无穷的奥秘 | 康托尔集合论与无穷算术

注英文引文机翻未校。如有内容异常请看原文。Unraveling the Mysteries of Infinity揭开无穷的奥秘Jimmie Lawson吉米·劳森Louisiana State University路易斯安那州立大学Infinity – p.无穷 —— 第 页Thinking about the Infinite / 思考无穷There is a rich history of study and reflection on the concept and mystery of the infinite in a variety of contexts. This contemplation has often inspired feelings of awe, mystery, bafflement, skepticism (either about the reality of the infinite or our ability to make any sense of it), even fear.在诸多领域中关于无穷这一概念与奥秘的研究和思考有着悠久的历史。这类思索常常引发人们的敬畏、神秘、困惑、怀疑怀疑无穷是否真实存在或怀疑我们能否理解它乃至恐惧。The French mathematician/philosopher/religious writer Blaise Pascal captures some of these feelings in the words: “When I consider the small span of my life absorbed in the eternity of all time, or the small part of space that I can touch or see engulfed by the infinite immensity of space, I am frightened and astonished.”法国数学家、哲学家、宗教作家布莱士·帕斯卡用一段话道出了其中部分感受当我思索自己短暂的生命消融于永恒的时间之中或是我所能触及与看见的狭小空间被无垠的宇宙所吞没时我既恐惧又震撼。The Absolute Infinity / 绝对无穷The term “Absolute Infinity” has been used to refer to beings who personify the infinite. Religions in general and theologians in particular, also philosophers, have contemplated beings who were infinite in various aspects of their beings. A most common aspect has been immortality, existing through time without beginning or end.“绝对无穷”一词常用来指代化身无穷的存在。各类宗教、神学家以及哲学家们都曾思索在存在的诸多层面上具备无穷属性的存在其中最常见的属性是不朽——在时间中无始无终地存在。The Hebrew Bible, for instance, gives vivid statement to this in Psalm 90:例如《希伯来圣经》在诗篇 90 篇中对此有生动表述Before the mountains were born or you brought forth the earth and the world, from everlasting to everlasting you are God.诸山未曾生出地与世界你未曾造成从亘古到永远你是神。The Physically Infinite / 物理层面的无穷Philosophers, scientists, and other thinkers from at least the days of the Greeks have contemplated the physically infinite.至少从古希腊时代起哲学家、科学家与其他思想家就开始思考物理层面的无穷。Is time infinite in duration, without beginning and ending? Can it be infinitely divided?时间在持续上是否无穷、无始无终时间能否被无限分割Is space infinite in extent? What about our universe? Are there an infinite number of heavenly bodies? Are there infinitely many (possible or actual) universes? Can matter be infinitely divided?空间在范围上是否无穷我们的宇宙如何天体的数量是否无穷可能或实际存在的宇宙是否有无穷多个物质能否被无限分割The Infinite in Mathematics / 数学中的无穷Realizing the logical pitfalls surrounding the idea of allowing infinite quantities, pitfalls dating back at least to the puzzles or paradoxes of Zeno, mathematicians shied away from the notion.数学家们意识到引入无穷量会伴随逻辑陷阱这类陷阱至少可追溯至芝诺悖论因此他们一度回避无穷这一概念。The rise of modern mathematics and modern science in Europe after the Renaissance revived interest in the subject, however, as, for example,然而文艺复兴后欧洲现代数学与现代科学的兴起重新唤起了对这一主题的兴趣例如Galileo considered motion as a function of time over continuously varying instances of time;伽利略将运动视为时间在连续变化时刻下的函数Newton and Leibnitz introduced the calculus with its calculations based on infinitesimally small, but non-zero, quantities.牛顿与莱布尼茨创立了微积分其计算基于无穷小但非零的量。George Cantor / 格奥尔格·康托尔The major mathematical breakthrough came in the late 19th century in groundbreaking work of George Cantor, who introduced the theory of sets as a foundation for mathematics and included a substantial mathematical theory of infinite sets. His highly original work was quite controversial in its day, but has made a major impact on modern mathematics.19 世纪末格奥尔格·康托尔的开创性工作带来了数学上的重大突破他提出集合论作为数学的基础并建立了系统的无穷集数学理论。他极具原创性的成果在当时备受争议却对现代数学产生了深远影响。One-to-One Correspondences / 一一对应A basic insight of Cantor was that two sets should be compared by the existence or non-existence of one-to-one correspondences (members of one set could be paired up with members of the other so that everyone had a “dancing partner”), but not by whether one was a smaller set than the other.康托尔的核心洞见是比较两个集合应依据是否存在一一对应一个集合的元素可与另一个集合的元素两两配对每个元素都有“舞伴”而非直观上谁比谁“更小”。1 2 3 4 5 6 ... ↕ ↕ ↕ ↕ ↕ ↕ ... 1 4 9 16 25 36 ...The set of perfect squares has the same number of elements as all the counting numbers though it is a much smaller subset.完全平方数集合虽是自然数集的真子集却与自然数集拥有相同的元素个数。The Infinite Numberℵ \alephℵ/ 无穷数ℵ \alephℵ阿列夫Rather than trying to define the number 3, we need to learn how to tell whether a collection of objects has 3 members or some other number of members. We do this by counting and seeing whether we use precisely the numbers 1,2, and 3. We use a similar approach forℵ \alephℵ(aleph). We count the objects and see whether we use precisely all the numbersN { 1 , 2 , 3 , 4 , … } \mathbb{N}\{1,2,3,4,\dots\}N{1,2,3,4,…}. (Note the resemblance betweenℵ \alephℵand N.)与其直接定义数字 3不如学会判断一组对象是否包含 3 个或其他数量的元素我们通过计数看是否恰好用到 1、2、3 这三个数。对ℵ \alephℵ阿列夫我们采用类似方法对对象计数看是否恰好用到全体自然数N { 1 , 2 , 3 , 4 , … } \mathbb{N}\{1,2,3,4,\dots\}N{1,2,3,4,…}注意ℵ \alephℵ与N \mathbb{N}N的形似。More precisely, we say a collection of objects has size or cardinalityℵ \alephℵ, or is countably infinite, if it can be put in one-to-one correspondence with the setN \mathbb{N}N. Intuitively this means that there is some method of pasting exactly one number on each of the objects.更严格地说若一个对象集合能与自然数集N \mathbb{N}N建立一一对应则称该集合的基数大小为ℵ \alephℵ或称其为可数无穷。直观理解就是存在一种方式给每个对象唯一贴上一个自然数编号。A New Arithmetic / 新的算术Cantor was able to show many surprising things about the arithmetic of infinite numbers. We will illustrate a few of these with a retelling of the story of Hercules cleaning the Augean stables, one of his twelve labors. Recall that Augeas was a man of vast herds of animals, and Hercules was able to cleanse the huge and filthy stables in one day by diverting a river through them.康托尔证明了无穷数算术的诸多惊人结论。我们借用赫拉克勒斯清洗奥革阿斯牛棚十二项功绩之一的改编故事来举例说明奥革阿斯拥有无数牲畜赫拉克勒斯曾引河水一日洗净肮脏的巨大牛棚。We modify the story so that the Augean herds are infinite in size and Hercules’ tasks now have a significant mental component (the twelve intellectual labors of Hercules?). He has to assist him the greatest of the Greek mathematical minds, none other than Archimedes himself.我们对故事做如下改编奥革阿斯的畜群为无穷规模赫拉克勒斯的任务包含大量数理思考可称为“赫拉克勒斯的十二项智力功绩”他将得到古希腊最伟大数学家阿基米德的协助。Task One / 任务一Augeas orders Hercules to place a newly acquired horse into his already full barn (withℵ \alephℵstalls), one horse to a stall. Archimedes advises Hercules to move each horse to the next higher stalln → n 1 n \to n1n→n1and then the first stall will be freed up for the new horse. Working at compound double speed, Hercules completes the task in eight minutes.奥革阿斯命令赫拉克勒斯将一匹新马放入已住满的马棚共有ℵ \alephℵ个马栏一马一栏。阿基米德建议将每匹马移至下一个编号更大的马栏n → n 1 n \to n1n→n1如此第一个马栏就会空出留给新马。赫拉克勒斯以复合双倍速度工作8 分钟完成任务。The Arithmetic ofN \mathbb{N}N/ 自然数集的算术If we add one horse ton nnhorses, we obtainn 1 n1n1horses. In the same way adding one horse toℵ \alephℵhorses givesℵ 1 \aleph1ℵ1horses.若给n nn匹马再加 1 匹总数为n 1 n1n1匹同理给ℵ \alephℵ匹马再加 1 匹总数为ℵ 1 \aleph1ℵ1匹。Applying the method used by Archimedes and Hercules, we observe thatℵ 1 ℵ \aleph1\alephℵ1ℵ.依据阿基米德与赫拉克勒斯的方法可得ℵ 1 ℵ \aleph1\alephℵ1ℵ。Task Two / 任务二In his full mare-with-colt barn, Augeas orders Hercules to separate the colts from their mothers and place each colt in its own stall.在住满母马与马驹的马棚中奥革阿斯命令赫拉克勒斯将马驹与母马分开每匹小马单独占一个马栏。Archimedes advises Hercules to move the mare in stalln nnto stall2 n 2n2nand put her colt just before her in stall2 n − 1 2n-12n−1. For example, the mare in stall 50 is moved to stall 100 and her colt to stall 99.阿基米德建议将第n nn栏母马移至第2 n 2n2n栏其马驹放在前一栏2 n − 1 2n-12n−1。例如第 50 栏母马移至第 100 栏马驹放在第 99 栏。Working at compound double speed, Hercules completes the task in one hour. At the end the mares occupy the even numbered stalls and the colts the odd numbered.赫拉克勒斯以复合双倍速度工作1 小时完成任务。最终母马占据偶数栏马驹占据奇数栏。More Arithmetic / 更多算术规则If we haven nncolts andn nnmares, then we have in totaln n nnnnhorses. In the same wayℵ \alephℵcolts plusℵ \alephℵmares givesℵ ℵ \aleph\alephℵℵhorses.若有n nn匹小马与n nn匹母马总数为n n nnnn匹同理ℵ \alephℵ匹小马加ℵ \alephℵ匹母马总数为ℵ ℵ \aleph\alephℵℵ匹。Again using the rearrangement method applied by Archimedes and Hercules, we observe thatℵ ℵ ℵ \aleph\aleph\alephℵℵℵ.再次使用阿基米德与赫拉克勒斯的重排方法可得ℵ ℵ ℵ \aleph\aleph\alephℵℵℵ。Task 3 / 任务三On the back part of his property Augeas has built a brand new stable. He wants Hercules to move all the horses from all of his stables (ℵ \alephℵstables withℵ \alephℵhorses in each) into the one new stable, with one horse per stable, a seemingly impossible task.奥革阿斯在场地后方新建了一座马棚。他要求赫拉克勒斯把所有马棚里的马共ℵ \alephℵ座马棚每座有ℵ \alephℵ匹马全部迁入这一座新马棚一马一栏——这看似是不可能完成的任务。Many Barns / 多座马棚Below is the scheme. Each row represents an old barn, with entry( 4 , 3 ) (4, 3)(4,3), for instance, representing the 3rd horse in the 4th barn.排布如下每一行代表一座旧马棚例如( 4 , 3 ) (4,3)(4,3)表示第 4 座马棚的第 3 匹马。(1,1) (1,2) (1,3) (1,4) ... (2,1) (2,2) (2,3) (2,4) ... (3,1) (3,2) (3,3) (3,4) ... (4,1) (4,2) (4,3) (4,4) ... (5,1) (5,2) ...可数无限集合的枚举方式康托尔配对函数。Stall 1 is Filled / 第 1 栏已安置After consider thought, Archimedes suggests a scheme.深思后阿基米德给出方案。The first horse in Barn 1 goes to Stall 1 in the new barn.第 1 座马棚的第 1 匹马迁入新马棚第 1 栏。(1,1)¹ → (1,2) (1,3) (1,4) ... (2,1) (2,2) (2,3) (2,4) ... (3,1) (3,2) (3,3) (3,4) ... (4,1) (4,2) (4,3) (4,4) ... (5,1) (5,2) ...康托尔对角线枚举法Stall 2 is Filled / 第 2 栏已安置The second horse in Barn 1 goes to Stall 2 in the new barn.第 1 座马棚的第 2 匹马迁入新马棚第 2 栏。(1,1)¹ → (1,2)² (1,3) (1,4) ... ↙ (2,1) (2,2) (2,3) (2,4) ... (3,1) (3,2) (3,3) (3,4) ... (4,1) (4,2) (4,3) (4,4) ... (5,1) (5,2) ...Stall 3 is Filled / 第 3 栏已安置The first horse in Barn 2 goes to Stall 3 in the new barn.第 2 座马棚的第 1 匹马迁入新马棚第 3 栏。(1,1)¹ → (1,2)² (1,3) (1,4) ... ↙ (2,1)³ (2,2) (2,3) (2,4) ... ↓ (3,1) (3,2) (3,3) (3,4) ... (4,1) (4,2) (4,3) (4,4) ... (5,1) (5,2) ...Stalls 4 through 6 / 第 4 至 6 栏We proceed along the next diagonal to place the next 3 horses.沿下一条对角线依次安置接下来的 3 匹马。(1,1)¹ → (1,2)² (1,3)⁶ → (1,4) ↘ ↗ (2,1)³ (2,2)⁵ (2,3) (2,4) ↓ ↗ (3,1)⁴ (3,2) (3,3) (3,4) ... (4,1) (4,2) (4,3) (4,4)... (5,1) (5,2) ...Stalls 7 through 10 / 第 7 至 10 栏We proceed along the next diagonal to place the next 4 horses.沿下一条对角线依次安置接下来的 4 匹马。(1,1)¹ → (1,2)² (1,3)⁶ → (1,4)⁷ ↘ ↗ ↘ (2,1)³ (2,2)⁵ (2,3)⁸ (2,4) ↓ ↗ ↘ (3,1)⁴ (3,2)⁹ (3,3) (3,4) ... ↘ (4,1)¹⁰ (4,2) (4,3) (4,4)... ↓ (5,1) (5,2) ...And So Forth依此类推We continue traversing the diagonals by this pattern, eventually reaching and assigning all horses to stalls.按此模式沿对角线遍历最终可将所有马匹逐一安置到马栏中。(1,1)¹ → (1,2)² (1,3)⁶ → (1,4)⁷ ↘ ↗ ↘ ↗ (2,1)³ (2,2)⁵ (2,3)⁸ (2,4)¹⁴ ↓ ↗ ↘ ↗ ↘ (3,1)⁴ (3,2)⁹ (3,3)¹³ (3,4) ↘ ↗ ↘ ↗ (4,1)¹⁰ (4,2)¹² (4,3) (4,4)... ↓ ↗ ↘ ↗ ↘ (5,1)¹¹ (5,2) ... (5,3) (5,4)...Multiplication by Infinity / 无穷的乘法If we had eight barns with 7 horses each, then we would have a total of8 × 7 56 8 \times 7568×756horses.若有 8 座马棚每座 7 匹马总数为8 × 7 56 8 \times 7 568×756匹。Applying the same method for multiplying infinite quantities, we have just derived the following multiplication fact:ℵ × ℵ ℵ \aleph \times \aleph\alephℵ×ℵℵ将同一方法用于无穷量的乘法可得如下结论ℵ × ℵ ℵ \aleph \times \aleph\alephℵ×ℵℵLarger Infinities? / 存在更大的无穷吗As we leave our heroes, we are left with a question. Are all infinite sets of sizeℵ \alephℵ? Cantor showed that the answer is no, which surprised many, who thought all infinities, were after all, just infinity.故事收尾我们留下一个问题所有无穷集的大小都是ℵ \alephℵ吗康托尔证明答案是否定的这令许多人震惊——人们曾以为所有无穷都只是“无穷”而已。The sets of sizeℵ \alephℵare the smallest of the infinite sets. For this reason,ℵ \alephℵis denotedℵ 0 \aleph_{0}ℵ0​, the first of the infinite cardinal numbers.基数为ℵ \alephℵ的集合是最小的无穷集因此ℵ \alephℵ记作ℵ 0 \aleph_0ℵ0​是第一个无穷基数。More Cowherds Than Cows / 牧人比牛更多Imagine a cow barn withℵ 0 \aleph_{0}ℵ0​cows, one in each stall1 , 2 , 3 , … 1, 2, 3, \dots1,2,3,…. Now imagine all possible different herds that could be formed by taking some of the cows into the pasture and leaving others behind, and suppose for each possible herd there was a different cowherd to attend that specific herd. Then the number of cowherds needed is a larger infinity thanℵ 0 \aleph_{0}ℵ0​.设想一座牛棚有ℵ 0 \aleph_0ℵ0​头牛分别在 1、2、3……号牛栏。现在考虑所有可能的牛群组合任选一些牛去牧场、留下另一些。假设每种组合都需要一位专属牧人那么所需牧人的总数是比ℵ 0 \aleph_0ℵ0​更大的无穷。In math jargon, the set of all possible subsets of the setN { 1 , 2 , 3 , … } \mathbb{N}\{1,2,3,\dots\}N{1,2,3,…}is strictly larger in size thanℵ 0 \aleph_{0}ℵ0​. Cantor gave a beautiful and surprising proof of this result called “Cantor’s diagonal argument.” You can google to find it at several sites on the web.用数学语言表述自然数集N { 1 , 2 , 3 , … } \mathbb{N}\{1,2,3,\dots\}N{1,2,3,…}的全体子集构成的集合其基数严格大于ℵ 0 \aleph_0ℵ0​。康托尔用优美而惊艳的对角线论证证明了这一结论你可在网上搜索查阅。Conclusion / 结语The study of large infinite cardinal numbers remains an active area of mathematical research to the current day.对大无穷基数的研究至今仍是数学领域的活跃方向。General References / 参考文献Rudy Rucker, Infinity and the Mind, Princeton University Press, 1982.鲁迪·拉克《无穷与心灵》普林斯顿大学出版社1982 年。N. Ya. Vilenkin, In Search of Infinity, Birkhäuser, 1995.N·亚·维连金《探索无穷》伯克霍夫出版社1995 年。referenceUnraveling the Mysteries of Infinity - Jimmie Lawson, Louisiana State Universityhttps://www.math.lsu.edu/~lawson/infinity.pdf