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非构造性证明.
非构造性证明是“表述存在性的命题或定理”的一种证明方式:证明的过程中,不举例而只证明语句是否正确。非构造性证明很多时候依赖于排中律。数学构成主义数学不允许非构造性证明。
A、B两人进行这样一个数学游戏:在黑板上轮流写下1到2000中的任意一个整数(含边界,A先写),但不能写下任何黑板上已存在的数的因子。当一方不能写出数字时该方则输。问:谁有必胜策略?
- 考虑一种新的游戏:A'、B'在黑板上轮流写下2到2000中的任意一个整数(含边界,A'先写),但不能写下任何黑板上已存在的数的因子。当一方不能写出数字时该方则输。在这个游戏中谁有必胜策略?
- 如果A'有必胜策略,那么A在原游戏中也采用这个策略。注意,1在以后的过程中再也不能写上了(因为它是任何数的因子)。由于在新游戏中A'有必胜策略,所以在原游戏中,A有必胜策略。
- 如果B'有必胜策略,那么A在原游戏中先写上1。这就相当于构建了上述新游戏,B是新游戏中的A',A是新游戏中的B'。由于在新游戏中B'有必胜策略,所以在原游戏中,A有必胜策略。
- 综上所述,A有必胜策略。
上述证明过程中并没有找出具体的必胜策略,但是仍然证明了A有必胜策略。
比如要证明一个简单的命题:
- 超越数存在。
- 因为全体实数不可数,而全体代数数可数,所以超越数作为全体代数数的补集肯定非空。证毕。
证明过程并没有找出任何一个超越数,但是依然证明了上述命题的正确性。
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