错在哪里?定理:所有的房子的颜色都是一样的。

来源: 2012-08-22 18:28:33 [] [旧帖] [给我悄悄话] 本文已被阅读: 次 (830 bytes)

Theorem. All horses are the same color.

Proof. We'll induct on the number of horses. Base case: 1 horse. Clearly with just 1 horse, all horses have the same color.

Now, for the inductive step: we'll show that if it is true for any group of N horses, that all have the same color, then it is true for any group of N+1 horses.

Well, given any set of N+1 horses, if you exclude the last horse, you get a set of N horses. By the inductive step these N horses all have the same color. But by excluding the first horse in the pack of N+1 horses, you can conclude that the last N horses also have the same color. Therefore all N+1 horses have the same color. QED.

Hmmn... clearly not all horses have the same color. So what's wrong with this proof by induction?

所有跟帖: 

先找出一条明显的错 -布衣之才- 给 布衣之才 发送悄悄话 布衣之才 的博客首页 (31 bytes) () 08/22/2012 postreply 19:01:27

haha...You are absolutely right. -wxcfan123- 给 wxcfan123 发送悄悄话 (0 bytes) () 08/22/2012 postreply 19:42:17

没什么,用等价的概念就过去了.搞数学的,是粗心的细心人. -jinjing- 给 jinjing 发送悄悄话 (0 bytes) () 08/23/2012 postreply 05:25:21

同色是比较概念,至少两个马.从2开始,如对,可推出所有马同色. -jinjing- 给 jinjing 发送悄悄话 (0 bytes) () 08/22/2012 postreply 19:30:47

当然,任意两匹马同色等价于所有的马同色。问题是上面的证明错在哪? -wxcfan123- 给 wxcfan123 发送悄悄话 (0 bytes) () 08/23/2012 postreply 18:01:21

错在同色的定义.一匹马同色,与其它马无关,这过程证明了每匹马与自己同色. -jinjing- 给 jinjing 发送悄悄话 (0 bytes) () 08/23/2012 postreply 20:00:24

Real Math is here: f(hi1,i2,...,hit)=b,meaning t horses have b c -jinjing- 给 jinjing 发送悄悄话 (148 bytes) () 08/24/2012 postreply 12:02:25

说说我的看法。这里的归纳推理的第二步有三个集合 -wxcfan123- 给 wxcfan123 发送悄悄话 (248 bytes) () 08/24/2012 postreply 18:01:09

谬在第二步。 -cantorian- 给 cantorian 发送悄悄话 (201 bytes) () 08/25/2012 postreply 18:57:23

这些集合并没有问题,它们是N+1匹马的两个子集。 -wxcfan123- 给 wxcfan123 发送悄悄话 (0 bytes) () 08/25/2012 postreply 20:41:07

这两个子集不是题目里的集。 -cantorian- 给 cantorian 发送悄悄话 (142 bytes) () 08/25/2012 postreply 22:19:22

有趣 -发奖- 给 发奖 发送悄悄话 (397 bytes) () 09/12/2012 postreply 16:36:43

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