In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in the theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets, by equinumerosity). The concepts are developed by defining equinumerosity in terms of functions and the concepts of one-to-one and onto (injectivity and surjectivity); this gives us a quasi-ordering relation
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| - Cardinal assignment (en)
- 基数指派 (zh)
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| - In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in the theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets, by equinumerosity). The concepts are developed by defining equinumerosity in terms of functions and the concepts of one-to-one and onto (injectivity and surjectivity); this gives us a quasi-ordering relation (en)
- 在集合论中，势的概念可以有相當的發展，而無需借助于定义基数为理论自身内的对象（这实际上是弗雷格采用的观点；基本上是指在等勢關係下，由在全集中的集合所組成的各個等价类）。勢的概念可以依据函数的单射、双射与满射概念來闡述；比如透過單射，可以给出在整个全集上通过大小比較的預序关系 是单射。 它不是真的排序，因为三分律不一定成立：如果 和 都为真，则通过 康托尔-伯恩斯坦-施罗德定理 为真，就是说 A 和 B 是等势的，但作為集合它们可以不是相等的；“三者至少一种情况成立”這一陳述等价于选择公理。 不过多数关于势和它的算术的有趣结果可以只通过 =c 来表达。 基数指派的目标是把每个集合 A 指派到特定的唯一的一个集合，所指派的集合只取決於 A 的势。这跟康托尔最初對基數的設想是一致的：取一个集合并把它的元素抽象为规范“单位”，再把这些单位收集到另一个集合中，使得有关这个集合唯一特殊的事情是它的大小。這類集合在 下會是全序的，而=c 會變成真正的等號。不過，如 Y. N. Moschovakis 所说，这只是作為體現数学簡潔性的一个練習，你不会得到更多东西除非你“对下标过敏”。但是在集合论的各种模型中，有“真实”基数的各种有价值的应用。 (zh)
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| - In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in the theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets, by equinumerosity). The concepts are developed by defining equinumerosity in terms of functions and the concepts of one-to-one and onto (injectivity and surjectivity); this gives us a quasi-ordering relation on the whole universe by size. It is not a true partial ordering because antisymmetry need not hold: if both and , it is true by the Cantor–Bernstein–Schroeder theorem that i.e. A and B are equinumerous, but they do not have to be literally equal (see isomorphism). That at least one of and holds turns out to be equivalent to the axiom of choice. Nevertheless, most of the interesting results on cardinality and its arithmetic can be expressed merely with =c. The goal of a cardinal assignment is to assign to every set A a specific, unique set that is only dependent on the cardinality of A. This is in accordance with Cantor's original vision of cardinals: to take a set and abstract its elements into canonical "units" and collect these units into another set, such that the only thing special about this set is its size. These would be totally ordered by the relation , and =c would be true equality. As Y. N. Moschovakis says, however, this is mostly an exercise in mathematical elegance, and you don't gain much unless you are "allergic to subscripts." However, there are various valuable applications of "real" cardinal numbers in various models of set theory. In modern set theory, we usually use the Von Neumann cardinal assignment, which uses the theory of ordinal numbers and the full power of the axioms of choice and replacement. Cardinal assignments do need the full axiom of choice, if we want a decent cardinal arithmetic and an assignment for all sets. (en)
- 在集合论中，势的概念可以有相當的發展，而無需借助于定义基数为理论自身内的对象（这实际上是弗雷格采用的观点；基本上是指在等勢關係下，由在全集中的集合所組成的各個等价类）。勢的概念可以依据函数的单射、双射与满射概念來闡述；比如透過單射，可以给出在整个全集上通过大小比較的預序关系 是单射。 它不是真的排序，因为三分律不一定成立：如果 和 都为真，则通过 康托尔-伯恩斯坦-施罗德定理 为真，就是说 A 和 B 是等势的，但作為集合它们可以不是相等的；“三者至少一种情况成立”這一陳述等价于选择公理。 不过多数关于势和它的算术的有趣结果可以只通过 =c 来表达。 基数指派的目标是把每个集合 A 指派到特定的唯一的一个集合，所指派的集合只取決於 A 的势。这跟康托尔最初對基數的設想是一致的：取一个集合并把它的元素抽象为规范“单位”，再把这些单位收集到另一个集合中，使得有关这个集合唯一特殊的事情是它的大小。這類集合在 下會是全序的，而=c 會變成真正的等號。不過，如 Y. N. Moschovakis 所说，这只是作為體現数学簡潔性的一个練習，你不会得到更多东西除非你“对下标过敏”。但是在集合论的各种模型中，有“真实”基数的各种有价值的应用。 在现代集合论中，我们通常使用冯·诺伊曼基数指派，它使用序数的理论与选择公理和替代公理的全部能力。基数指派需要完全的选择公理，如果我们想要像样的基数算术和对所有集合的基數指派。 (zh)
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