The goal of a cardinal assignment is to assign to every set A a specific, unique set which is only dependent on the cardinality of A. This is in accordance with Cantor's original vision of a 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 <math>\leq_c</math> and =c would be true equality. As Y. N.

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  • The goal of a cardinal assignment is to assign to every set A a specific, unique set which is only dependent on the cardinality of A. This is in accordance with Cantor's original vision of a 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 <math>\leq_c</math> 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. More on this (and much more good set theory in general!) can be found in Moschovakis' excellent introduction to set theory.
  • 基数指派的目标是对每个集合 A 指派只由 A 的势决定的特定的唯一的一个集合。这和谐于康托尔最初版本的势: 采用一个集合并抽象它的元素为规范“单位”并收集这些单位到另一个集合中，使得有关这个集合唯一特殊的事情是它的大小。它们将按 <math>\leq_c</math> 而是全序的，=c 将是真正的等式。如 Y. N. Moschovakis 所说这只是在数学高雅上的一个实验，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。基数指派不需要完全的选择公理，如果我们想要象样的基数算术和对所有集合的指派。可以参见 Moschovakis 对集合论的卓越介绍。
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  • The goal of a cardinal assignment is to assign to every set A a specific, unique set which is only dependent on the cardinality of A. This is in accordance with Cantor's original vision of a 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 <math>\leq_c</math> and =c would be true equality. As Y. N.
  • 基数指派的目标是对每个集合 A 指派只由 A 的势决定的特定的唯一的一个集合。这和谐于康托尔最初版本的势: 采用一个集合并抽象它的元素为规范“单位”并收集这些单位到另一个集合中，使得有关这个集合唯一特殊的事情是它的大小。它们将按 <math>\leq_c</math> 而是全序的，=c 将是真正的等式。如 Y. N.
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  • Cardinal assignment
  • 基数指派
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