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The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely: where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal.

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  • Von Neumann cardinal assignment
  • 冯·诺伊曼基数指派
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  • 冯·诺伊曼基数指派是使用序数的基数指派。对于良序集合 U,我们定义它的基数为等势(equinumerous)于 U 的最小序数。更加精确的, , 當中: * 是单射 * 和 都为真為真 * 是序数的类。 这个序数也叫做这个基数的初始序数。使用替代公理,U 是良序的和序数的类是良序的的事实保证这样一个序数存在并且是唯一的。通过完全选择公理,所有集合都是可良序的,所以所有集合都有一个基数;我们使用从序数继承来的次序排序基数。容易发现这与通过 的排序相符。这是基数的良序排序。
  • The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely: where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal.
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  • The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely: where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal. That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤c. This is a well-ordering of cardinal numbers.
  • 冯·诺伊曼基数指派是使用序数的基数指派。对于良序集合 U,我们定义它的基数为等势(equinumerous)于 U 的最小序数。更加精确的, , 當中: * 是单射 * 和 都为真為真 * 是序数的类。 这个序数也叫做这个基数的初始序数。使用替代公理,U 是良序的和序数的类是良序的的事实保证这样一个序数存在并且是唯一的。通过完全选择公理,所有集合都是可良序的,所以所有集合都有一个基数;我们使用从序数继承来的次序排序基数。容易发现这与通过 的排序相符。这是基数的良序排序。
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