\u81ea\u7136\u6570\u4e0ePeano\u516c\u7406<\/title> <\/head> <body> <\/p>\n<div style=\"padding:20px;\">\n<center><\/p>\n<h1>\u81ea\u7136\u6570\u4e0ePeano\u516c\u7406<\/h1>\n\u6458\u8981<\/b> \n\u8fd9\u8282\u8bfe\u8bb2\u8ff0\u4e86\u81ea\u7136\u6570\u4ee5\u53ca\u5b83\u4eec\u662f\u5982\u4f55\u901a\u8fc7Peano\u516c\u7406\u6765\u5b9a\u4e49\u7684\uff1a\u4e00\u7cfb\u5217\u5efa\u7acb\u5176\u57fa\u672c\u6027\u8d28\u7684\u6570\u5b66\u539f\u5219\u3002\u5b83\u8fd8\u89e3\u91ca\u4e86\u5982\u4f55\u4f7f\u7528\u7b26\u53f7\u6765\u8868\u793a\u81ea\u7136\u6570\u7684\u540e\u7eed\u6570\uff0c\u5982\u4f55\u7b26\u53f7\u5316\u5730\u8868\u793a\u5b83\u4eec\uff0c\u4ee5\u53ca\u5982\u4f55\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8fdb\u884c\u5f52\u7eb3\u8bc1\u660e\u3002<\/em><\/p>\n\u5b66\u4e60\u76ee\u6807<\/b><\/p>\n<\/center><\/p>\n<ol>\n<li>\u7406\u89e3<\/strong> Peano\u516c\u7406\u5bf9\u81ea\u7136\u6570\u7684\u9610\u8ff0\u3002<\/li>\n<li>\u7406\u89e3<\/strong> \u81ea\u7136\u6570\u7684\u7b26\u53f7\u8868\u793a\u7684\u5236\u5b9a\u3002<\/li>\n<\/ol>\n<center><\/p>\n\u76ee\u5f55<\/strong><\/p>\n<a href=\"#1\">Peano\u516c\u7406\u4e0e\u81ea\u7136\u6570<\/strong><\/a> \n<a href=\"#2\">\u81ea\u7136\u6570\u7684\u5f52\u7eb3\u539f\u7406<\/strong><\/a> \n<a href=\"#3\">\u5173\u4e8e\u8bc1\u660e\u7684\u8bc4\u8bba<\/a><\/p>\n<\/center><\/p>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/w-BznjX88No\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/div>\n<a name=\"1\"><\/a><\/p>\n<h2>\u76ae\u4e9a\u8bfa\u516c\u7406\u4e0e\u81ea\u7136\u6570<\/h2>\n\u6240\u8c13\u7684\u81ea\u7136\u6570<\/em>\uff0c\u4e5f\u88ab\u79f0\u4e3a\u6b63\u6574\u6570<\/em>\uff0c\u662f\u6211\u4eec\u7528\u6765\u8ba1\u6570\u548c\u6d4b\u91cf\u7684\u6570\u3002\u5728\u7b97\u672f\u4e2d\u6700\u7b80\u5355\u7684\u64cd\u4f5c\u662f\u8ba1\u6570\u3002\u8fd9\u4e9b\u6570\u5b57\u662f\u901a\u8fc7\u76ae\u4e9a\u8bfa\u516c\u7406<\/em><\/strong>\u5b9a\u4e49\u7684\uff0c\u8fd9\u662f\u4e00\u7cfb\u5217\u5efa\u7acb\u8fd9\u4e9b\u6570\u5b57\u5982\u4f55\u5de5\u4f5c\u7684\u6570\u5b66\u539f\u5219\u3002<\/p>\n<ol style=\"text-align: justify;\">\n<li>\u00ab1<\/span>\u00ab\u662f\u4e00\u4e2a\u81ea\u7136\u6570<\/li>\n<li>\u5982\u679cn<\/span>\u662f\u81ea\u7136\u6570\uff0c\u90a3\u4e48\u5b83\u7684\u540e\u7ee7S(n)<\/span>\u4e5f\u662f\u3002<\/li>\n<li>\u00ab1<\/span>\u00ab\u4e0d\u662f\u4efb\u4f55\u81ea\u7136\u6570\u7684\u540e\u7ee7\u3002<\/li>\n<li>\u5982\u679cS(n) = S(m)<\/span>\uff0c\u90a3\u4e48n=m<\/span>\u3002<\/li>\n<li>\u5982\u679c1<\/span>\u5c5e\u4e8e\u67d0\u4e2a\u96c6\u5408A<\/span>\uff1b\u5e76\u4e14\u7ed9\u5b9a\u4e00\u4e2a\u5728A<\/span>\u4e2d\u7684\u4efb\u610fk<\/span>\uff0cS(k)<\/span>\u4e5f\u5728A<\/span>\u4e2d\uff0c\u90a3\u4e48A<\/span>\u5c31\u662f\u81ea\u7136\u6570\u7684\u96c6\u5408\uff0c\u8868\u793a\u4e3a\\mathbb{N}<\/span>\u3002<\/li>\n<\/ol>\n\u5f53\u6211\u4eec\u7814\u7a76\u76ae\u4e9a\u8bfa\u7684\u516c\u7406\u65f6\uff0c\u6211\u4eec\u610f\u8bc6\u5230\u7b26\u53f7\u00bb1<\/span>\u00ab\u5b9e\u9645\u4e0a\u53ea\u662f\u4e00\u4e2a\u8868\u793a\u7b26\u53f7\uff0c\u7528\u4e8e\u6307\u4ee3\u4e00\u4e2a\u7279\u5b9a\u7684\u81ea\u7136\u6570\u3002\u8fd9\u4e2a\u6570\u5b57\u6ee1\u8db3\u8fd9\u4e9b\u5c5e\u6027\u3002\u5c31\u50cf1<\/span>\u8868\u793a\u00bb\u7b2c\u4e00\u4e2a\u81ea\u7136\u6570\u00bb\uff0c\u6211\u4eec\u4e5f\u4f7f\u7528\u7b26\u53f7\uff08\u5bf9\u6211\u4eec\u6765\u8bf4\u662f\u719f\u6089\u7684\uff09\u6765\u8868\u793a\u5b83\u7684\u540e\u7ee7\u3002<\/p>\n<ul style=\"text-align: justify;\">\n<li>2=S(1)<\/span><\/li>\n<li>3=S(2)<\/span><\/li>\n<li>4=S(3) \\\\ \\vdots<\/span><\/li>\n<\/ul>\n\u4f9d\u6b64\u7c7b\u63a8\u3002\u8fd9\u6837\uff0c\u7b26\u53f71<\/span>, 2<\/span>, 3<\/span>, \u7b49\u7b49… \u5b9e\u9645\u4e0a\u662f\u4ee3\u88681<\/span>\u7684\u4e0d\u540c\u540e\u7ee7\u7684\u62bd\u8c61\u5b9e\u4f53\u3002\u6240\u6709\u8fd9\u4e9b\u5bf9\u8c61\u7684\u96c6\u5408\u5c31\u662f\u81ea\u7136\u6570\uff0c\u6211\u4eec\u901a\u8fc7\u4ee5\u4e0b\u65b9\u5f0f\u8868\u793a\u5b83\u4eec:<\/p>\n\\mathbb{N}=\\{1,2,3,4,\\cdots \\}<\/span>\n\u6211\u4eec\u4e5f\u8bf4\u81ea\u7136\u6570\u6309\u7167\u987a\u5e8f\u6392\u5217\uff0c\u5373\u81ea\u7136\u6570\u7684\u987a\u5e8f:<\/p>\n1,2,3,4,5,6,7,8,9,10,11,12, \\cdots <\/span>\n<a name=\"2\"><\/a><\/p>\n<h2>\u81ea\u7136\u6570\u7684\u5f52\u7eb3\u539f\u7406<\/h2>\n\u81ea\u7136\u6570\u7684\u4e00\u4e2a\u91cd\u8981\u7279\u70b9\u662f\uff0c\u6bcf\u4e00\u4e2a\u6570\u5b57\u540e\u9762\u90fd\u6709\u4e00\u4e2a\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6709\u65e0\u9650\u7684\u81ea\u7136\u6570\u3002\u6211\u4eec\u53ef\u4ee5\u4ece\u7b2c\u4e94\u516c\u7406\uff0c\u6216\u5f52\u7eb3\u539f\u7406<\/strong>\uff0c\u6765\u611f\u77e5\u8fd9\u4e00\u70b9\uff0c\u5b83\u7684\u8868\u8ff0\u5982\u4e0b\uff1a<\/p>\n\u5982\u679c\u4e00\u4e2a\u6027\u8d28\u5bf9\u4e8e1<\/span>\u6210\u7acb\uff1b\u5982\u679c\u627f\u8ba4\u5b83\u5bf9\u4e8e\u4efb\u4f55\u81ea\u7136\u6570k<\/span>\u90fd\u6210\u7acb\uff0c\u90a3\u4e48\u5b83\u5bf9\u4e8e\u4e0b\u4e00\u4e2aS(k)<\/span>\u4e5f\u6210\u7acb\uff1b\u90a3\u4e48\u8fd9\u4e2a\u6027\u8d28\u5bf9\u4e8e\u6240\u6709\u81ea\u7136\u6570\u90fd\u6210\u7acb\u3002<\/em><\/p>\n\u5f52\u7eb3\u539f\u7406\u4e0d\u4ec5\u4e3a\u81ea\u7136\u6570\u63d0\u4f9b\u4e86\u4e00\u4e2a\u57fa\u7840\uff0c\u800c\u4e14\u662f\u4e00\u4e2a\u6709\u7528\u7684\u5de5\u5177\uff0c\u7528\u6765\u8bc1\u660e\u81ea\u7136\u6570\u7684\u67d0\u4e2a\u6027\u8d28\u662f\u5426\u6210\u7acb\u3002\u4e3a\u4e86\u67e5\u770b\u8fd9\u4e00\u70b9\uff0c\u8ba9\u6211\u4eec\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\uff1a<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify;\">\n\u793a\u4f8b\uff1a<\/strong> \u901a\u8fc7\u5f52\u7eb3\u539f\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\u6bcf\u4e2a\u81ea\u7136\u6570\u90fd\u4e0e\u5176\u540e\u7ee7\u4e0d\u540c\u3002<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\">\n\u867d\u7136\u8fd9\u662f\u663e\u800c\u6613\u89c1\u7684\uff0c\u4f46\u5b83\u6709\u52a9\u4e8e\u7406\u89e3\u5f53\u901a\u8fc7\u5f52\u7eb3\u6cd5\u8bc1\u660e\u65f6\u7684\u6b65\u9aa4\u3002<\/p>\n\u8bc1\u660e<\/strong><\/p>\n<ul>\n<li>\n\u5f88\u660e\u663e\uff0c1<\/span>\u4e0eS(1)=2<\/span>\u4e0d\u540c\u3002\u8fd9\u662f\u521d\u59cb\u6b65\u9aa4\uff0c<\/strong>\u6211\u4eec\u9a8c\u8bc1\u6027\u8d28\u5bf9\u7b2c\u4e00\u4e2a\u5143\u7d20\u6210\u7acb\u3002<\/p>\n<\/li>\n<li>\n\u5047\u8bbe\u6027\u8d28\u5bf9\u4efb\u610f\u7684k<\/span>\u6210\u7acb\uff0c\u5373k\\neq S(k)<\/span>\uff0c\u6211\u4eec\u8981\u505a\u7684\u662f\u8bc1\u660e\u57fa\u4e8e\u6b64\u53ef\u8bc1\u660e\u5b83\u5bf9S(k)<\/span>\u4e5f\u6210\u7acb\uff08\u5373\uff0c\u8bc1\u660eS(k)\\neq S(S(k))<\/span>\uff09\u3002\u8fd9\u662f\u5f52\u7eb3\u6b65\u9aa4\u3002<\/strong>\u5982\u679c\u8fd9\u4e24\u6b65\u5b8c\u6210\uff0c\u90a3\u4e48\u53ef\u4ee5\u8bf4\u5f52\u7eb3\u662f\u5b8c\u6574\u7684\uff0c\u5e76\u4e14\u6027\u8d28\u5bf9\u6240\u6709\u81ea\u7136\u6570\u90fd\u6210\u7acb\u3002<\/p>\n<\/p>\n[1]<\/strong> \u9996\u5148\uff0c\u6ce8\u610fS(k) \\neq k,<\/span>\u7b49\u4ef7\u4e8e\u8bf4\\neg [k=S(k)]<\/span>\u3002<\/p>\n[2]<\/strong> \u4f46\u7531\u4e8ek<\/span>\u548cS(k)<\/span>\u90fd\u662f\u81ea\u7136\u6570\uff0c\u6839\u636e\u7b2c\u4e8c\u516c\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u8bf4\u5b83\u4eec\u90fd\u6709\u540e\u7ee7\uff1aS(k)<\/span>\u548cS(S(k))<\/span>\uff0c\u5b83\u4eec\u4e5f\u90fd\u662f\u81ea\u7136\u6570\u3002<\/p>\n[3]<\/strong> \u7136\u540e\uff0c\u6839\u636e\u7b2c\u56db\u516c\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u8bf4\uff1aS(k) = S(S(k))<\/span>\u610f\u5473\u7740k = S(k)<\/span>\u3002\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u5199\uff1a<\/p>\n\\left[ S(k) = S(S(k)) \\right] \\rightarrow \\left[k = S(k)\\right]<\/span>\n\u7531\u4e8e \u9006\u5426\u547d\u9898\uff0c\u8fd9\u7b49\u4ef7\u4e8e\uff1a<\/p>\n\\neg \\left[k = S(k)\\right] \\rightarrow \\neg \\left[ S(k) = S(S(k)) \\right] <\/span>\n[4]<\/strong> \u6700\u540e\uff0c\u4f7f\u7528 modus ponens \u5bf9\u4e0a\u8ff0\u8868\u8fbe\u5f0f\u548c[1]<\/strong>\u6b65\u9aa4\u4e2d\u5f97\u5230\u7684\u8868\u8fbe\u5f0f\u8fdb\u884c\u63a8\u5bfc\uff0c\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n\\neg \\left[ S(k) = S(S(k)) \\right] <\/span>\n\u8fd9\u4e0e\u4ee5\u4e0b\u8bf4\u6cd5\u76f8\u540c\uff1a<\/p>\n S(k) \\neq S(S(k)) <\/span>\n\u56e0\u6b64\uff0c\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\uff0c\u5982\u679cS(k) \\neq k,<\/span>\uff0c\u90a3\u4e48 S(k) \\neq S(S(k))<\/span>\u4e5f\u6210\u7acb\uff1b\u56e0\u4e3a1\\neq 2<\/span>\u662f\u663e\u800c\u6613\u89c1\u7684\uff0c\u6240\u4ee5\u5f52\u7eb3\u662f\u5b8c\u6574\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u5199\u4f5c\uff1a<\/p>\n\\left(\\forall n\\in\\mathbb{N}\\right)\\left(n \\neq S(n)\\right) <\/span>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<a name=\"3\"><\/a><\/p>\n<h3>\u5173\u4e8e\u8bc1\u660e\u7684\u8bc4\u8bba<\/h3>\n\u5c3d\u7ba1\u793a\u4f8b\u4e2d\u6240\u9648\u8ff0\u7684\u5c5e\u6027\u975e\u5e38\u660e\u663e\uff0c\u4f46\u5728\u6570\u5b66\u4e2d\uff0c\u8bc1\u660e\u901a\u5e38\u4e0d\u4f1a\u4fdd\u6301\u8fd9\u79cd\u660e\u663e\u6027\u3002\u6211\u4eec\u521a\u521a\u770b\u5230\u7684\u8fd9\u4e2a\u8bc1\u660e\u662f\u4e00\u4e2a\u4f8b\u5b50\uff0c\u5c55\u793a\u4e86\u5728\u8fdb\u884c\u6570\u5b66\u5de5\u4f5c\u65f6\u901a\u5e38\u4f1a\u505a\u4ec0\u4e48\u3002\u4e3a\u4e86\u5e2e\u52a9\u4f60\u7406\u89e3\u6570\u5b66\u7279\u6709\u7684\u63a8\u7406\u6280\u672f\uff0c\u6211\u5efa\u8bae\u4f60\u67e5\u770b\u4e3a<a href=\"http:\/\/toposuranos.com\/material\/zh\/category\/%e6%95%b0%e5%ad%a6\/%e6%95%b0%e7%90%86%e9%80%bb%e8%be%91\/%e5%91%bd%e9%a2%98%e9%80%bb%e8%be%91\/\" target=\"_blank\" rel=\"noopener\">\u6570\u5b66\u903b\u8f91<\/strong><\/a>\u8bfe\u7a0b\u63d0\u4f9b\u7684\u6750\u6599\u3002<\/p>\n<\/body> 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