Talk:Class (set theory)

message: Why is this a proper class (ie not a set)? Shouldn't it be both a class and an infinite set, at least according to naive set theory?
Brianjd 07:50, 2004 Nov 7 (UTC)
:Because then it would be a member of itself. Whilst this is at first sight not a problem under naive set theory, a subset of this set of all sets must be the set of all sets which do not contain themselves, which leads you to Russell's paradox, which blows naive set theory to pieces. Rebuilding set theory axiomatically using the ZF axioms solves Russell's paradox whilst producing a theory which is almost identical with naive set theory, but has extra constraints such as requiring sets not to be members of themselves. -- The Anome 12:57, Nov 7, 2004 (UTC)
After reading the whole article I understood. I think the article's introduction is a bit misleading - it should clarify that the comments made only apply to axiomatic set theory, which I expect most people are not familiar with (while most people are familiar with naive set theory - at least with the definition of a set according to this theory).
Brianjd 03:09, 2004 Nov 9 (UTC)
Wow. The above comment by Anome explains this whole concept better than anything on the actual article page itself. I was just confused after reading the article, but Anome's comment made it crystal clear. I am strongly tempted to include it (suitably paraphrased) in the article. --dmmaus (talk) 04:25, 7 October 2015 (UTC)
thread_title: Class of all sets

message: The article defines classes in terms of collections.  What are collections?
--Roderick Bloem, 16 June 2005
: Nothing. Think about it as an informal description, not definition. In NBG and related theories, a class is a primitive notion, and thus cannot be defined. In ZFC, classes formally do not exist, they are just shortcuts for their defining formulas on metalevel. -- EJ 13:36, 29 August 2005 (UTC)
replies: text: primitive notions are defined by the form of the axioms. I believe collection is a synonym for class.--MarSch 10:06, 26 October 2005 (UTC)
thread_title: What are Collections?

message: Is there a standard name for collections of classes? Perhaps metaclass, hyperclass, quasiclass? I've seen the word conglomerate used for this concept. These things do show up in category theory (as is the metaclass of all categories, or the metaclass of all functors from one category to another) and it would be nice to refer to them by some name; and maybe even have an article about them. Also, is there any set theory which talks about such objects? -- Fropuff (talk) 04:39, 9 January 2008 (UTC)
replies: text: One way of handling this is with Grothendieck universes. Sam Staton (talk) 13:27, 2 April 2008 (UTC)
thread_title:  Metaclasses 

message: The articles of sets states, that
*“A set is a collection of distinct objects”.
This article states, that
*“A class that is not a set is called a proper class” → To not be a set, it has to not be “a collection of distinct objects”.
*“and a class that is a set is sometimes called a small class” → So through above statements, that also makes a set a small class.
But then it states, that
*“For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.” → Which would mean, that “the class of all ordinal numbers, and the class of all sets” are not sets.
Which obviously can’t be, as “the class of all ordinal numbers, and the class of all sets” obviously are sets, since they are “collection[s] of distinct objects”, and thereby sets.
Which means, that “the class of all ordinal numbers, and the class of all sets” are small classes, as “a class that is a set is sometimes called a small class”.
□
— 94.220.250.151 (talk) 10:55, 4 November 2009 (UTC)
replies: text: Good point. The statement “A set is a collection of distinct objects” is an informal statement. When you come to carefully formalize set theory, you find that this definition of set doesn't make for a very nice theory. Consider the collection of all sets that do not contain themselves. Is that a set? If so, does it contain itself? Sam (talk) 17:46, 4 November 2009 (UTC)
thread_title:  Contradictory “explanation” of what classes are! 

message: An anonymous editor added the sentence "It was introduced by John von Neumann in 1925." I assume "it" means "classes", and the 1925 reference is a reference to his thesis. My understanding is that von Neumann's thesis introduces a formalization of classes in axiomatic set theory (leading to NBG). But did it really introduce the vague, informal concept of "class" at this point? wasn't that already around? Feel free to reinstate the sentence if need be. It was a bit heavy handed of me to remove it, but I wanted to say a bit more than "citation needed". ComputScientist (talk) 15:15, 24 October 2011 (UTC)
thread_title:  Classes "introduced" by von Neumann? 

message: 1)Is there an "increasing" sequence:sets, proper classes, proper conglomerates,...., that goes on for a "long time", that is a long finite sequence, or even an infinite sequence of more general and larger "lumps of stuff"?
2)On proper classes:Is a modification of cardinality available that measures the size of proper classes?76.218.104.120 (talk) 04:54, 5 July 2012 (UTC)
replies: text: I'd say it's up to you how you set up your foundations. One option is to assume Grothendieck's universes axiom (see Grothendieck universe), in which case the answer to both questions is "yes". ComputScientist (talk) 19:05, 11 July 2012 (UTC)
thread_title: A couple questions

message: From the first paragraph of article:
"...whereas other set theories, such as Von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity."
Suppose i'm a math major and my courses this semester are vector spaces, set theory, and ring theory. Couldn't I say my coursework is the set: {the class of vector spaces, the class of sets, the class of rings}? Just like we can't in general trisec an angle, but can trisect 90 degrees, perhaps the article should say proper classes cannot always be elements of a set?76.218.104.120 (talk) 00:46, 20 February 2013 (UTC)
replies: text: The paragraph is not discussing any informal notions of sets and classes, but axiomatic theories, and in all axiomatic theories of classes I am aware of, proper classes cannot be members of sets (or classes), and this is in fact their defining property. In any case, the paragraph says “e.g.”, so it allows for a different setup in principle.—Emil J. 13:45, 22 February 2013 (UTC)
thread_title: proper classes can't be memebers of a set??

message: In the following:
\forall x.(x\in A\leftrightarrow x=x). 
\lambda x.\phi
I am confused by the period after the x used these math expressions.
I don't recall seeing a period used this way, I've searched for references on math notation but found nothing.
Mathwikigeek (talk) 19:43, 8 October 2018 (UTC)
thread_title:  Notation used in the section, Classes in formal set theories, the period . 

message: I do not think it belongs in this article.  If it does, 
#it requires a source relevant to set theory (for which none has been provided in the conglomerate (category theory) article), 
#it should only be in one of the sections, 
#and the link should not be conglomerate (set theory), but conglomerate (category theory).  — Arthur Rubin  (talk) 10:02, 21 May 2019 (UTC)
thread_title:  [[Conglomerate (set theory)]]