{"id":569,"date":"2022-09-04T23:35:36","date_gmt":"2022-09-04T15:35:36","guid":{"rendered":"https:\/\/damakey.com\/?p=569"},"modified":"2023-05-24T16:01:36","modified_gmt":"2023-05-24T08:01:36","slug":"%e6%95%b8%e5%ad%b8%e5%a5%b3%e5%ad%a9-%e5%93%a5%e5%be%b7%e7%88%be%e4%b8%8d%e5%ae%8c%e5%82%99%e5%ae%9a%e7%90%86","status":"publish","type":"post","link":"https:\/\/damakey.com\/?p=569","title":{"rendered":"\u6578\u5b78\u5973\u5b69 \u54e5\u5fb7\u723e\u4e0d\u5b8c\u5099\u5b9a\u7406"},"content":{"rendered":"

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\u6731\u585e\u4f69·\u76ae\u4e9e\u8afe<\/span>(Giuseppe Peano<\/span>\uff0c<\/span>1858-1932)<\/span>\u662f\u7fa9\u5927\u5229\u7684\u6578\u5b78\u5bb6\u3002\u4ed6\u7279\u5225\u7684\u8ca2\u737b\uff0c\u662f\u523b\u610f\u5340\u5225\u4e86\u547d\u984c\u6f14\u7b97\u548c\u985e\u6f14\u7b97\uff0c\u4e26\u8a8d\u70ba\u547d\u984c\u6f14\u7b97\u66f4\u70ba\u57fa\u672c\u3002\u4ed6\u5728<\/span>1889<\/span>\u5e74\u63d0\u51fa\u81ea\u7136\u6578\u7684\u4e94\u689d\u516c\u7406\uff0c\u76ae\u4e9e\u8afe\u516c\u7406\uff0c\u5efa\u7acb\u4e86\u81ea\u7136\u6578\u7684\u7406\u8ad6\uff0c\u6a19\u5fd7\u8457\u7576\u6642\u6578\u5b78\u5206\u6790\u7b97\u8853\u5316\u7684\u7d42\u7d50\u3002<\/span>(<\/span>\u7dad\u57fa\u767e\u79d1<\/span>)<\/span><\/p>\n

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\u76ae\u4e9e\u8afe\u516c\u7406<\/span>*<\/span><\/p>\n

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PA1 1<\/span>\u662f\u81ea\u7136\u6578\u3002<\/span><\/p>\n

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PA2 <\/span>\u6bcf\u4e00\u500b\u78ba\u5b9a\u81ea\u7136\u6578<\/span>n<\/span>\uff0c\u6703\u6709\u4e00\u500b\u78ba\u5b9a\u5f8c\u7e7c\u6578<\/span>n’ <\/span>\u4e5f\u662f\u81ea\u7136\u6578\u3002<\/span><\/p>\n

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PA3 <\/span>\u5c0d\u4efb\u4f55\u81ea\u7136\u6578<\/span>n<\/span>\uff0c<\/span>n’≠ 1 <\/span>\u6703\u6210\u7acb\uff0c\u5373<\/span>1<\/span>\u4e0d\u662f\u4efb\u4f55\u81ea\u7136\u6578\u7684\u5f8c\u7e7c\u6578\u3002<\/span><\/p>\n

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PA4 <\/span>\u5c31\u5f8c\u7e7c\u6578\u800c\u8a00\uff0c\u5c0d\u4efb\u4f55\u81ea\u7136\u6578<\/span>m<\/span>\uff0c<\/span>n<\/span>\uff0c\u82e5<\/span>m’=n’<\/span>\uff0c\u5247<\/span>m=n<\/span>\u3002<\/span><\/p>\n

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PA5 <\/span>\u5c31\u4e00\u500b\u95dc\u65bc\u81ea\u7136\u6578\u7684\u547d\u984c<\/span>P(n)<\/span>\uff0c\u5018\u82e5\u4ee5\u4e0b<\/span>(a)<\/span>\u8207<\/span>(b)<\/span>\u5169\u500b\u689d\u4ef6\u6210\u7acb\u3002<\/span><\/p>\n

(a) P(1)<\/span>\u6210\u7acb\u3002<\/span><\/p>\n

(b) <\/span>\u5c0d\u4efb\u4f55\u81ea\u7136\u6578<\/span>k<\/span>\uff0c\u82e5\u7576<\/span>P(k)<\/span>\u6210\u7acb\uff0c\u5247<\/span>P(k’)<\/span>\u4e5f\u6210\u7acb\u3002<\/span><\/p>\n

\u5247\u5c0d\u4efb\u4f55\u81ea\u7136\u6578<\/span>n<\/span>\uff0c<\/span>P(n)<\/span>\u90fd\u6703\u6210\u7acb\u3002<\/span><\/p>\n

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\u524d\u8ff0\u7684\u76ae\u4e9e\u8afe<\/span>PA5<\/span>\u516c\u7406\uff0c\u4e5f\u7a31\u70ba\u6578\u5b78\u6b78\u7d0d\u6cd5\u3002<\/span><\/p>\n

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\u61c9\u7528\uff1a<\/span><\/p>\n

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\u8a66\u8b49\u660e\u4efb\u4f55\u81ea\u7136\u6578<\/span>n<\/span>\u4e0b\u9762\u6578\u5f0f\u6703\u6210\u7acb\u3002<\/span><\/p>\n

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1+3+5+…+(2n-1)=n^2<\/span><\/p>\n

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\u904b\u7528\u76ae\u4e9e\u8afe\u516c\u7406<\/span>PA5(<\/span>\u6578\u5b78\u6b78\u7d0d\u6cd5<\/span>)<\/span><\/p>\n

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P(1)=1=1^2=1<\/span>\u3002\u6210\u7acb\u3002<\/span><\/p>\n

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P(k)=1+3+5+…+(2k-1)=k^2<\/span><\/p>\n

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P(k’)=P(k+1)= 1+3+5+…+(2(k+1)-1)<\/span><\/p>\n

= 1+3+5+…+(2k+1)<\/span><\/p>\n

= 1+3+5+…+(2k-1)+(2k+1)<\/span><\/p>\n

= k^2+2k+1<\/span><\/p>\n

=(k+1)^2<\/span><\/p>\n

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P(k’)=P(k+1)= (k+1)^2 <\/span>\u4ea6\u6210\u7acb<\/span><\/p>\n

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\u6240\u4ee5\uff0c\u5c0d\u4efb\u4f55\u81ea\u7136\u6578<\/span>n<\/span>\uff0c<\/span>P(n)=n^2 <\/span>\u90fd\u6703\u6210\u7acb\u3002<\/span><\/p>\n

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Q.E.D. (Quod Erat Demonstrardum)**<\/span><\/p>\n

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\u5f8c\u4f86\u6578\u5b78\u754c\u6709\u4e00\u500b\u96c4\u5fc3\u52c3\u52c3\u7684\u300c\u5e0c\u4f2f\u7279\u8a08\u5283\u300d\uff0c\u5176\u4e2d\u95dc\u65bc\u81ea\u7136\u6578\u65b9\u9762\uff0c\u5c31\u662f\u5e0c\u671b\u80fd\u5920\u5efa\u7acb\u4e00\u500b\u5167\u90e8\u6c92\u6709\u76f8\u4e92\u77db\u76fe\u6027\u7684\u7cfb\u7d71<\/span>(<\/span>\u5177\u6709\u76f8\u5bb9\u6027<\/span>)<\/span>\uff0c\u5f9e\u516c\u7406\u51fa\u767c\u4e00\u6b65\u6b65\u53ef\u4ee5\u8b49\u660e\u81ea\u7136\u6578\u7684\u6240\u6709\u76f8\u95dc\u7684\u547d\u984c<\/span>(<\/span>\u5177\u5b8c\u5099\u6027<\/span>)<\/span>\u3002<\/span><\/p>\n

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\u4f46\u662f\u5f88\u4e0d\u5e78\u7684\uff0c\u4f9d\u7167\u300c\u54e5\u5fb7\u723e\u7b2c\u4e00\u4e0d\u5b8c\u5099\u5b9a\u7406\u300d\uff0c\u6709\u4e9b\u81ea\u7136\u6578\u76f8\u95dc\u7684\u547d\u984c\u662f\u771f\u7684\uff0c\u537b\u7121\u6cd5<\/span>(<\/span>\u5f9e\u76ae\u4e9e\u8afe\u516c\u7406\u6240\u9010\u6b65\u5efa\u7acb\u7684\u7cfb\u7d71<\/span>)<\/span>\u5f97\u5230\u8b49\u660e\uff0c\u6240\u4ee5\u8a72\u516c\u7406\u7cfb\u7d71\u662f\u300c\u4e0d\u5b8c\u5099\u300d\u7684\u3002\u800c\u300c\u54e5\u5fb7\u723e\u7684\u7b2c\u4e8c\u4e0d\u5b8c\u5099\u5b9a\u7406\u300d\u662f\u95dc\u65bc\u300c\u76f8\u5bb9\u6027\u300d\u7684\uff0c\u610f\u601d\u662f\u4e00\u500b\u5f37\u5ea6\u8db3\u4ee5\u8b49\u660e\u81ea\u8eab\u76f8\u5bb9\u6027\u7684\u81ea\u7136\u6578\u516c\u7406\u7cfb\u7d71\uff0c\u90a3\u9ebc\u5b83\u5c31\u6703\u662f\u4e0d\u76f8\u5bb9\u7684\uff01<\/span><\/p>\n

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\u53e4\u5fb7\u65af\u5766\u5b9a\u7406<\/span>(Goodstein’s theorem)<\/span>\u7684\u8b49\u660e\uff0c\u5c31\u662f\u4e00\u500b\u300c\u54e5\u5fb7\u723e\u7b2c\u4e00\u4e0d\u5b8c\u5099\u5b9a\u7406\u300d\u8af8\u591a\u4f8b\u5b50\u4e4b\u4e00\u3002\u76ae\u4e9e\u8afe\u516c\u7406\u4e0d\u8db3\u4ee5\u7528\u4f86\u8b49\u660e\u53e4\u5fb7\u65af\u5766\u5b9a\u7406\uff0c\u5fc5\u9808\u52a0\u4e0a\u5176\u4ed6\u6280\u5de7\u624d\u8fa6\u5f97\u5230<\/span>(<\/span>\u8a73\u898b\u7dad\u57fa\u767e\u79d1<\/span>)<\/span>\u3002<\/span><\/p>\n

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\u5947\u6578\u6578\u5217\u7684\u548c\uff0c<\/span>1+3+5+…+(2n-1)=n^2<\/span>\uff0c\u662f\u4e00\u500b\u95dc\u65bc\u81ea\u7136\u6578\u7684\u547d\u984c\uff0c\u5247\u662f\u53ef\u4ee5\u88ab\u8f15\u9b06\u5bb9\u6613\u904b\u7528\u76ae\u4e9e\u8afe\u516c\u7406\u8b49\u660e<\/span>(<\/span>\u70ba\u771f<\/span>)<\/span>\u7684\u3002\u771f\u597d\uff01<\/span><\/p>\n

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*<\/span>\uff1a\u300a\u6578\u5b78\u5973\u5b69<\/span> <\/span>\u54e5\u5fb7\u723e\u4e0d\u5b8c\u5099\u5b9a\u7406\u300b\uff0c\u7d50\u57ce\u6d69<\/span> <\/span>\u8457\uff0c\u937e\u9713<\/span> <\/span>\u8b6f<\/span><\/p>\n

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**<\/span>\uff1a<\/span>Q.E.D. <\/span>\u662f\u62c9\u4e01\u7247\u8a9e\u300c<\/span>Quod Erat Demonstrandum<\/span>\u300d<\/span>(<\/span>\u8b49\u660e\u5b8c\u7562<\/span>)<\/span>\u7684\u7e2e\u5beb\uff0c\u8b6f\u81ea\u5e0c\u81d8\u8a9e\u300c<\/span>\u1f45<\/span>περ <\/span>\u1f14<\/span>δει δε<\/span>\u1fd6<\/span>ξαι(hóper édei deîxai)<\/span>\u300d\uff0c\u4e26\u7d93\u5e38\u88ab\u60e1\u641e\u70ba\u300c<\/span>quite easily done<\/span>\u300d<\/span>(<\/span>\u7dad\u57fa\u767e\u79d1<\/span>)<\/span><\/p>\n

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P.S. <\/span>\u300c<\/span>0 <\/span>\u300d\u662f\u4e0d\u662f\u81ea\u7136\u6578\u5462\uff1f\u6c92\u6709\u60f3\u5230\u90a3\u9ebc\u7c21\u55ae\u7684\u554f\u984c\uff0c\u5728\u7db2\u8def\u4e0a\u9084\u66fe\u5f15\u8d77\u5f88\u71b1\u70c8\u7684\u8a0e\u8ad6\u5462\uff01<\/span><\/p>\n

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2022\/9\/3 <\/span>\u6578\u5b78\u5973\u5b69<\/span> <\/span>\u54e5\u5fb7\u723e\u4e0d\u5b8c\u5099\u5b9a\u7406<\/span> Damakey<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\u6731\u585e\u4f69·\u76ae\u4e9e\u8afe(Giuseppe Peano\uff0c…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"content-type":"","footnotes":""},"categories":[3],"tags":[],"class_list":["post-569","post","type-post","status-publish","format-standard","hentry","category-3"],"_links":{"self":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/569","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=569"}],"version-history":[{"count":1,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/569\/revisions"}],"predecessor-version":[{"id":19570,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/569\/revisions\/19570"}],"wp:attachment":[{"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=569"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=569"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=569"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}