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假设有一个电灯，你开1/2分钟，然后关1/4分钟，再开1/8分钟，再关1/16分钟，再开。。。，一直进行下去。

问题：电灯在1分钟整是亮的还是关的？

这个脑筋实验问题与Zeno's Dichotomy Paradox不太一样。

张旺教授 写了： 今天 08:10

假设有一个电灯，你开1/2分钟，然后关1/4分钟，再开1/8分钟，再关1/16分钟，再开。。。，一直进行下去。

问题：电灯在1分钟整是亮的还是关的？

这个脑筋实验问题与Zeno's Dichotomy Paradox不太一样。

条件不cover 1分钟啊。

TheMatrix 写了： 今天 08:21

条件不cover 1分钟啊。

开关次数趋于无穷时，就cover一分钟了。

一个脑残题。

TheMatrix 写了： 今天 08:21

条件不cover 1分钟啊。

1/2+1/4+1/8+ \cdots =1.

还是有限无限的问题。

下面是该问题的英文原版，与我凭记忆写的set-up 有点不同，但问题大同小异。

Thomson's lamp is a philosophical thought experiment created by British philosopher James F. Thomson in 1954.

You have a lamp with a toggle switch.

The action: You turn the lamp on. Wait 1 minute, then turn it off.

The acceleration: Wait 30 seconds, turn it on. Wait 15 seconds, turn it off. Each time you flip the switch, you cut the waiting time in half.

The timeline: The intervals get smaller and smaller (1 minute, 1/2, 1/4, 1/8, and so on). The total time required to complete this infinite number of flips adds up to exactly 2 minutes.

At the exact 2-minute mark, you must ask a simple question: Is the light bulb on or off?

这都是啥。利用常识语言和数学语言的gap问假问题。可以转为数学语言：
把灯的状态记为0/1，每次开关更新状态。这问题在问（1，0,1,0…) 这个序列的极限。 当然不存在。

t=1对应n趋向于无穷的情况，但是[0,1,0,1,0,1,...]这个序列不收敛。这个极限不存在，所以结果是不存在的。

rgg 写了： 今天 11:30

这都是啥。利用常识语言和数学语言的gap问假问题。可以转为数学语言：
把灯的状态记为0/1，每次开关更新状态。这问题在问（1，0,1,0…) 这个序列的极限。 当然不存在。

楼主说他证明过数学猜想。一篇论文引用一千多。

张旺教授 写了： 今天 08:10

假设有一个电灯，你开1/2分钟，然后关1/4分钟，再开1/8分钟，再关1/16分钟，再开。。。，一直进行下去。

问题：电灯在1分钟整是亮的还是关的？

这个脑筋实验问题与Zeno's Dichotomy Paradox不太一样。

这个达不到吧？1/2+1/4+1/8....才是1，你去掉一些，应该少了

DIYer 写了： 今天 11:38

这个达不到吧？1/2+1/4+1/8....才是1，你去掉一些，应该少了

“1分钟整”——这个不能测量，此句就不该在科学讨论中出现。

测量问题由ISO和相关机构统一address。
ISO 9000之外则没有科学。目前最高精度1e-19秒（ai说的）

我认为楼主看科普太多，被witten之流弄傻了。

其实这是英国哲学家Thomson在1954年提出的脑筋试验的Thomson's lamp paradox. 在哲学界比较有名，涉及到无限。

搞到逻辑这块，哲学家跟数学家交集比较大。罗素，哥德尔都是跨越哲学、数学两界的。

看了跟帖，搞CS的马上belittle这个问题，进一步人身攻击，令我很吃惊。

下面取自维基：

“Thomson's lamp is a philosophical thought experiment based on infinites. It was devised in 1954 by British philosopher James F. Thomson, who used it to analyze the possibility of a supertask, which is the completion of an infinite number of tasks.

Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being who is able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously.[1] The sum of this infinite series of time intervals is exactly two minutes.[2]

The following question is then considered: Is the lamp on or off at two minutes?[1] Thomson reasoned that this supertask creates a contradiction:

It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction。”

geniushanbiao 写了： 今天 11:36

t=1对应n趋向于无穷的情况，但是[0,1,0,1,0,1,...]这个序列不收敛。这个极限不存在，所以结果是不存在的。

这是一种解决方案。

张旺教授 写了： 今天 13:25

其实这是英国哲学家Thomson在1954年提出的脑筋试验的Thomson's lamp paradox. 在哲学界比较有名，涉及到无限。

搞到逻辑这块，哲学家跟数学家交集比较大。罗素，哥德尔都是跨越哲学、数学两界的。

看了跟帖，搞CS的马上belittle这个问题，进一步人身攻击，令我很吃惊。

下面取自维基：

“Thomson's lamp is a philosophical thought experiment based on infinites. It was devised in 1954 by British philosopher James F. Thomson, who used it to analyze the possibility of a supertask, which is the completion of an infinite number of tasks.

Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being who is able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously.[1] The sum of this infinite series of time intervals is exactly two minutes.[2]

The following question is then considered: Is the lamp on or off at two minutes?[1] Thomson reasoned that this supertask creates a contradiction:

It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction。”

找什么什么有名来瞎扯。菌斑就这特色。自己的思考呢

天天讲数学和科学的人里头。我看就你一个题不做，一本书籍不读。回菌斑去吧。或者你先找下弃婴的垃圾题看看。

张旺教授 写了： 今天 08:10

假设有一个电灯，你开1/2分钟，然后关1/4分钟，再开1/8分钟，再关1/16分钟，再开。。。，一直进行下去。

问题：电灯在1分钟整是亮的还是关的？

这个脑筋实验问题与Zeno's Dichotomy Paradox不太一样。

这个问题本身有问题。 你在 1/2 分钟的时候关灯， 那么在 1/2 分钟的时候灯算开还是关呢？

你问的是一分钟的时候的情况对吧。

qxc 写了： 今天 13:42

这个问题本身有问题。 你在 1/2 分钟的时候关灯， 那么在 1/2 分钟的时候灯算开还是关呢？

你问的是一分钟的时候的情况对吧。

这个是好问题。

你可以看看Thomson的原paradox。我主帖是凭记忆写的，与原题基本一致，只是时间从2分钟变成1分钟。

再重复一遍，Thomson这个paradox与Zeno's Dichotomy Paradox有点类似，我觉得二者并不一样。

qxc 写了： 今天 13:42

这个问题本身有问题。 你在 1/2 分钟的时候关灯， 那么在 1/2 分钟的时候灯算开还是关呢？

你问的是一分钟的时候的情况对吧。

测量就只到1e-21精度。完美的测量没有意义。都是分布。
什么1分钟的情况，与喜怒哀乐这些范畴没区别。不属于stem

哲学家Paul Benacerraf’s解决方案：

In his 1962 paper "Tasks, Super-Tasks and the Modern Eleatics," Benacerraf argued that Thomson made an incorrect assumption that the rules of the setup dictate a mandatory final state.

Indeterminate State: The rules only define the state of the lamp at every finite fraction of time before the limit, but they are completely silent on what happens at the limit itself.

No Contradiction: Either being on or being off is fully compatible with the prior infinite sequence of switches; the lack of a determined final state is just a missing boundary condition, not a logical contradiction.

Thomson later conceded that Benacerraf's critique was correct.

刚查到Benacerraf’s solution.
归根到底还是无限和有限的鸿沟不可跨越。哲学家也不行。