理发师悖论 (The Barber’s Paradox)

[pplar]

https://plus.maths.org/content/mathemat ... 0the%20set.

"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself? If he does, then he mustn't, because he doesn't shave men who shave themselves, but then he doesn't, so he must, because he shaves every man who doesn't shave himself... and so on. Both possibilities lead to a contradiction.

This is the Barber's Paradox, discovered by mathematician, philosopher and conscientious objector Bertrand Russell, at the begining of the twentieth century. As stated, it seems simple, and you might think a little thought should show you the way around it. At worst, you can just say "Well, the barber's condition doesn't work! He's just going to have to decide who to shave in some different way." But in fact, restated in terms of so-called "naïve" set theory, the Barber's paradox exposed a huge problem, and changed the entire direction of twentieth century mathematics.

[forecasting]

https://plus.maths.org/content/mathemat ... 0the%20set.

"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself? If he does, then he mustn't, because he doesn't shave men who shave themselves, but then he doesn't, so he must, because he shaves every man who doesn't shave himself... and so on. Both possibilities lead to a contradiction.

This is the Barber's Paradox, discovered by mathematician, philosopher and conscientious objector Bertrand Russell, at the begining of the twentieth century. As stated, it seems simple, and you might think a little thought should show you the way around it. At worst, you can just say "Well, the barber's condition doesn't work! He's just going to have to decide who to shave in some different way." But in fact, restated in terms of so-called "naïve" set theory, the Barber's paradox exposed a huge problem, and changed the entire direction of twentieth century mathematics.

没正经学过公理集合论，说这个悖论改变了二十世纪数学的方向，夸大其词了吧。记得公理集合论区别类class和集合set绕过了这个问题。集合论的中心问题还是连续统假设吧，啥时候罗素悖论成了中心？

[pplar]

集合论是抽象数学的基础。

理发师悖论的确给naive集合论开了一个天窗，即，集合的集合的存在性问题。公理集合论由此诞生。

[forecasting]

集合论是抽象数学的基础。

理发师悖论的确给naive集合论开了一个天窗，即，集合的集合的存在性问题。公理集合论由此诞生。

你学过公理集合论吗？我没正经学过，咋感觉你这些话怪怪的？

[pplar]

你学过公理集合论吗？我没正经学过，咋感觉你这些话怪怪的？

我知道一点点ZFC。事实上，只有极少数研究集合论公理体系的数学工作者，才会去真正学习这些乏味无趣的东西

[forecasting]

我知道一点点ZFC。事实上，只有极少数研究集合论公理体系的数学工作者，才会去真正学习这些乏味无趣的东西

没入门罢了，倒不觉得乏味无趣，就是觉得用到的可能性太小，只有理论趣味，所以不肯学。比如力迫法，比如大基数。
这是数理逻辑四个方向（证明论，模型论，递归论，集合论）之一。数理逻辑专业大概要具备基本的知识。

[pplar]

没入门罢了，倒不觉得乏味无趣，就是觉得用到的可能性太小，只有理论趣味，所以不肯学。比如力迫法，比如大基数。
这是数理逻辑四个方向（证明论，模型论，递归论，集合论）之一。数理逻辑专业大概要具备基本的知识。

你说对了，数理逻辑。

我是改行学习理论计算机科学时去学了些皮毛。