac00 \ub3d9\uc2dc\uc5d0 \\(x\\in y\\)\uc640 \\(y\\in x\\)\ub97c \ub9cc\uc871\uc2dc\ud0ac \uc218 \uc5c6\uc74c\uc744 \uc815\uce59\uc131 \uacf5\ub9ac\ub85c \uc99d\uba85\ud558\uc2dc\uc624.<\/li>\n
        3. \ud2b9\ud788 \\(x\\in x\\)\uc778 \uc9d1\ud569\uc774 \uc874\uc7ac\ud560 \uc218 \uc5c6\uc74c\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \uacf5\ub9ac 9.12. (\uc120\ud0dd\uacf5\ub9ac)<\/span><\/p>\n

          \\(F\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc815\uc758\uc5ed\uc774 \\(F\\)\uc774\uace0 \uac01 \\(A\\in F\\)\uc5d0 \ub300\ud558\uc5ec \\(f(A)\\in A\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(f\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc55e\uc758 \uc57d\uc2dd \ud45c\uae30\ub97c \uc0ac\uc6a9\ud558\uba74
          \n\\[
          \n\\forall F\\left(
          \n \\forall A\\in F\\,(A\\ne\\varnothing)
          \n \\to
          \n \\exists f\\bigl(\\operatorname{dom}(f)=F\\wedge
          \n \\forall A\\in F\\,(f(A)\\in A)\\bigr)
          \n\\right)
          \n\\]
          \n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n

          \uc774\ub7ec\ud55c \\(f\\)\ub97c \uc120\ud0dd\ud568\uc218<\/span>(choice function)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub294 ZF\ub85c\ubd80\ud130 \uc99d\uba85\ub418\uc9c0 \uc54a\uc73c\uba70 \uadf8 \ubd80\uc815\ub3c4 ZF\ub85c\ubd80\ud130 \uc99d\uba85\ub418\uc9c0 \uc54a\ub294\ub2e4. \uc815\ud655\ud788 \ub9d0\ud558\uba74 ZF\uac00 \ubb34\ubaa8\uc21c\uc774\ub77c\uace0 \uac00\uc815\ud560 \ub54c ZFC\uc640 \\(\\mathrm{ZF}+\\neg\\mathrm{AC}\\)\ub3c4 \uac01\uac01 \ubb34\ubaa8\uc21c\uc774\ub77c\ub294 \uc0c1\ub300\uc801 \ubb34\ubaa8\uc21c\uc131 \uacb0\uacfc\uac00 \uc54c\ub824\uc838 \uc788\ub2e4. \uc120\ud0dd\uacf5\ub9ac\uc640 \uc5ec\ub7ec \ub3d9\uce58\uba85\uc81c\ub294 \ub2e4\uc74c \uc7a5<\/a>\uc5d0\uc11c \uc790\uc138\ud788 \ub2e4\ub8ec\ub2e4.<\/p>\n

          \n

          \ubb38\uc81c 9.5.<\/span>
          \n\\(F=\\bigl\\{\\{0,1\\},\\{2,3,4\\},\\{\\varnothing\\}\\bigr\\}\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

            \n
          1. \\(F\\) \uc704\uc758 \uc120\ud0dd\ud568\uc218 \ud558\ub098\ub97c \uad6c\uccb4\uc801\uc73c\ub85c \uc4f0\uc2dc\uc624.<\/li>\n
          2. \\(F\\) \uc704\uc758 \uc120\ud0dd\ud568\uc218\ub294 \ubaa8\ub450 \uba87 \uac1c\uc778\uc9c0 \uad6c\ud558\uc2dc\uc624.<\/li>\n
          3. \uc774 \uc608\uc5d0\uc11c\ub294 \uc120\ud0dd\uacf5\ub9ac\ub97c \ub530\ub85c \uac00\uc815\ud558\uc9c0 \uc54a\uc544\ub3c4 \uc120\ud0dd\ud568\uc218\ub97c \uc9c1\uc811 \ub9cc\ub4e4 \uc218 \uc788\ub294 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

            3. \uc9d1\ud569 \uad6c\uc131\uc758 \uc608<\/h3>\n

            ZFC \uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc55e\uc5d0\uc11c \uc9c1\uad00\uc801\uc73c\ub85c \uc0ac\uc6a9\ud574 \uc628 \ub300\uc0c1\ub4e4\uc774 \uc2e4\uc81c \uc9d1\ud569\uc73c\ub85c \uad6c\uc131\ub428\uc744 \ud655\uc778\ud558\uc790.<\/p>\n

            (1) \uc21c\uc11c\uc30d\uc758 \uad6c\uc131<\/h4>\n

            \ucfe0\ub77c\ud1a0\ud504\uc2a4\ud0a4(Kuratowski) \uc815\uc758\uc5d0 \ub530\ub77c
            \n\\[(a,b)=\\{\\{a\\},\\{a,b\\}\\}\\]
            \n\ub85c \ub454\ub2e4. \uc9dd \uacf5\ub9ac\ub97c \ub450 \ubc88 \uc0ac\uc6a9\ud558\uba74 \uc774 \uc9d1\ud569\uc758 \uc874\uc7ac\ub97c \uc5bb\ub294\ub2e4. \uc774 \uc815\uc758\ub294 \uc21c\uc11c\uc30d\uc758 \uae30\ubcf8 \uc131\uc9c8
            \n\\[(a,b)=(c,d)\\quad\\Longleftrightarrow\\quad a=c\\text{\uc774\uace0 }b=d\\]
            \n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n

            \n

            \ubb38\uc81c 9.6.<\/span>
            \n\\(0=\\varnothing\\), \\(1=\\{0\\}\\)\uc774\ub77c \ud558\uc790. \ucfe0\ub77c\ud1a0\ud504\uc2a4\ud0a4 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \uc21c\uc11c\uc30d\uc744 \uc2e4\uc81c \uc9d1\ud569\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/p>\n

              \n
            1. \\((0,0)\\)<\/li>\n
            2. \\((0,1)\\)<\/li>\n
            3. \\((1,0)\\)<\/li>\n<\/ol>\n

              \ub610\ud55c \\((0,1)\\ne(1,0)\\)\uc784\uc744 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 9.7.<\/span>
              \n\uc21c\uc11c\uc30d\uc758 \uc131\uc9c8
              \n\\[(a,b)=(c,d)\\quad\\Longleftrightarrow\\quad a=c\\text{\uc774\uace0 }b=d\\]
              \n\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 9.8.<\/span>
              \n\\(3\\)-\uc21c\uc11c\uc30d\uc744
              \n\\[(a,b,c)=((a,b),c)\\]
              \n\ub85c \uc815\uc758\ud558\uace0,
              \n\\[(a,b,c)=(a’,b’,c’)\\quad\\Longleftrightarrow\\quad a=a’,\\ b=b’,\\ c=c’\\]
              \n\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624. \ub610\ud55c \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc720\ud55c \\(n\\)-\uc21c\uc11c\uc30d\uc744 \uc7ac\uadc0\uc801\uc73c\ub85c \uc815\uc758\ud558\ub294 \ubc29\ubc95\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              \uc774 \uc815\uc758\ub294 3\uc7a5<\/a>\uc5d0\uc11c \uc0ac\uc6a9\ud55c \\(n\\)-\uc21c\uc11c\uc30d\uc744 \uc2e4\uc81c \uc9d1\ud569\uc73c\ub85c \ubd80\ud638\ud654\ud558\ub294 \ud55c \ubc29\ubc95\uc774\ub2e4. \uad04\ud638\ub97c \uce58\ub294 \ubc29\uc2dd\uc5d0 \ub530\ub77c \uc5bb\ub294 \uc9d1\ud569 \uc790\uccb4\ub294 \ub2ec\ub77c\uc9c8 \uc218 \uc788\uc9c0\ub9cc, \uc131\ubd84\uc744 \ubcf4\uc874\ud558\ub294 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc788\ub2e4.<\/p>\n

              (2) \ub370\uce74\ub974\ud2b8 \uacf1\uc758 \uad6c\uc131<\/h4>\n

              \\(a\\in A\\), \\(b\\in B\\)\uc774\uba74 \ucfe0\ub77c\ud1a0\ud504\uc2a4\ud0a4 \uc21c\uc11c\uc30d\uc5d0 \ub300\ud558\uc5ec
              \n\\[(a,b)\\subseteq\\mathcal P(A\\cup B),\\quad (a,b)\\in\\mathcal P(\\mathcal P(A\\cup B))\\]
              \n\uc774\ub2e4. \ub530\ub77c\uc11c
              \n\\[A\\times B
              \n=\\{p\\in\\mathcal P(\\mathcal P(A\\cup B))\\mid
              \n\\exists a\\in A\\,\\exists b\\in B\\,(p=(a,b))\\}\\]
              \n\ub85c \uc4f8 \uc218 \uc788\ub2e4. \\(A\\cup B\\)\ub294 \uc9dd \uacf5\ub9ac\uc640 \ud569\uc9d1\ud569 \uacf5\ub9ac\ub85c, \uadf8 \ub450 \ubc88\uc758 \uba71\uc9d1\ud569\uc740 \uba71\uc9d1\ud569 \uacf5\ub9ac\ub85c \ub9cc\ub4e4 \uc218 \uc788\uace0, \ub9c8\uc9c0\ub9c9 \ubd80\ubd84\uc9d1\ud569\uc740 \ubd84\ub9ac \uacf5\ub9ac\uaf34\ub85c \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \\(A\\times B\\)\ub294 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

              (3) \uad00\uacc4\uc640 \ud568\uc218\uc758 \uad6c\uc131<\/h4>\n

              \uad00\uacc4 \\(R\\)\uc740 \\(A\\times B\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ubbc0\ub85c \uba71\uc9d1\ud569 \uacf5\ub9ac\uc5d0 \uc758\ud574 \uc9d1\ud569\uc73c\ub85c \uc874\uc7ac\ud55c\ub2e4. \ud568\uc218\uc758 \uadf8\ub798\ud504\ub3c4 \ud2b9\ubcc4\ud55c \uad00\uacc4\uc774\ubbc0\ub85c \uc9d1\ud569\uc774\ub2e4.<\/p>\n

              4\uc7a5<\/a>\uc5d0\uc11c \uc774 \ucc45\uc758 \ud568\uc218 \\(f\\colon A\\to B\\)\ub294 \uc815\uc758\uc5ed \\(A\\), \uacf5\uc5ed \\(B\\), \ub300\uc751 \uad00\uacc4\ub97c \ud568\uaed8 \uc9c0\uc815\ud55c \uac83\uc73c\ub85c \uc815\uc758\ud558\uc600\ub2e4. \uc774 \uc138 \uc790\ub8cc \uc790\uccb4\ub97c \ud558\ub098\uc758 \uc9d1\ud569\uc73c\ub85c \ubd80\ud638\ud654\ud558\uace0 \uc2f6\ub2e4\uba74 \uc55e\uc758 \uc21c\uc11c\uc30d \uad6c\uc131\uc744 \ubc18\ubcf5\ud558\uc5ec \uc608\ub97c \ub4e4\uc5b4
              \n\\[(A,B,G_f)=((A,B),G_f)\\]
              \n\uc640 \uac19\uc740 \\(3\\)-\uc21c\uc11c\uc30d\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \\(G_f\\subseteq A\\times B\\)\ub294
              \n\\[\\forall a\\in A\\,\\exists!b\\in B\\,((a,b)\\in G_f)\\]
              \n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc774\ub2e4.<\/p>\n

              (4) \ucd08\uae30\uc11c\uc218\uc758 \uc874\uc7ac<\/h4>\n
              \n

              \uc815\ub9ac 9.13. (\ucd08\uae30\uc11c\uc218\uc758 \uc874\uc7ac)<\/span><\/p>\n

              \uc9d1\ud569 \\(A\\)\uac00 \uc815\ub82c \uac00\ub2a5\ud558\uba74 \\(A\\)\uc640 \ub300\ub4f1\ud55c \ucd08\uae30\uc11c\uc218\uac00 \uc815\ud655\ud788 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/span>
              \n\\(A\\) \uc704\uc758 \ubaa8\ub4e0 \uc774\ud56d\uad00\uacc4\ub294 \\(\\mathcal P(A\\times A)\\)\uc758 \uc6d0\uc18c\uc774\ubbc0\ub85c, \\(A\\)\ub97c \uc815\ub82c\uc2dc\ud0a4\ub294 \uad00\uacc4\ub4e4\uc758 \ubaa8\uc784
              \n\\[\\mathcal W=\\{R\\in\\mathcal P(A\\times A)\\mid R\\text{\ub294 }A\\text{\uc758 \uc815\ub82c\uc21c\uc11c}\\}\\]
              \n\uc740 \ubd84\ub9ac \uacf5\ub9ac\uaf34\uc5d0 \uc758\ud574 \uc9d1\ud569\uc774\ub2e4. \\(A\\)\uac00 \uc815\ub82c \uac00\ub2a5\ud558\ubbc0\ub85c \\(\\mathcal W\\ne\\varnothing\\)\uc774\ub2e4.<\/p>\n

              \uc815\ub9ac 8.6<\/a>\uc5d0 \ub530\ub77c \uac01 \\(R\\in\\mathcal W\\)\uc5d0\ub294 \\((A,R)\\)\uacfc \uc21c\uc11c\ub3d9\ud615\uc778 \uc720\uc77c\ud55c \uc11c\uc218 \\(\\alpha_R\\)\uac00 \ub300\uc751\ud55c\ub2e4. \ucd08\ud55c\uc7ac\uadc0\uc640 \uce58\ud658 \uacf5\ub9ac\uaf34\uc744 \uc774\uc6a9\ud558\uba74
              \n\\[T=\\{\\alpha_R\\mid R\\in\\mathcal W\\}\\]
              \n\ub3c4 \uc9d1\ud569\uc774 \ub41c\ub2e4. \\(T\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc11c\uc218\ub4e4\uc758 \uc9d1\ud569\uc774\ubbc0\ub85c \ucd5c\uc18c\uc6d0\uc18c \\(\\kappa\\)\ub97c \uac16\ub294\ub2e4. \uac01 \\(\\alpha_R\\)\ub294 \\(A\\)\uc640 \ub300\ub4f1\ud558\ubbc0\ub85c \\(\\kappa\\sim A\\)\uc774\ub2e4. \ub9cc\uc57d \\(\\beta<\\kappa\\)\uc774\uace0 \\(\\beta\\sim A\\)\uc778 \uc11c\uc218 \\(\\beta\\)\uac00 \uc788\ub2e4\uba74, \uc77c\ub300\uc77c\ub300\uc751 \\(A\\to\\beta\\)\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\beta\\)\uc758 \uc21c\uc11c\ub97c \\(A\\)\ub85c \uc62e\uaca8 \\(A\\) \uc704\uc758 \uc815\ub82c\uc21c\uc11c \\(R\\)\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uadf8\ub7ec\uba74 \\(\\alpha_R=\\beta\\)\uc774\ubbc0\ub85c \\(\\beta\\in T\\)\uac00 \ub418\uc5b4 \\(\\kappa\\)\uc758 \ucd5c\uc18c\uc131\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\kappa\\)\ub294 \\(A\\)\uc640 \ub300\ub4f1\ud55c \ucd08\uae30\uc11c\uc218\uc774\ub2e4. \ucd08\uae30\uc11c\uc218\uc758 \ucd5c\uc18c\uc131\uc5d0\uc11c \uc720\uc77c\uc131\ub3c4 \ub530\ub978\ub2e4.<\/span><\/p>\n<\/div>\n

              \ub530\ub77c\uc11c 8\uc7a5<\/a>\uc5d0\uc11c \uc0ac\uc6a9\ud55c \ucd08\uae30\uc11c\uc218\ub294 \uc815\ub82c \uac00\ub2a5\ud55c \uc9d1\ud569\uc5d0 \ub300\ud574\uc11c ZF \uc548\uc5d0\uc11c \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \ubaa8\ub4e0 \uc9d1\ud569\uc774 \uc815\ub82c \uac00\ub2a5\ud558\ub2e4\ub294 \uc815\ub82c \uc815\ub9ac\ub294 \uc120\ud0dd\uacf5\ub9ac\uc640 \ub3d9\uce58\uc774\ubbc0\ub85c, \ubaa8\ub4e0 \uae30\uc218\ub97c \ucd08\uae30\uc11c\uc218\ub85c \ub300\ud45c\ud558\ub824\uba74 \uc120\ud0dd\uacf5\ub9ac\uac00 \ud544\uc694\ud558\ub2e4.<\/p>\n

              \ub610\ud55c \uc815\ub9ac 8.12<\/a>\uc640 \uac19\uc740 \ucd08\ud55c\uc7ac\uadc0\uc758 \uc644\uc804\ud55c \uc9d1\ud569\ub860\uc801 \uc99d\uba85\uc5d0\ub294 \uce58\ud658 \uacf5\ub9ac\uaf34\uc774 \uc0ac\uc6a9\ub41c\ub2e4. \uc774 \uc810\uc740 \uc790\uc5f0\uc218 \uc704\uc758 \uc7ac\uadc0\uac00 \ubb34\ud55c \uacf5\ub9ac \uc704\uc5d0\uc11c \uc774\ub8e8\uc5b4\uc9c0\ub294 \uac83\uc744 \ucd08\ud55c\uc7ac\uadc0\ub97c \uc0ac\uc6a9\ud558\uc5ec \ud655\uc7a5\ud55c \uac83\uc73c\ub85c \ubcfc \uc218 \uc788\ub2e4.<\/p>\n

              4. \ub7ec\uc140\uc758 \uc5ed\uc124<\/h3>\n

              \uc18c\ubc15\ud55c \uc9d1\ud569\ub860\uc5d0\uc11c
              \n\\[\\mathcal R=\\{x\\mid x\\notin x\\}\\]
              \n\ub97c \uc9d1\ud569\uc774\ub77c\uace0 \ud5c8\uc6a9\ud558\uba74
              \n\\[\\mathcal R\\in\\mathcal R\\quad\\Longleftrightarrow\\quad \\mathcal R\\notin\\mathcal R\\]
              \n\ub77c\ub294 \ubaa8\uc21c\uc774 \uc0dd\uae34\ub2e4.<\/p>\n

              ZFC\uc758 \ubd84\ub9ac \uacf5\ub9ac\uaf34\uc740 \uc774\ubbf8 \uc874\uc7ac\ud558\ub294 \uc9d1\ud569 \uc548\uc5d0\uc11c\ub9cc \ubd80\ubd84\uc9d1\ud569\uc744 \ub9cc\ub4e4\uac8c \ud558\ubbc0\ub85c \uc774\ub7ec\ud55c \\(\\mathcal R\\)\uc758 \uc874\uc7ac\ub97c \ud5c8\uc6a9\ud558\uc9c0 \uc54a\ub294\ub2e4. \ud074\ub798\uc2a4 \ud45c\uae30\ubc95\uc73c\ub85c\ub294 \\(\\mathcal R\\)\uc744 \ub9d0\ud560 \uc218 \uc788\uc9c0\ub9cc \uc774\ub294 \uace0\uc720\ud074\ub798\uc2a4\uc774\ub2e4. \ub354\uad6c\ub098 \uc815\uce59\uc131 \uacf5\ub9ac\uc5d0 \uc758\ud574 \ubaa8\ub4e0 \uc9d1\ud569 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\notin x\\)\uc774\ubbc0\ub85c, \ud074\ub798\uc2a4 \\(\\mathcal R\\)\uc740 \uc0ac\uc2e4 \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4 \\(V\\)\uc640 \uac19\ub2e4.<\/p>\n

              \n

              \ubb38\uc81c 9.9.<\/span>
              \n\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \ubd84\ub9ac \uacf5\ub9ac\uaf34\ub85c
              \n\\[R_A=\\{x\\in A\\mid x\\notin x\\}\\]
              \n\ub97c \uc815\uc758\ud558\uc790.<\/p>\n

                \n
              1. \\(R_A\\)\uac00 \uc9d1\ud569\uc784\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n
              2. \\(R_A\\in A\\)\ub77c\uace0 \uac00\uc815\ud558\uba74 \ubaa8\uc21c\uc774 \uc0dd\uae40\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
              3. \ub530\ub77c\uc11c \uc784\uc758\uc758 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(R_A\\notin A\\)\uc784\uc744 \ubcf4\uc774\uace0, \uc774\uac83\uc774 \ub7ec\uc140\uc758 \uc5ed\uc124\uacfc \ubaa8\uc21c\ub418\uc9c0 \uc54a\ub294 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

                \uc5f0\uc2b5\ubb38\uc81c<\/h3>\n
                \n

                \ubb38\uc81c 9.10.<\/span>
                \n\ub2e4\uc74c \uc9d1\ud569\uc774\ub098 \ud568\uc218\ub97c \uad6c\uc131\ud560 \ub54c \ud575\uc2ec\uc801\uc73c\ub85c \uc0ac\uc6a9\ud558\ub294 ZFC\uc758 \uacf5\ub9ac \ub610\ub294 \uacf5\ub9ac\uaf34\uc744 \uac01\uac01 \ub9d0\ud558\uc2dc\uc624. \ud544\uc694\ud55c \uacbd\uc6b0 \ub458 \uc774\uc0c1\uc758 \uacf5\ub9ac\ub97c \uc801\uc73c\uc2dc\uc624.<\/p>\n

                  \n
                1. \\(\\{x,y\\}\\)<\/li>\n
                2. \\(A\\cup B\\)<\/li>\n
                3. \\(\\mathcal P(A)\\)<\/li>\n
                4. \\(\\{x\\in A\\mid\\phi(x)\\}\\)<\/li>\n
                5. \uc9d1\ud569 \\(A\\)\uc758 \uac01 \\(x\\)\uc5d0 \uc720\uc77c\ud55c \\(y\\)\ub97c \ub300\uc751\uc2dc\ud0a4\ub294 \uc815\uc758 \uac00\ub2a5\ud55c \uaddc\uce59\uc758 \uac12\ub4e4\uc744 \ubaa8\uc740 \uc9d1\ud569<\/li>\n
                6. \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569 \\(F\\)\uc5d0\uc11c \uac01 \uc6d0\uc18c \ud558\ub098\uc529\uc744 \uace0\ub974\ub294 \ud568\uc218<\/li>\n<\/ol>\n<\/div>\n
                  \n

                  \ubb38\uc81c 9.11.<\/span>
                  \n\uc678\uc5f0 \uacf5\ub9ac\uc640 \uacf5\uc9d1\ud569\uc758 \uc874\uc7ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uacf5\uc9d1\ud569\uc774 \uc720\uc77c\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                  \n

                  \ubb38\uc81c 9.12.<\/span>
                  \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\\}\\)\uac00 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

                  \n

                  \ubb38\uc81c 9.13.<\/span>
                  \n\uc9d1\ud569 \\(A=\\{0,1,2\\}\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218\uc801 \uc131\uc9c8 \\(\\psi(x,y)\\)\ub97c \u201c\\(y=2x\\)\u201d\ub77c\uace0 \ud558\uc790. \uce58\ud658 \uacf5\ub9ac\uaf34\ub85c \uc5bb\ub294 \uc0c1\uc758 \uc9d1\ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                  \n

                  \ubb38\uc81c 9.14.<\/span>
                  \n\uc9d1\ud569 \\(A\\), \\(B\\)\uc5d0 \ub300\ud558\uc5ec
                  \n\\[A\\times B\\subseteq\\mathcal P(\\mathcal P(A\\cup B))\\]
                  \n\uc784\uc744 \ud655\uc778\ud558\uace0, \uc9dd \uacf5\ub9ac, \ud569\uc9d1\ud569 \uacf5\ub9ac, \uba71\uc9d1\ud569 \uacf5\ub9ac, \ubd84\ub9ac \uacf5\ub9ac\uaf34\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(A\\times B\\)\uac00 \uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                  \n

                  \ubb38\uc81c 9.15.<\/span>
                  \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

                    \n
                  1. \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4<\/li>\n
                  2. \ubaa8\ub4e0 \ub2e8\uc6d0\uc18c \uc9d1\ud569\uc758 \ud074\ub798\uc2a4<\/li>\n
                  3. \ubaa8\ub4e0 \uc720\ud55c\uc9d1\ud569\uc758 \ud074\ub798\uc2a4<\/li>\n
                  4. \uc790\uae30 \uc790\uc2e0\uc744 \uc6d0\uc18c\ub85c \uac16\ub294 \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ud074\ub798\uc2a4<\/li>\n<\/ol>\n<\/div>\n
                    \n

                    \ubb38\uc81c 9.16.<\/span>
                    \nZFC \uacf5\ub9ac\uacc4\uc5d0\uc11c \ub2e4\uc74c \uc9d1\ud569\uc774 \uc874\uc7ac\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

                      \n
                    1. \\(\\{1,2,3\\}\\)<\/li>\n
                    2. \\(\\{\\varnothing,\\{\\varnothing\\},\\{\\{\\varnothing\\}\\}\\}\\)<\/li>\n
                    3. \ub450 \uc9d1\ud569 \\(A\\), \\(B\\)\uc758 \uad50\uc9d1\ud569 \\(A\\cap B\\)<\/li>\n<\/ol>\n<\/div>\n
                      \n

                      \uc9d1\ud569\uacfc \uc218\ub9ac\ub17c\ub9ac \uccab\uac78\uc74c \ubaa9\ucc28 \ubcf4\uae30<\/a><\/p>\n

                      \uba85\uc81c\uc640 \ub17c\ub9ac<\/a><\/span>
                      \n                      \uc9d1\ud569\uc758 \uac1c\ub150<\/a><\/span>
                      \n                      \ub2e4\uc591\ud55c \uc9d1\ud569\uc758 \uc5f0\uc0b0<\/a><\/span>
                      \n                      \uad00\uacc4\uc640 \ud568\uc218<\/a><\/span>
                      \n                      \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/span>
                      \n                      \uc790\uc5f0\uc218<\/a><\/span>
                      \n                      \uc9d1\ud569\uc758 \uae30\uc218<\/a><\/span>
                      \n                      \uc9d1\ud569\uc758 \uc11c\uc218<\/a><\/span>
                      \n                      \uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/a><\/span>
                      \n                      \uc120\ud0dd \uacf5\ub9ac<\/a><\/span>
                      \n                      \ud615\uc2dd\ub17c\ub9ac<\/a><\/span>
                      \n                      \uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150<\/a><\/span>
                      \n                      \uba85\uc81c\ub17c\ub9ac\uc758 \uac74\uc804\uc131\uacfc \uc644\uc804\uc131<\/a><\/span>
                      \n                      \uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860<\/a><\/span>
                      \n                      \uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a><\/span>
                      \n                      \uc77c\uacc4\ub17c\ub9ac\uc758 \ucd94\ub860\uaddc\uce59<\/a><\/span>
                      \n                      \uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131<\/a><\/span>
                      \n                      \ud398\uc544\ub178 \uc0b0\uc220<\/a><\/span>
                      \n                      \ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/a><\/span>\n<\/div>\n<\/div>\n

                      <\/p>\n","protected":false},"excerpt":{"rendered":"

                      \uc55e \uc7a5\uae4c\uc9c0\ub294 \uc9d1\ud569\uc744 \uc9c1\uad00\uc801\uc73c\ub85c \uc0ac\uc6a9\ud558\uba74\uc11c \uc790\uc5f0\uc218, \uae30\uc218, \uc11c\uc218\ub97c \uad6c\uc131\ud558\uc600\ub2e4. \uadf8\ub7ec\ub098 \u201c\uc5b4\ub5a4 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub300\uc0c1\ub4e4\uc744 \ubaa8\ub450 \ubaa8\uc73c\uba74 \uc9d1\ud569\uc774 \ub41c\ub2e4\u201d\uace0 \ubb34\uc81c\ud55c\uc73c\ub85c \ud5c8\uc6a9\ud558\uba74 \ub7ec\uc140\uc758 \uc5ed\uc124\uacfc \uac19\uc740 \ubaa8\uc21c\uc774 \uc0dd\uae34\ub2e4. \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \uc5b4\ub5a4 \uc9d1\ud569\uc758 \uc874\uc7ac\ub97c \uc778\uc815\ud560 \uac83\uc778\uc9c0 \uacf5\ub9ac\ub85c \uc81c\ud55c\ud558\uc5ec \uc774\ub7ec\ud55c \ubb38\uc81c\ub97c \ud53c\ud55c\ub2e4. \uc774 \ubd80\uc5d0\uc11c\ub294 \ud604\ub300 \uc218\ud559\uc758 \ud45c\uc900\uc801\uc778 \uae30\ucd08 \uac00\uc6b4\ub370 \ud558\ub098\uc778 ZFC\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. ZFC\ub294 \uccb4\ub974\uba5c\ub85c-\ud504\ub81d\ucf08 \uc9d1\ud569\ub860(ZF)\uc5d0 \uc120\ud0dd\uacf5\ub9ac\ub97c \ub354\ud55c \uccb4\uacc4\uc774\ub2e4. \ud2b9\ud788 \uc55e \uc7a5\uc5d0\uc11c \uc0ac\uc6a9\ud55c \uc790\uc5f0\uc218 \uc9d1\ud569\uc758 \uc874\uc7ac, \ucd08\ud55c\uc7ac\uadc0, \ucd08\uae30\uc11c\uc218\uc758 \uc874\uc7ac\uac00 \uc5b4\ub5a4…<\/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":6,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9264\/revisions"}],"predecessor-version":[{"id":10085,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9264\/revisions\/10085"}],"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}]}}