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.

  • 4
    $\begingroup$ @MeeSeongIm this is not an answer to what was asked $\endgroup$ – rschwieb Nov 24 '17 at 16:50
  • 2
    $\begingroup$ 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

In set theory the class of all sets is often denoted $V$.

  • $\begingroup$ 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
  • 2
    $\begingroup$ 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

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.

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.

  • 1
    $\begingroup$ 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
  • 2
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.