Philosophy of mathematics: What is Cantor's continuum problem?
Philosophy of mathematics: What is Cantor's continuum problem?
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数学哲学：什么是康托连续统问题？
DOI：
10.1017/cbo9781139171519.025
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
K. Gödel
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文献类型：
--
作者：
K. Gödel
当然，这个问题只有在”数”的概念被扩展到无限集合之后才可能出现;因此，这种扩展是否能以唯一确定的方式实现，以及是否因此用上面使用的简单术语来陈述这个问题是合理的，可能会受到怀疑。然而，更仔细的研究表明，康托的无限数定义确实具有这种独特性，而且是以一种非常惊人的方式。因为无论应用于无限集合的”数”是什么意思，我们当然希望它具有这样的性质，即属于某类的对象的数不会改变，如果保持对象不变，人们以任何方式改变它们的性质或相互关系（例如，它们的颜色或它们在空间中的分布）。然而，从这一点可以立即得出结论，如果两个集合（至少是时空世界中两个可变对象的集合）的元素可以一一对应，那么它们将具有相同的基数，这就是康托尔对数之间相等的定义。因为如果两个集合A和Bit存在这样的对应关系，那么（至少在理论上）就有可能将A的每个元素的性质和关系改变为B的相应元素的性质和关系，从而A被变换为与B完全不可区分的集合，因此具有相同的基数。例如，假设一个正方形和一条线段都充满了质点，（因此，在它们的每一点上都恰好有一个质点），由于可以证明的事实，即正方形和直线段的点之间存在一一对应关系，因此，也存在一一对应关系。相应的质点，正方形的质点可以重新排列，正好填满线段，反之亦然。诚然，这样的考虑只直接适用于物理对象，但取决于被编号的对象的种类的”数”概念的定义很难被认为是合理的。这可以很容易地扩展到”更大”和”更小”的定义。对于无限数，规定集合A的基数M被称为小于集合B的基数N，如果M不同于N，但等于B的某个子集的基数。在这些定义的基础上，可以证明存在无穷多个不同的无穷基数或”幂”，特别是，一个集合的子集的数量总是大于它的元素的数量;此外，可以将算术运算扩展（再次没有任何任意性）到无穷多个（包括与任何无穷大的和和积）。
This question, of course, could arise only after the concept of" number" had been extended to infinite sets; hence it might be doubted if this extension can be effected in a uniquely determined manner and if, therefore, the statement of the problem in the simple terms used above is justified. Closer examination, however, shows that Cantor's definition of infinite numbers really has this character of uniqueness, and that in a very striking manner. For whatever" number" as applied to infinite sets may mean, we certainly want it to have the property that the number of objects belonging to some class does not change if, leaving the objects the same, one changes in any way whatsoever their properties or mutual relations (eg, their colors or their distribution in space). From this, however, it follows at once that two sets (at least two sets of changeable objects of the space-time world) will have the same cardinal number if their elements can be brought into a one-to-one correspondence, which is Cantor's definition of equality between numbers. For if there exists such a correspondence for two sets A and Bit is possible (at least theoretically) to change the properties and relations of each element of A into those of the corresponding element of B, whereby A is transformed into a set completely indistinguishable from B, hence of the same cardinal number. For example, assuming a square and a line segment both completely filled with mass points (so that at each point of them exactly one mass point is situated), it follows owing to the demonstrable fact that there exists a one-to-one correspondence between the points of a square and of a line segment, and, therefore, also between the corresponding mass points, that the mass points of the square can be so rearranged as exactly to fill out the line segment, and vice versa. Such considerations, it is true, apply directly only to physical objects, but a definition of the concept of" number" which would depend on the kind of objects that are numbered could hardly be considered as satisfactory.So there is hardly any choice left but to accept Cantor's definition of equality between numbers, which can easily be extended to a definition of" greater" and" less" for infinite numbers by stipulating that the cardinal number M of a set A is to be called less than the cardinal number N of a set B if M is different from N but equal to the cardinal number of some subset of B. On the basis of these definitions it becomes possible to prove that there exist infinitely many different infinite cardinal numbers or" powers," and that, in particular, the number of subsets of a set is always greater than the number of its elements; furthermore it becomes possible to extend (again without any arbitrariness) the arithmetical operations to infinite numbers (including sums and products with any infinite 515