Talk:Universal set

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On 22 October 2025, it was proposed that this article be moved to Universe of sets. The result of [[Special:Permalink/1320723229#Rename to Universe of sets?|the discussion]] was not moved.

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)Reply

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)Reply

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)Reply

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)Reply

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)Reply

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)Reply

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)Reply

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)Reply

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

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.

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

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

"With ${\displaystyle \varphi (x)}$ chosen as ${\displaystyle x\notin x}$, it follows that the subset ${\displaystyle B=\{x\in A\mid x\not \in x\}}$ is never a member of ${\displaystyle A}$, since, as Bertrand Russell observed, the alternative is paradoxical: If ${\displaystyle B}$ 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)Reply

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)Reply