Talk:Universal set

Merge suggestion with Set of all sets

I’ve undone an anonymous deletion of the content of this page. It may have been intended as implementing the proposed merger of Set of all sets into this page; but it was anonymous, nearly commentless, and got the direction wrong, and there was no corresponding entry in the discussion. Since “Universal set” is the term used by the standard work on the subject, it seems to me the merging, if done, should be in the direction originally suggested, though there may well be arguments to the contrary. —FlashSheridan (talk) 20:54, 26 May 2008 (UTC)

Universal set in probability theory

It is claimed in the article that the sample space in probability theory contains all events and is a universal set. My understanding of probability theory (which is admittedly limited) is that the sample space is the set of all outcomes, not of all events. Besides, I don't see how either the sample space or the set of events are universal sets. Am I missing something or should this be removed? HerrEkberg (talk) 11:34, 26 July 2011 (UTC)

Intro is dubious

The intro claims that the problem with a universal set stems from the axiom of comprehension.

But shortly afterwards it points out that the axiom of comprehension causes a problem even for sets which aren't the universal set. So it isn't the universal set which is the problem at all; it's the axiom of comprehension.

With the axiom of comprehension removed (and the axiom of separation substituted for instance), it is not clear why the universal set causes problems. --131.111.248.243 (talk) 04:18, 26 November 2012 (UTC)

Too technical

I have tagged this article as currently being too "technical". It needs to take more account of the Wikipedia:WikiProject Mathematics "style guidelines" http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics#Some_issues_to_think_about :

if it makes sense, add a very quick (and naive) explanation/description 

There is need of an introduction to the article that sets out the subject in simple language (not the mathematical formulae currently used), and in this case this would include Zermelo's axiom of comprehension which would need to be outlined.

In the maths there is there perhaps an error in the grammar: "which contains exactly those elements x of A that satisfy \varphi exists" It doesn't seem to quite make sense as it stands. --LookingGlass (talk) 12:12, 27 November 2012 (UTC)

A necessary circularity?

I am confused by the first sentence:

In set theory, a universal set is a set which contains all objects, including itself.

Why does it of necessity need to be defined in this way? Doing so introduces the circularity which appears unnecessary so what in the theory of sets (as opposed to the reality that sets describe) requires such circularity to be set up? It would seem easy to subsitute "wxcluding" for "including, so there must be a solid reason. Also the word "objects" isn presumably defined in a specuialist way as I imagine it is possible to have a set of all the primary colours for example. --LookingGlass (talk) 12:21, 27 November 2012 (UTC)

If the set does not contain itself, then it's not the Universal set because there is a set which it does not contain. More importantly, defining U as the set which contains every set except itself does not resolve the paradoxes, since Russell's paradox does not involve U itself, but merely R (defined as all sets which do not contain themselves). Informally, when sets get "too inclusive", paradoxes result. The Universal set is as inclusive as you can get, but less inclusive sets such as R still raise the paradoxes. Mnudelman (talk) 16:22, 1 April 2013 (UTC)

Thanks Mnudelman. I understand what you say, however now I find that the concept of a thing (a set) which doesn't contain itself but only those things (sets) that it's made up of hard to get my head around. As I don't speak "set theory" the explanations in the article are all Greek to me. Is there a "translation" of Russell's Paradox? LookingGlass (talk) 12:10, 2 April 2013 (UTC)

When I went to school, "universal set" meant the set of all objects in the domain under consideration. A related concept, the set complement, tends to rely on this definition. If the domain under consideration doesn't contain sets, "including itself" ceases to be satisfied. Furthermore, of the textbooks I used, the one I remember mentioning the term had a symbol not mentioned here for universal set: essentially a Latin epsilon which loops round and intersects itself at the bottom. — Smjg (talk) 13:37, 13 May 2022 (UTC)

You're thinking of Universe (mathematics), a different topic from the one in this article. —David Eppstein (talk) 16:13, 19 May 2022 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Universal set/CommentsTalk:Universal set/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Just a comment on the "sets of all sets which are not members of themselves" (which Russell made up when he created the paradox). Frege said that all sets are members of themselves - thus by definition it makes no sense to conceive of a set not being a member of itself. Or is there something I have missed? Mezzof (talk) 19:59, 5 January 2009 (UTC)

Substituted at 18:42, 17 July 2016 (UTC)

Error?

In the section Russell's paradox one sentence reads as follows:

"With $\varphi (x)$ chosen as $x\notin x$, it follows that the subset $B=\{x\in A\mid x\not \in x\}$ is never a member of $A$, since, as Bertrand Russell observed, the alternative is paradoxical: If $B$ contains itself, then it should not contain itself, and vice versa."

This seems to have one or more mistakes, since I find it difficult or impossible to follow. (Even though I am thoroughly familiar with Russell's paradox.)

And if it is correct, then I hope someone knowledgeable about this subject will rewrite it to make it far, far easier to follow. 2601:200:C000:1A0:8C8B:E7C5:297C:FFFB (talk) 17:18, 23 October 2021 (UTC)

I think the proof is correct, and I made an attempt to explain it in a better way - not sure that it succeeded. - Jochen Burghardt (talk) 19:23, 23 October 2021 (UTC)

Rename to Universe of sets?

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: not moved. Favonian (talk) 18:18, 12 November 2025 (UTC)

---

Universal set → Universe of setsUniverse of sets – I'm proposing a minor topic change from "Set of all sets" to "Universe of sets", having the non-existent object be a section of the article. There's a reasonable amount to say about the universe of sets in terms of model theory and philosophy. Specifically, in a Platonist view and the role of large cardinals, compared to a multiversalist view. The broader concept is already discussed here, and, honestly, it seems weird to have the non-existent object be the main topic of the article. – Farkle Griffen (talk) 05:01, 22 October 2025 (UTC) — Relisting. Jeffrey34555 (talk) 05:24, 29 October 2025 (UTC) — Relisting. CNC (talk) 21:54, 5 November 2025 (UTC)

Oppose. We already have an article Universe (mathematics) that appears to be what you want the article to be about. This one is about a different topic, non-standard set theories in which the universe is an actual set, and the problems that causes in formulating a consistent set theory. —David Eppstein (talk) 06:20, 22 October 2025 (UTC)

Universe (mathematics) appears to be about an internally-defined set under investigation, eg. $\mathbb {N} ,\mathbb {R} ,{\text{Ob}}({\mathcal {C}})$, etc. Domain of discourse might be closer, but it's too broad. There is no article about the "Universe of sets" $V$ in the usual sense, which is the article I find myself wanting to link to most often. The concept comes up often enough that it certainly deserves its own article, but I would really hate to add a fourth article to the mix. – Farkle Griffen (talk) 13:02, 22 October 2025 (UTC)

"NBG set theory" is an elaborate article where the class of all sets is introduced in section NBG set theory#Class existence axioms and axiom of regularity. Since its history section handles several similar systems, it might be considered an alternative to link $V$ to. However, I can't judge whether there are fundamentally different approaches that provide a class of all sets. - Jochen Burghardt (talk) 15:13, 22 October 2025 (UTC)

Isn't V the topic of Von Neumann universe? At least for well-founded set theories? —David Eppstein (talk) 06:34, 29 October 2025 (UTC)

That's probably where it came from, but it's used more generally as just any universe of sets. It's even used here in § Restricted comprehension. – Farkle Griffen (talk) 18:34, 4 November 2025 (UTC)

That section refers to universal set; it does not use the word universe at all. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:57, 6 November 2025 (UTC)

Note: WikiProject Mathematics has been notified of this discussion. CNC (talk) 21:54, 5 November 2025 (UTC)

Oppose. Universe of sets is a concept in model theory, not a concept of set theory. The univeral set exists in non-well-founded set theories such as New Foundations. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:57, 6 November 2025 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.