{"id":9256,"date":"2025-10-17T20:01:18","date_gmt":"2025-10-17T11:01:18","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9256"},"modified":"2025-10-20T18:48:41","modified_gmt":"2025-10-20T09:48:41","slug":"ch06-natural-numbers","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\/","title":{"rendered":"\uc790\uc5f0\uc218"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p>\uc9d1\ud569\uc758 \uae30\ubcf8 \uc131\uc9c8 \uc911 \ud558\ub098\ub294 \uc9d1\ud569\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\uc774\ub2e4. \uc720\ud55c\uc9d1\ud569\uc758 \uacbd\uc6b0 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \uc790\uc5f0\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc9c0\ub9cc, \ubb34\ud55c\uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ub2e4\ub8e8\ub824\uba74 \ub354 \uc815\uad50\ud55c \uac1c\ub150\uc774 \ud544\uc694\ud558\ub2e4. \uce78\ud1a0\uc5b4\ub294 19\uc138\uae30 \ub9d0 \ubb34\ud55c\uc758 \ud06c\uae30\ub97c \uccb4\uacc4\uc801\uc73c\ub85c \uc5f0\uad6c\ud558\uba74\uc11c \uae30\uc218\uc640 \uc11c\uc218\ub77c\ub294 \ud601\uc2e0\uc801\uc778 \uac1c\ub150\uc744 \ub3c4\uc785\ud588\ub2e4. \uc774\uac83\uc740 \uc218\ud559\uc5d0\uc11c \ubb34\ud55c\uc744 \ub2e4\ub8e8\ub294 \ubc29\uc2dd\uc744 \uc644\uc804\ud788 \ubc14\uafb8\uc5b4 \ub193\uc558\ub2e4.<\/p>\n<p>\uae30\uc218(cardinal number)\ub294 \uc9d1\ud569\uc758 \ud06c\uae30 \ub610\ub294 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac1c\ub150\uc774\ub2e4. \ub450 \uc9d1\ud569 \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud558\uba74 \uac19\uc740 \uae30\uc218\ub97c \uac00\uc9c4\ub2e4\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc740 \uc720\ud55c\uc9d1\ud569\ubfd0\ub9cc \uc544\ub2c8\ub77c \ubb34\ud55c\uc9d1\ud569\uc5d0\ub3c4 \uc801\uc6a9\ub418\uc5b4, \uc11c\ub85c \ub2e4\ub978 \ud06c\uae30\uc758 \ubb34\ud55c\uc774 \uc874\uc7ac\ud568\uc744 \ubcf4\uc5ec\uc900\ub2e4. \ud55c\ud3b8 \uc11c\uc218(ordinal number)\ub294 \uc9d1\ud569\uc758 \uc21c\uc11c \uad6c\uc870\ub97c \ub098\ud0c0\ub0b4\ub294 \uac1c\ub150\uc73c\ub85c, \ubb34\ud55c\uc9d1\ud569\ub3c4 \uc801\uc808\ud788 \uc21c\uc11c\ub97c \ubd80\uc5ec\ud558\uba74 \uadf8 \uc21c\uc11c \uc720\ud615\uc744 \uc11c\uc218\ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\ud765\ubbf8\ub86d\uac8c\ub3c4 \ubaa8\ub4e0 \uc790\uc5f0\uc218\ub294 \uc720\ud55c\uae30\uc218\uc778 \ub3d9\uc2dc\uc5d0 \uc720\ud55c\uc11c\uc218\uc774\ub2e4. \uc608\ub97c \ub4e4\uba74, \uc790\uc5f0\uc218 \\(3\\)\uc740 \uc138 \uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\uc218\uc774\uba74\uc11c, \ub3d9\uc2dc\uc5d0 \uc138 \ubc88\uc9f8 \uc704\uce58\uae4c\uc9c0\uc758 \uc21c\uc11c\ub97c \ub098\ud0c0\ub0b4\ub294 \uc11c\uc218\uc774\uae30\ub3c4 \ud558\ub2e4. \uc774 \ubd80\uc5d0\uc11c\ub294 \uba3c\uc800 \uc790\uc5f0\uc218\ub97c \uc9d1\ud569\ub860\uc801\uc73c\ub85c \uad6c\uc131\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8 \ud6c4, \uc774\uac83\uc744 \ubc14\ud0d5\uc73c\ub85c \uae30\uc218\uc640 \uc11c\uc218\uc758 \uac1c\ub150\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<p><!-- \n\n<h2>6. \uc790\uc5f0\uc218<\/h2>\n\n --><\/p>\n<p>\uc790\uc5f0\uc218\ub294 \uc218\ud559\uc758 \uac00\uc7a5 \uae30\ubcf8\uc801\uc778 \ub300\uc0c1\uc774\uc9c0\ub9cc, \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud558\uae30\ub294 \uc27d\uc9c0 \uc54a\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc9d1\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc790\uc5f0\uc218\ub97c \uad6c\uc131\ud558\uace0, \uc790\uc5f0\uc218\uc758 \uc5f0\uc0b0\uc744 \uc815\uc758\ud558\uba70, \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc758 \uc6d0\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h3>1. \uc9d1\ud569\uc744 \uc0ac\uc6a9\ud55c \uc790\uc5f0\uc218\uc758 \uc815\uc758<\/h3>\n<p><span class=\"defined\">\ud3f0 \ub178\uc774\ub9cc<\/span>(von Neumann)\uc758 \ubc29\ubc95\uc744 \ub530\ub77c \uc790\uc5f0\uc218\ub97c \uad6c\uc131\ud574 \ubcf4\uc790. \uba3c\uc800 \\(0\\)\uc744 \uacf5\uc9d1\ud569\uc73c\ub85c \uc815\uc758\ud558\uace0, \uac01 \uc790\uc5f0\uc218\ub97c \uc790\uc2e0\ubcf4\ub2e4 \uc791\uc740 \ubaa8\ub4e0 \uc790\uc5f0\uc218\uc758 \uc9d1\ud569\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n0 &#038;= \\varnothing = \\{\\},\\\\[6pt]<br \/>\n1 &#038;= \\{0\\} = \\{\\varnothing\\},\\\\[6pt]<br \/>\n2 &#038;= \\{0,\\, 1\\} = \\{\\varnothing,\\, \\{\\varnothing\\}\\},\\\\[6pt]<br \/>\n3 &#038;= \\{0,\\, 1,\\, 2\\} = \\{\\varnothing,\\, \\{\\varnothing\\},\\, \\{\\varnothing,\\, \\{\\varnothing\\}\\}\\},\\\\[6pt]<br \/>\n&#038;\\vdots<br \/>\n\\end{aligned}\\]<br \/>\n\uc77c\ubc18\uc801\uc73c\ub85c, \uc790\uc5f0\uc218 \\(n\\)\uc758 <span class=\"defined\">\ub530\ub984\uc218<\/span>(successor) \\(S(n)\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[S(n) = n \\cup \\{n\\}.\\]<br \/>\n\uc9c1\uad00\uc801\uc73c\ub85c \\(S(n)\\)\uc740 \\(n\\)\uc758 \ub2e4\uc74c \uc218, \uc989 \\(n+1\\)\uc744 \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc0dd\uac01\ud574\ub3c4 \ubb34\ubc29\ud558\ub2e4. \uc608\ub97c \ub4e4\uc5b4,<\/p>\n<ul>\n<li>\\(S(0) = \\varnothing \\cup \\{\\varnothing\\} = \\{\\varnothing\\} = 1\\),<\/li>\n<li>\\(S(1) = \\{0\\} \\cup \\{\\{0\\}\\} = \\{0,\\, 1\\} = 2\\),<\/li>\n<li>\\(S(2) = \\{0,\\, 1\\} \\cup \\{\\{0,\\, 1\\}\\} = \\{0,\\, 1,\\, 2\\} = 3\\).<\/li>\n<\/ul>\n<p>\uc774 \uc815\uc758\uc5d0 \uc758\ud558\uba74 \uac01 \uc790\uc5f0\uc218\ub294 \\(n = \\{0,\\, 1,\\, 2,\\, \\ldots,\\, n-1\\}\\)\uc774\uba70, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(m < n\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(m \\in n\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\(m \\leq n\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(m \\subseteq n\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc9d1\ud569 \\(N\\)\uc774 \ub450 \uc870\uac74<\/p>\n<ul>\n<li>\\(0\\in N\\)\uc774\ub2e4,<\/li>\n<li>\\(n\\in N\\)\uc77c \ub54c\ub9c8\ub2e4 \\(S(n)\\in N\\)\uc774\ub2e4<\/li>\n<\/ul>\n<p>\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(N\\)\uc744 <span class=\"defined\">\uadc0\ub0a9\uc801 \uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \ubaa8\ub4e0 \uadc0\ub0a9\uc801 \uc9d1\ud569\ub4e4\uc758 \uad50\uc9d1\ud569\uc744 <span class=\"defined\">\uc790\uc5f0\uc218 \uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\mathbb{N}\\) \ub610\ub294 \\(\\omega\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. [\ucd08\uc911\ub4f1 \uad50\uc721\uacfc\uc815\uc5d0\uc11c\ub294 \\(1\\) \uc774\uc0c1\uc778 \uc815\uc218\ub97c \uc790\uc5f0\uc218\ub77c\uace0 \ubd80\ub974\uc9c0\ub9cc \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \\(0\\) \uc774\uc0c1\uc778 \uc815\uc218\ub97c \uc790\uc5f0\uc218\ub77c\uace0 \ubd80\ub978\ub2e4.] \uc989 \uc790\uc5f0\uc218 \uc9d1\ud569\uc744 \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\left\\{ 0,\\, 1,\\, 2,\\, 3,\\, 4,\\, \\ldots \\right\\}.\\]<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 6.1.<\/span><br \/>\n\uadc0\ub0a9\uc801 \uc9d1\ud569\uc758 \uad50\uc9d1\ud569\uc774 \uadc0\ub0a9\uc801 \uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>2. \uc218\ud559\uc801 \uadc0\ub0a9\ubc95<\/h3>\n<p><span class=\"defined\">\uc218\ud559\uc801 \uadc0\ub0a9\ubc95<\/span>(mathematical induction)\uc740 \uc790\uc5f0\uc218\uc5d0 \ub300\ud55c \uba85\uc81c\ub97c \uc99d\uba85\ud558\ub294 \uac15\ub825\ud55c \ubc29\ubc95\uc774\ub2e4. \uc989, \uc790\uc5f0\uc218\uc5d0 \ub300\ud55c \uba85\uc81c \\(P(n)\\)\uc774 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \ubaa8\ub4e0 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud574 \\(P(n)\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li><strong>\uae30\ucd08 \ub2e8\uacc4<\/strong>: \\(P(0)\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/li>\n<li><strong>\uadc0\ub0a9 \ub2e8\uacc4<\/strong>: \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud574, \\(P(k)\\)\uac00 \uc131\ub9bd\ud558\uba74 \\(P(k+1)\\)\ub3c4 \uc131\ub9bd\ud55c\ub2e4.<\/li>\n<\/ol>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 6.2.<\/span><br \/>\n\uc704 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc740 \uc99d\uba85\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc815\uc758\uc5d0\ub3c4 \uc0ac\uc6a9\ub41c\ub2e4. \uc774\uac83\uc744 <span class=\"defined\">\uadc0\ub0a9\uc801 \uc815\uc758<\/span>(recursive definition)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc790\uc5f0\uc218\uc758 \ub367\uc148, \uacf1\uc148, \uac70\ub4ed\uc81c\uacf1\uc744 \uc815\uc758\ud560 \ub54c \uadc0\ub0a9\uc801 \uc815\uc758\uac00 \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<h3>3. \uc790\uc5f0\uc218\uc758 \ub367\uc148<\/h3>\n<p>\uc790\uc5f0\uc218\uc758 \ub367\uc148\uc744 \uadc0\ub0a9\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc989, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(m\\), \\(n\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nm + 0 &#038;= m ,\\\\[6pt]<br \/>\nm + S(n) &#038;= S(m + n) .<br \/>\n\\end{aligned}\\]<br \/>\n\uc608\ub97c \ub4e4\uc5b4, \\(2 + 2\\)\ub97c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n2 + 2 &#038;= 2 + S(1)<br \/>\n= S(2 + 1)<br \/>\n= S(2 + S(0))\\\\[6pt]<br \/>\n&#038;= S(S(2 + 0))<br \/>\n= S(S(2))<br \/>\n= S(3)<br \/>\n= 4 .<br \/>\n\\end{aligned}\\]<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 6.3.<\/span><br \/>\n\ub367\uc148\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(3+2\\)<\/li>\n<li>\\(2+3\\)<\/li>\n<li>\\(4+1\\)<\/li>\n<li>\\(1+4\\)<\/li>\n<\/ol>\n<\/div>\n<p>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \ub367\uc148\uc758 \uae30\ubcf8 \uc131\uc9c8\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li><strong>\uacb0\ud569\ubc95\uce59<\/strong>: \\((m + n) + p = m + (n + p)\\)<\/li>\n<li><strong>\uad50\ud658\ubc95\uce59<\/strong>: \\(m + n = n + m\\)<\/li>\n<li><strong>\uc18c\uac70\ubc95\uce59<\/strong>: \\(m + p = n + p\\)\uc774\uba74 \\(m = n\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 6.4.<\/span><br \/>\n\uc790\uc5f0\uc218 \\(n\\), \\(m\\), \\(p\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624. (\uac8c\uc2dc\uae00 \uc544\ub798\ucabd\uc758 \u201c7. \uc5f0\uc0b0 \ubc95\uce59\uc758 \uc99d\uba85\u201d \ubd80\ubd84\uc744 \ucc38\uc870\ud558\uc2dc\uc624.)<\/p>\n<ol class=\"parenthesis\">\n<li>\\((n+m)+p = n+(m+p)\\)<\/li>\n<li>\\(n+S(p) = S(n) +p\\)<\/li>\n<li>\\(p+0 = 0+p\\)<\/li>\n<li>\\(n+1 = S(n)\\)<\/li>\n<li>\\(n+p = p+n\\)<\/li>\n<\/ol>\n<\/div>\n<h3>4. \uc790\uc5f0\uc218\uc758 \uacf1\uc148<\/h3>\n<p>\uc790\uc5f0\uc218\uc758 \uacf1\uc148\ub3c4 \uadc0\ub0a9\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc989, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(m\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nm \\cdot 0 &#038;= 0 ,\\\\[6pt]<br \/>\nm \\cdot S(n) &#038;= m \\cdot n + m .<br \/>\n\\end{aligned}\\]<br \/>\n\uc608\ub97c \ub4e4\uc5b4, \\(2 \\cdot 3\\)\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n2 \\cdot 3 &#038;= 2 \\cdot S(2)<br \/>\n= 2 \\cdot 2 + 2\\\\[6pt]<br \/>\n&#038;= 2 \\cdot S(1) + 2<br \/>\n= (2 \\cdot 1 + 2) + 2\\\\[6pt]<br \/>\n&#038;= (2 \\cdot S(0) + 2) + 2<br \/>\n= ((2 \\cdot 0 + 2) + 2) + 2\\\\[6pt]<br \/>\n&#038;= (0 + 2) + 2 + 2<br \/>\n= 6 .<br \/>\n\\end{aligned}\\]<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 6.5.<\/span><br \/>\n\uacf1\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(3\\cdot 2\\)<\/li>\n<li>\\(4\\cdot 1\\)<\/li>\n<li>\\(1\\cdot 3\\)<\/li>\n<li>\\(3\\cdot 3\\)<\/li>\n<\/ol>\n<\/div>\n<p>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uacf1\uc148\uc758 \uae30\ubcf8 \uc131\uc9c8\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li><strong>\uacb0\ud569\ubc95\uce59<\/strong>: \\((m \\cdot n) \\cdot p = m \\cdot (n \\cdot p)\\)<\/li>\n<li><strong>\uad50\ud658\ubc95\uce59<\/strong>: \\(m \\cdot n = n \\cdot m\\)<\/li>\n<li><strong>\ubd84\ubc30\ubc95\uce59<\/strong>: \\(m \\cdot (n + p) = m \\cdot n + m \\cdot p\\)<\/li>\n<li><strong>\ud56d\ub4f1\uc6d0<\/strong>: \\(m \\cdot 1 = m\\)<\/li>\n<li><strong>\uc601\uc6d0<\/strong>: \\(m \\cdot 0 = 0\\)<\/li>\n<\/ul>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 6.6.<\/span><br \/>\n\uc790\uc5f0\uc218 \\(n\\), \\(m\\), \\(p\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(n\\cdot 1 =n\\)<\/li>\n<li>\\(n\\cdot (m+p) = n\\cdot m + n\\cdot p\\)<\/li>\n<li>\\((n\\cdot m)\\cdot p = n \\cdot (m\\cdot p)\\)<\/li>\n<li>\\(0 \\cdot p = 0\\)<\/li>\n<li>\\(1 \\cdot p = p\\)<\/li>\n<li>\\((1+n) \\cdot p = 1\\cdot p + n\\cdot p\\)<\/li>\n<li>\\(n\\cdot p = p\\cdot n\\)<\/li>\n<li>\\((m+p) \\cdot n = m\\cdot n + p\\cdot n\\)<\/li>\n<\/ol>\n<\/div>\n<h3>5. \uc790\uc5f0\uc218\uc758 \uac70\ub4ed\uc81c\uacf1<\/h3>\n<p>\uc790\uc5f0\uc218\uc758 \uac70\ub4ed\uc81c\uacf1\ub3c4 \uadc0\ub0a9\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc989, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(m\\), \\(n\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nm^0 &#038;= 1 ,\\\\[6pt]<br \/>\nm^{S(n)} &#038;= m^n \\cdot m .<br \/>\n\\end{aligned}\\]<br \/>\n\ud2b9\ubcc4\ud788 \\(0^0 = 1\\)\ub85c \uc815\uc758\ud55c\ub2e4\ub294 \uc810\uc5d0 \uc8fc\ubaa9\ud558\uc790. \uc774\uac83\uc740 \uc870\ud569\ub860\uacfc \uc9d1\ud569\ub860\uc5d0\uc11c \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc815\uc758\uc774\ub2e4.<\/p>\n<p>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uac70\ub4ed\uc81c\uacf1\uc758 \uae30\ubcf8 \uc131\uc9c8\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li>\\(m^{a+b} = m^a \\cdot m^b\\)<\/li>\n<li>\\((m^a)^b = m^{a \\cdot b}\\)<\/li>\n<li>\\((m \\cdot n)^a = m^a \\cdot n^a\\)<\/li>\n<li>\\(1^n = 1\\)<\/li>\n<li>\\(m^1 = m\\)<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 6.7.<\/span><br \/>\n\uc790\uc5f0\uc218\uc758 \uac70\ub4ed\uc81c\uacf1\uc758 \uc131\uc9c8\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 6.8.<\/span><br \/>\n\uc790\uc5f0\uc218 \\(m\\)\uacfc \\(n\\)\uc744 \uc9d1\ud569\uc73c\ub85c \ubcf4\uc558\uc744 \ub54c, \\(m^n\\)\uc740 \uc815\uc758\uc5ed\uc774 \\(n\\)\uc774\uace0 \uacf5\uc5ed\uc774 \\(m\\)\uc778 \ud568\uc218\uc758 \ubaa8\uc784\uc774\ub2e4. \uc774\ub54c \\(m^n\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>6. \ud398\uc544\ub178 \uacf5\ub9ac<\/h3>\n<p><span class=\"defined\">\ud398\uc544\ub178 \uacf5\ub9ac<\/span>(Peano axioms)\ub294 \uc790\uc5f0\uc218\uc758 \ubcf8\uc9c8\uc801 \uc131\uc9c8\uc744 \uc124\uba85\ud558\ub294 \uacf5\ub9ac \uccb4\uacc4\uc774\ub2e4. \uc9d1\ud569 \\(\\mathbb{N}\\), \uc6d0\uc18c \\(0 \\in \\mathbb{N}\\), \ud568\uc218 \\(S: \\mathbb{N} \\to \\mathbb{N}\\)\uc774 \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c &#8216;\ud398\uc544\ub178 \uacf5\ub9ac\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4&#8217;\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(0 \\in \\mathbb{N}\\) (\uc601\uc740 \uc790\uc5f0\uc218\uc774\ub2e4)<\/li>\n<li>\\(n \\in \\mathbb{N}\\)\uc774\uba74 \\(S(n) \\in \\mathbb{N}\\)\uc774\ub2e4. (\uc790\uc5f0\uc218\uc758 \ub530\ub984\uc218\ub294 \uc790\uc5f0\uc218\uc774\ub2e4.)<\/li>\n<li>\ubaa8\ub4e0 \\(n \\in \\mathbb{N}\\)\uc5d0 \ub300\ud574 \\(S(n) \\neq 0\\)\uc774\ub2e4. (\uc601\uc740 \uc5b4\ub5a4 \uc790\uc5f0\uc218\uc758 \ub530\ub984\uc218\ub3c4 \uc544\ub2c8\ub2e4.)<\/li>\n<li>\\(S(m) = S(n)\\)\uc774\uba74 \\(m = n\\)\uc774\ub2e4. (\ub530\ub984\uc218 \ud568\uc218\ub294 \uc77c\ub300\uc77c\ud568\uc218\uc774\ub2e4.)<\/li>\n<li><strong>\uadc0\ub0a9 \uacf5\ub9ac<\/strong>: \\(A \\subseteq \\mathbb{N}\\)\uc774 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(A = \\mathbb{N}\\)\uc774\ub2e4.<br \/>\n&nbsp;&nbsp;&nbsp;&#8211; \\(0 \\in A\\);<br \/>\n&nbsp;&nbsp;&nbsp;&#8211; \\(n \\in A\\)\uc774\uba74 \\(S(n) \\in A\\)\uc774\ub2e4.\n<\/li>\n<\/ol>\n<p>\uc9d1\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uad6c\uc131\ud55c \uc790\uc5f0\uc218\ub294 \ud398\uc544\ub178 \uacf5\ub9ac\ub97c \ub9cc\uc871\ud55c\ub2e4.<\/p>\n<ul>\n<li>\uacf5\ub9ac 1: \\(0 = \\varnothing\\)\ub294 \uc815\uc758\uc5d0 \uc758\ud574 \uc790\uc5f0\uc218\uc774\ub2e4.<\/li>\n<li>\uacf5\ub9ac 2: \uc790\uc5f0\uc218 \\(n\\)\uc758 \ub530\ub984\uc218 \\(S(n) = n \\cup \\{n\\}\\)\ub3c4 \uc790\uc5f0\uc218\uc774\ub2e4.<\/li>\n<li>\uacf5\ub9ac 3: \\(S(n) = n \\cup \\{n\\}\\)\ub294 \ud56d\uc0c1 \ube44\uc5b4\uc788\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(0\\)\uc774 \ub420 \uc218 \uc5c6\ub2e4.<\/li>\n<li>\uacf5\ub9ac 4: \\(S(m) = S(n)\\)\uc774\uba74 \\(m \\cup \\{m\\} = n \\cup \\{n\\}\\)\uc774\ubbc0\ub85c \\(m = n\\)\uc774\ub2e4.<\/li>\n<li>\uacf5\ub9ac 5: \uc790\uc5f0\uc218 \uc9d1\ud569\uc774 \ubaa8\ub4e0 \uadc0\ub0a9\uc801 \uc9d1\ud569\uc758 \uad50\uc9d1\ud569\uc774\ubbc0\ub85c, \uacf5\ub9ac 5\uac00 \uc131\ub9bd\ud55c\ub2e4. (\ubb38\uc81c 6.2 \ucc38\uc870)<\/li>\n<\/ul>\n<h3>7. \uc5f0\uc0b0 \ubc95\uce59\uc758 \uc99d\uba85<\/h3>\n<p>\uc790\uc5f0\uc218\uc758 \uc5f0\uc0b0 \ubc95\uce59\ub4e4\uc740 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \ub367\uc148\uc758 \uacb0\ud569\ubc95\uce59<br \/>\n\\[(m + n) + p = m + (n + p)\\]<br \/>\n\ub294 \\(p\\)\uc5d0 \ub300\ud55c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li><strong>\uae30\ucd08 \ub2e8\uacc4<\/strong>: \\(p = 0\\)\uc77c \ub54c, \\((m + n) + 0 = m + n = m + (n + 0)\\).<\/li>\n<li><strong>\uadc0\ub0a9 \ub2e8\uacc4<\/strong>: \\((m + n) + p = m + (n + p)\\)\ub77c\uace0 \uac00\uc815\ud558\uba74,<br \/>\n\\[\\begin{aligned}<br \/>\n(m + n) + S(p) &#038;= S((m + n) + p) &#038;\\quad \\text{(\ub367\uc148\uc758 \uc815\uc758)}\\\\[6pt]<br \/>\n&#038;= S(m + (n + p)) &#038;\\quad \\text{(\uadc0\ub0a9 \uac00\uc815)}\\\\[6pt]<br \/>\n&#038;= m + S(n + p) &#038;\\quad \\text{(\ub367\uc148\uc758 \uc815\uc758)}\\\\[6pt]<br \/>\n&#038;= m + (n + S(p)). &#038;\\quad \\text{(\ub367\uc148\uc758 \uc815\uc758)}<br \/>\n\\end{aligned}\\]<\/li>\n<\/ul>\n<p>\ub2e4\ub978 \ubc95\uce59\ub4e4\ub3c4 \uc720\uc0ac\ud55c \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \uc99d\uba85\uc740 \uc790\uc5f0\uc218\uc758 \uadc0\ub0a9\uc801 \uad6c\uc870\ub97c \ud65c\uc6a9\ud55c\ub2e4.<\/p>\n<h3>8. \uc21c\uc11c \uad00\uacc4<\/h3>\n<p>\uc790\uc5f0\uc218\uc758 \uc21c\uc11c \uad00\uacc4\ub3c4 \ub2e4\uc74c\uacfc \uac19\uc774 \uc9d1\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc790\uc5f0\uc2a4\ub7fd\uac8c \uc815\uc758\ub41c\ub2e4. [\uc138 \uc2dd \uc911 \ud558\ub098\ub97c \uc815\uc758\ub85c \uc0bc\uc73c\uba74, \ub2e4\ub978 \ub450 \uc2dd\uc740 \uc815\ub9ac\uac00 \ub41c\ub2e4.]<\/p>\n<ul>\n<li>\\(m < n \\quad \\Longleftrightarrow \\quad m \\in n\\)<\/li>\n<li>\\(m \\leq n \\quad \\Longleftrightarrow \\quad m \\subseteq n\\)<\/li>\n<li>\\(m < n \\quad \\Longleftrightarrow \\quad \\exists k \\neq 0 : m + k = n\\)<\/li>\n<\/ul>\n<p>\uc774 \uc21c\uc11c \uad00\uacc4\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li><strong>\uc804\uc21c\uc11c<\/strong>: \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(m,\\, n\\)\uc5d0 \ub300\ud558\uc5ec \\(m < n\\), \\(m = n\\), \\(m > n\\) \uc911 \ub531 \ud558\ub098\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/li>\n<li><strong>\uc815\ub82c\uc131<\/strong>: \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc790\uc5f0\uc218\uc758 \ubd80\ubd84\uc9d1\ud569\uc740 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<li><strong>\uc544\ub974\ud0a4\uba54\ub370\uc2a4 \uc131\uc9c8<\/strong>: \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud574 \\(n < m\\)\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 6.9.<\/span><br \/>\n\uc790\uc5f0\uc218\uc758 \uc815\ub82c\uc131\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 6.10.<\/span><br \/>\n\ub3d9\uce58\uad00\uacc4\uc640 \uc0c1\uc9d1\ud569(quotient set)\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc790\uc5f0\uc218\ub97c \uc815\uc218\ub85c \ud655\uc7a5\ud558\ub294 \ubc29\ubc95\uc744 \ucc3e\uc544\ubcf4\uc2dc\uc624.<\/p>\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>\uc9d1\ud569\uc758 \uae30\ubcf8 \uc131\uc9c8 \uc911 \ud558\ub098\ub294 \uc9d1\ud569\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\uc774\ub2e4. \uc720\ud55c\uc9d1\ud569\uc758 \uacbd\uc6b0 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \uc790\uc5f0\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc9c0\ub9cc, \ubb34\ud55c\uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ub2e4\ub8e8\ub824\uba74 \ub354 \uc815\uad50\ud55c \uac1c\ub150\uc774 \ud544\uc694\ud558\ub2e4. \uce78\ud1a0\uc5b4\ub294 19\uc138\uae30 \ub9d0 \ubb34\ud55c\uc758 \ud06c\uae30\ub97c \uccb4\uacc4\uc801\uc73c\ub85c \uc5f0\uad6c\ud558\uba74\uc11c \uae30\uc218\uc640 \uc11c\uc218\ub77c\ub294 \ud601\uc2e0\uc801\uc778 \uac1c\ub150\uc744 \ub3c4\uc785\ud588\ub2e4. \uc774\uac83\uc740 \uc218\ud559\uc5d0\uc11c \ubb34\ud55c\uc744 \ub2e4\ub8e8\ub294 \ubc29\uc2dd\uc744 \uc644\uc804\ud788 \ubc14\uafb8\uc5b4 \ub193\uc558\ub2e4. \uae30\uc218(cardinal number)\ub294 \uc9d1\ud569\uc758 \ud06c\uae30 \ub610\ub294 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac1c\ub150\uc774\ub2e4. \ub450 \uc9d1\ud569 \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud558\uba74 \uac19\uc740 \uae30\uc218\ub97c \uac00\uc9c4\ub2e4\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc740 \uc720\ud55c\uc9d1\ud569\ubfd0\ub9cc \uc544\ub2c8\ub77c&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":106,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9256","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9256","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=9256"}],"version-history":[{"count":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9256\/revisions"}],"predecessor-version":[{"id":9436,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9256\/revisions\/9436"}],"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=9256"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}