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I know that it doesn't make sense mathematically (Russell's paradox), but is there any nice way or notation to express the set of all sets?

Edit: I want the set to contain itself, even though it might break some definition.

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    $\begingroup$ @MeeSeongIm this is not an answer to what was asked $\endgroup$ – rschwieb Nov 24 '17 at 16:50
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    $\begingroup$ As for how to notate the universal set itself, wikipedia suggests that there is no standard notation, but I like the look of $\mathcal{U}$ or $\mathbb{U}$. @rschwieb yes it is, as the "set" spydon describes would be the power set of the universal set, using this notation: $\mathcal{P}(\mathcal{U})$ $\endgroup$ – JMoravitz Nov 24 '17 at 16:50
  • $\begingroup$ @rschwieb Thanks. I see the subtle difference. The power set of the universal set notation is what spydon is looking for. $\endgroup$ – Mee Seong Im Nov 24 '17 at 16:52
  • $\begingroup$ @MeeSeongIm not really... note that it says inclusion of a universal set is part of some nonstandard set theories. $\endgroup$ – rschwieb Nov 24 '17 at 16:56
  • $\begingroup$ If we had a universal set, it would already contain all sets: we wouldn't need to take its power set. $\endgroup$ – Misha Lavrov Nov 24 '17 at 16:58
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In set theory the class of all sets is often denoted $V$.

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  • $\begingroup$ I see the question was a bit unclear from the beginning, I want the set to contain itself too. I have updated the question. $\endgroup$ – spydon Nov 25 '17 at 9:46
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    $\begingroup$ Well, if you want the set to contain itself too you're not just asking about notation, you're asking for a major revision in set theory. There is no notation for this in standard set theory because there's no such thing. $\endgroup$ – David C. Ullrich Nov 25 '17 at 15:39
  • $\begingroup$ Alright, I guess this is the closest to what I want then, thanks! $\endgroup$ – spydon Nov 29 '17 at 13:29
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This concept is usually referred to as a "class". This concept is formalized in Von Neumann–Bernays–Gödel set theory which is essentially the usual Zermelo–Fraenkel set theory (ZF) + classes. The basic rule is a class is just some predicate. A set is a predicate restricted to a set. We also allow ourselves the axioms of ZF to define sets to avoid having a self-referential definition.

Every set is a class because if we have a set $\{x \in A : \phi(x)\}$ (i.e. the predicate $\phi$ restricted to the set $A$) then we have a class $\{x : x \in A \wedge \phi(x)\}$. What distinguishes a set from a "proper class" (a class that is not a set) is that sets are allowed to be members of other classes. That is, for a set $A$ we are allowed to talk about $A \in B$ where $B$ is a class. The class of all sets may be defined as $$ C = \{A : A = A\}. $$

Russel's paradox tells us that the statement $C \in C$ leads to a contradiction. Since it doesn't make sense to talk about whether or not $C$ is a member of something, that makes $C$ a proper class.

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  • $\begingroup$ Or put $C=\{S\colon S=S\}$ and it works even for non-well-founded set theories. $\endgroup$ – ziggurism Nov 24 '17 at 17:17
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    $\begingroup$ For a number analogy, consider the question "Notation the greatest natural number?" -- clearly there's no natural number greater than all natural numbers. But you can define '∞' to be a number greater than all naturals, by making it not natural. $\endgroup$ – Real Nov 24 '17 at 22:52
  • $\begingroup$ I see that the question was a bit unclear from the beginning, I want the set to contain itself too. I have updated the question. $\endgroup$ – spydon Nov 25 '17 at 9:46
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In category theory, you can refer to the category of all sets as “the category Set,” and its objects are precisely the “class of all sets”. I can’t remember what is popular for denoting the objects of a category, but I think obj(Set) is one option.

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    $\begingroup$ There are multiple "element of" notations acceptable among mathematicians, e.g., if $A$ is an object in the category $\textsf{Set}$, then $A\in \textsf{Set}$ or $A\in \text{ob}(\textsf{Set})$ are both acceptable. $\endgroup$ – Mee Seong Im Nov 24 '17 at 16:55
  • $\begingroup$ It doesn't contain itself though, or does it? $\endgroup$ – spydon Nov 24 '17 at 16:55
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    $\begingroup$ It doesn't contain itself, because it's not a set (it's a class) and it only contains sets. $\endgroup$ – Misha Lavrov Nov 24 '17 at 17:01
  • $\begingroup$ @spydon I don’t see that “containing itself” is a requirement of the question, rather I think it is a sign the user is groping for the right word “class of all sets” $\endgroup$ – rschwieb Nov 24 '17 at 17:05
  • $\begingroup$ @rschwieb I thought that was clear from referring to the paradox, but I have clarified it in the question now! $\endgroup$ – spydon Nov 24 '17 at 18:08

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