Is the concept of a Universal Set an example of Russell's paradox?

Asked Jul 17 '15 at 22:35

Active Jul 19 '15 at 13:02

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In my book on functional analysis it is defined that a Universal Set is the union of all sets.

Is "union" used in the same sense as "set containing all sets"?

In other words, is the "set of all unions of sets" equivalent to "set containing all sets"?

edited Jul 19 '15 at 13:02

asked Jul 17 '15 at 22:35

Fraïssé

As far as we don't have other things but sets, yes: Let $U$ be the union of all sets and $V$ be the set of all sets. If $x$ is a set, then so is ${x}$, and hence $x\in{x}\subseteq U$. On the other hand, if $x\in U$, then $x$ is an element of a set, hence is a set by initial hypothesis, so $x\in V$. – Berci Jul 18 '15 at 01:11

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