Talk:Class (set theory)

Class of all sets

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)

Moved from Class

I moved the following from Class

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[ ] In some abstract algebra literature it has been found that a collection is sometimes required "to be a set and not a proper class". When might a proper class not be a set? How is a class not a set? A critical facing of this question would look at the major set theories with their various axioms. Notice also the possible confusion of words as noted in one axiom set discussion at Knowledge Interchange Format (its first intent is not for open human language, see provisos) site: http://logic.stanford.edu/kif/Hypertext/node21.html where one finds "An important word of warning for mathematicians. In KIF, certain words are used nontraditionally. Specifically, the standard notion of class is here called a set; the standard notion of set is replaced by the notion of bounded set; and the standard notion of proper class is replaced by unbounded set." Such matters are going to force a close look at definitions and axioms for clear resolution. Logicians and mathematicians may be needed to help sort this challenge's intricacies.

One set of sources for set theories having a universal set: http://math.boisestate.edu/~holmes/holmes/setbiblio.html

But attention will be needed for each major set theory. And from New Foundations set theory.

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I hope that the article explains that some classes are not sets. For instance, the collection of all sets which don't contain themselves as an element is a proper class, but not a set.

The KIF terminology is indeed non-standard, and nobody except them uses it. In Wikipedia, every set is a class, but not every class is a set. Those classes which are not sets (because they are too "big"), are called proper classes. --AxelBoldt

This was my first time being exposed to the concept of a class versus a set. I was confused by the introduction, and feel that it should be rewritten to be friendlier to non-techies. In particular, please do not give examples of sets versus classes until you have pointed out that this is different from classical ZFC set theory.

What are Collections?

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)

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)

Class of all classes?

It's unclear to me why the class of all classes can't exist. According to the article, the only requirement imposed on a class's elements is an unique shared property. So shouldn't I be able to create a class of all x, where x is a class?

As clearly stated in the article, only sets can be elements of a class. -- EJ 16:19, 25 October 2005 (UTC)

What about "or sometimes other mathematical objects"? A class is a "mathematical object", isn't it? Thanks much!

It comes down to this: if you allow a class of classes, you can define the class of all classes that don't contain themselves, giving you Russell's paradox again, exactly what this construct is attempting to avoid. Mark Hurd 11:33, 16 January 2006 (UTC)

You may define a hyperclass that can contain proper classes. And while you're at it, a hyper-hyperclass containing hyperclasses, and so on. Chithanh 02:19, 9 June 2006 (UTC)

There's a level of class for every ordinal number. Actually, there's a bunch of somethings like ordinal numbers for every level of class, and a level of class for every something like an ordinal number. This means the number of ordinal-number-like things is quite large. --Ihope127 20:41, 9 October 2006 (UTC)

Hmm, if my current understanding is correct, it's not possible to speak of a "powerclass" of a proper class, that is, the class of all its subclasses. Is it even possible to speak about subclasses? - Saibod 22:23, 24 March 2007 (UTC)

Yes, in NBG at least, for any proper class X there's a power-class P(X); of course not all subclasses are elements of P(X), only those which are sets - Gödel called them "subsets". In particular, for the universal class U, any set is a subset of U, and any subset of U is a set therefore an element of U ; by the extensionality axiom, P(U)=U. But all classical mathematics work in NBG the same way as in ZF.- Michel42 22:58, 27 May 2007 (UTC)

Metaclasses

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)

One way of handling this is with Grothendieck universes. Sam Staton (talk) 13:27, 2 April 2008 (UTC)

Contradictory “explanation” of what classes are!

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”.
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— 94.220.250.151 (talk) 10:55, 4 November 2009 (UTC)

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)