Ordinal: Difference between revisions
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(Created page with "In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the natural numbers that describe the order types of well-ordered sets. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. ==Von Neumann definition== The Von Neumann definition of ordinals defines ordinals as objects in ZFC. Each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \).") |
(ZFC not only setting of pure sets. There are also theories with urelements →Von Neumann definition) |
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==Von Neumann definition== |
==Von Neumann definition== |
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In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). |
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Revision as of 00:35, 16 October 2022
In set theory, the ordinal numbers or ordinals are an extension of the natural numbers that describe the order types of well-ordered sets. A set \( S \) is well-ordered if each non-empty \( T \subseteq S \) has a least element.
Von Neumann definition
In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \).