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 给小看的讲演助威：希尔伯特的“大酒店悖论” |
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作者：Anonymous 在 罕见奇谈 发贴, 来自 http://www.hjclub.org
给小看的“无穷大”讲演助威：希尔伯特的“大酒店悖论”
正如小看所说，任何两个有穷大或无穷大的基数都可以相加或相乘，只不过无穷大基数的
加法和乘法规则比较新鲜，结果对不搞数学的人有时显得“不可思议”。下面是一则有名
的数学小品：
Hilbert's paradox of the Grand Hotel
David Hilbert presented the following paradox about infinity.
In a hotel with a finite number of rooms, once it is full, no more guests can be
accommodated. Now imagine a hotel with an infinite number of rooms. You might assume that the same problem will arise when all the rooms are taken.
However, there is a way to solve this: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1. And you can even make place for an infinite number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, etc., and all the odd-numbered new rooms will be free for the new guests.
If an infinite number of coaches arrive, each with an infinite number of passengers, you can even deal with that: first empty the odd numbered rooms as above, then put the first coach's load in rooms 3^n (the nth power of 3) for n = 1, 2, 3, ..., the second coach's load in rooms 5^n for n = 1, 2, ... and so on; for coach number i we use the rooms pn where p is the i+1-st prime number.
This state of affairs is not really paradoxical but just profoundly counterintuitive. It is difficult to come to grips with infinite 'collections of things', as their properties are quite different from the properties of ordinary 'collections of things'. In an ordinary hotel, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel the 'number' of odd-numbered rooms is as 'large' as the total 'number' of rooms. In mathematical terms, this would be expressed as follows: the
cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. In fact, infinite sets are characterized as sets that have proper subsets of the same cardinality.
作者：Anonymous 在 罕见奇谈 发贴, 来自 http://www.hjclub.org |
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