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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4@MeeSeongIm this is not an answer to what was asked – rschwieb Nov 24 '17 at 16:50
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2As 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: $\mathcal{P}(\mathcal{U})$ – JMoravitz Nov 24 '17 at 16:50
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@rschwieb Thanks. I see the subtle difference. The power set of the universal set notation is what spydon is looking for. – Mee Seong Im Nov 24 '17 at 16:52
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@MeeSeongIm not really... note that it says inclusion of a universal set is part of some nonstandard set theories. – rschwieb Nov 24 '17 at 16:56
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If we had a universal set, it would already contain all sets: we wouldn't need to take its power set. – Misha Lavrov Nov 24 '17 at 16:58
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@MishaLavrov I think if we have a universal set $U$ then the sets in consideration would be $\wp(U)$. – Trevor Gunn Nov 24 '17 at 17:00
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That's not false, but if we have a universal set $U$ then $U = \wp(U)$, so it's redundant. – Misha Lavrov Nov 24 '17 at 17:01
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@MishaLavrov $U = \wp(U)$ never holds because $|U| < |\wp(U)|$. Think of $U$ as the base set and $\wp(U)$ as a $\sigma$-algebra on $U$. – Trevor Gunn Nov 24 '17 at 17:04
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@TrevorGunn That's a reason the universal set doesn't exist. If it did, it would contain all sets; every element of $\wp(U)$ is a set; so it would contain all of $\wp(U)$. – Misha Lavrov Nov 24 '17 at 17:06
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@MishaLavrov I see. You were talking about a universal set as a "set of all sets" in some set theory whereas I was thinking of a universal set as is taught in intro to probability: the sample space. – Trevor Gunn Nov 24 '17 at 17:12
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For @TrevorGunn and Mee Seong Im: If we consider the universe of sets to be a collection of sets $U$ that is itself a set, then Cantor's theorem applies and $\lvert U\rvert < \lvert \wp(U)\rvert.$ But then $U$ is not really universal, since it does not contain itself. On the other hand, if we consider a collection $U$ of all sets that is a proper class, then Cantor's theorem doesn't apply and in fact $\lvert U\rvert = \lvert \wp(U)\rvert.$ The powerset of a proper class is itself a proper class that only contains those subcollections that are sets, not classes. – ziggurism Nov 24 '17 at 17:13
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In set theory the class of all sets is often denoted $V$.
David C. Ullrich
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I see the question was a bit unclear from the beginning, I want the set to contain itself too. I have updated the question. – spydon Nov 25 '17 at 09:46
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2Well, 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. – David C. Ullrich Nov 25 '17 at 15:39
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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.
Trevor Gunn
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Or put $C={S\colon S=S}$ and it works even for non-well-founded set theories. – ziggurism Nov 24 '17 at 17:17
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1For 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. – Real Nov 24 '17 at 22:52
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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. – spydon Nov 25 '17 at 09:46
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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.
rschwieb
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1There 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. – Mee Seong Im Nov 24 '17 at 16:55
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2It doesn't contain itself, because it's not a set (it's a class) and it only contains sets. – Misha Lavrov Nov 24 '17 at 17:01
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@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” – rschwieb Nov 24 '17 at 17:05
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@rschwieb I thought that was clear from referring to the paradox, but I have clarified it in the question now! – spydon Nov 24 '17 at 18:08
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@spydon OK: I thought you were looking for the term in standard set theory. I figured wrong, sorry. You’ll notice that all the answers are along the same lines... – rschwieb Nov 24 '17 at 22:26