# Difference between revisions of "Universal Set"

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A Universe Set is a set that contains all Things, including itself.

## References

### 2011

In set theory, a universal set is a set which contains all objects, including itself.[1] In set theory as usually formulated, the conception of a set of all sets leads to a paradox. The reason for this lies with Zermelo's axiom of separation: for any formula $\varphi(x)$ and set $A$, the set $\{x \in A \mid \varphi(x)\}$ which contains exactly those elements $x$ of $A$ that satisfy $\varphi$ exists. If the universal set $V$ existed and the axiom of separation applied to it, then Russell's paradox would arise from $\{x \in V\mid x\not\in x\}$. More generally, for any set $A$ we can prove that $\{x \in A\mid x\not\in x\}$ is not an element of A.

A second issue is that the power set of the set of all sets would be a subset of the set of all sets, providing that both exist. This conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality than the set itself.

### 2019

• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Universal_set Retrieved:2019-11-10.
• In set theory, a universal set is a set which contains all objects, including itself. [2] In set theory as usually formulated, the conception of a universal set leads to Russell's paradox and is consequently not allowed. However, some non-standard variants of set theory include a universal set.
• Forster 1995 p. 1.
• Forster 1995 p. 1.