つづき

https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%83%E3%82%BB%E3%83%AB%E3%81%AE%E3%83%91%E3%83%A9%E3%83%89%E3%83%83%E3%82%AF%E3%82%B9
ラッセルのパラドックス（英: Russell's paradox）とは、素朴集合論において、自身を要素として持たない集合全体からなる集合の存在を認めると矛盾が導かれるというパラドックス
パラドックスの回避については、様々な方法が提案されている。詳細は矛盾の解消を参照
矛盾の解消
公理的集合論によって何をもって集合とするかについての形式的な整備が進められ
いかなる形の集合が存在するかを公理によって規定する
1.公理的集合論による解消[注 1]
2.単純型理論による解消[注 2]

https://en.wikipedia.org/wiki/Russell%27s_paradox
Russell's paradox （ここから下記へ）

https://en.wikipedia.org/wiki/Universal_set
Universal set
In set theory, a universal set is a set which contains all objects, including itself.[1] In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set.

Reasons for nonexistence
Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory.

Theories of universality
The difficulties associated with a universal set can be avoided either by using a variant of set theory in which the axiom of comprehension is restricted in some way, or by using a universal object that is not considered to be a set.

Universal objects that are not sets
Main article: Universe (mathematics)
The idea of a universal set seems intuitively desirable in the Zermelo–Fraenkel set theory, particularly because most versions of this theory do allow the use of quantifiers over all sets (see universal quantifier). One way of allowing an object that behaves similarly to a universal set, without creating paradoxes, is to describe V and similar large collections as proper classes rather than as sets. Russell's paradox does not apply in these theories because the axiom of comprehension operates on sets, not on classes.

The category of sets can also be considered to be a universal object that is, again, not itself a set. It has all sets as elements, and also includes arrows for all functions from one set to another. Again, it does not contain itself, because it is not itself a set.
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