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universal set
In set theory, a universal set is a set which contains all objects, including itself.〔Forster 1995 p. 1.〕 In set theory as usually formulated, the conception of a universal set leads to a paradox (Russell's paradox) and is consequently not allowed. However, some non-standard variants of set theory include a universal set.
It is often symbolized by the Greek letter xi.
==Reasons for nonexistence==
Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, do not allow for the existence of a universal set. Its existence would cause paradoxes which would make the theory inconsistent.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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