第七讲集合势.pdf

绂 鏁 鏁 瀛 Discrete Mathematics 绗竷璁诧細闆嗗悎鐨勫娍 鍗椾含澶у璁＄畻鏈虹瀛︿笌鎶€鏈郴 2012骞?鏈?2 鏃 鍓嶆儏鎻愯 飦 鍑芥暟鐨勫畾涔変笌澶栧欢鍘熷垯 飦 鍍忎笌瀹屽叏鍘熷儚 飦 鍑犵鐗规畩鐨勫嚱鏁扮被鍨 飩 婊″皠銆佸崟灏勩€佸弻灏 飦 鐗瑰緛鍑芥暟涓庤嚜鐒舵槧灏 飦 鍑芥暟鐨勫鍚 飦 鍙嶅嚱鏁 鍓 鎯 鎻 瑕 2 鏈涓昏鍐呭 飦 鑷劧鏁颁笌鏃犵┓鍏悊 飦 鏈夌┓闆嗕笌鏃犵┓闆 飦 闆嗗悎鐨勫娍 飦 闆嗗悎鐨勭瓑鍔垮叧绯 飦 Cantor 瀹氱悊 飦 闆嗗悎鐨勪紭鍔垮叧绯 鏈涓昏鍐呭 3 鏈夌┓ vs.鏃犵┓ 鏈夌┓ vs.鏃犵┓ 4 鑷劧鏁颁笌鏃犵┓闆嗗悎 God made the integers; all else is the work of man. 鈥斺€ Leopold Kronecker 鑷劧鏁颁笌鏃犵┓闆嗗悎 5 浠庨泦鍚堟瀯閫犺嚜鐒舵暟 飦 璁卷饱 涓洪泦鍚堬紝肀 鐨勫悗缁?succeed)肀 + 鎸図饱 鈭 {肀 锛 饊仜 浠? = 鈭? 1 = 0+, 2 = 0++, 鈰 , 肀 = 0+ 鈰? 锛岃繖 鏄痸on Neumann鐨勫畾涔 飦 璁卷按 涓洪泦鍚堬紝绉绊按 涓哄綊绾抽泦(inductive set)鎸囷細 鈭 鈭 戆 鈭 鈭€肀 鈭 戆 肀 + 鈭 戆 鑷劧鏁颁笌鏃犵┓闆嗗悎 6 褰掔撼闆嗘槸鍚﹀瓨鍦紵 飦 鑻ュ瓨鍦ㄥ綊绾抽泦戆 锛屽垯鈭 鈭 戆? 鈭? 鈭 戆? 鈭?+ 鈰? 鈭 戆 锛 浠庤€岉按 鏄棤绌烽泦 飦 Russell璇达細 鈥滀簨瀹炰笂锛屽湪杩欎釜涓栫晫涓槸鍚︽湁鏃 绌烽泦鍚堬紝鎴戜滑杩樹笉鑳界‘瀹氥€傗€ 飦 鎹锛岃繕涓嶈兘纭畾褰掔撼闆嗘槸鍚﹀瓨鍦ㄣ€傚ぇ澶氭暟浜 璁や负瀹囧畽鏄棤绌风殑 锛圚ilbert 鍒欏惁锛変负浜嗕繚璇 褰掔撼闆嗙殑瀛樺湪锛屽紩鍏ユ棤绌峰叕鐞 鑷劧鏁颁笌鏃犵┓闆嗗悎 7 鏃犵┓鍏悊 飦 鏃犵┓鍏悊 锛圓xiom of Infinity 锛屾棤绌锋€у師鍒欙級锛 鈭冺按 鈭 鈭 戆 鈭 鈭€肀 鈭 戆 肀 + 鈭 戆 飦 浠ュ線鎸夌収von Neumann 鐨勫畾涔夛紝0 = 鈭 锛岉睕 + 1 = 肀? 锛屼粠鑰屽彲浠ュ畾涔夊嚭鍗曚釜鐨勮嚜鐒舵暟锛屼絾 涓嶈兘璇存槑鍏ㄤ綋鑷劧鏁伴泦鍚堚剷鐨勫瓨鍦ㄦ€э紝鑰岀敱鏃 绌峰叕鐞嗗彲浠ュ畾涔夆剷 飦 鈩 =鈭 {戆 |戆翠负褰掔撼闆唥 鑷劧鏁颁笌鏃犵┓闆嗗悎 8 鑷劧鏁扮殑Peano鍏悊鍖栦綋绯 飦 鑷劧鏁扮殑Peano鍏悊鍖栦綋