Talk:Class (set theory)

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 Field: Foundations, logic, and set theory


Class of all sets[edit]

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[edit]

I moved the following from Class


[ ] 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.


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?[edit]

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?[edit]

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[edit]

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![edit]

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)

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)

ZF[edit]

I found the description "equivalence classes of logical formulas in the metalanguage" confusing, so I propose to expand the description like this:

In ZF set theory, classes exist only in the metalanguage, as equivalence classes of logical formulas for predicates. For example, if \mathcal A is a structure interpreting ZF, then the metalanguage expression \{x\in \mathcal A \mid x=x \} is interpreted by the collection of all the elements from the domain of \mathcal A; that is, all the sets in \mathcal A. So we can identify the "class of all sets" with the predicate x=x or any equivalent predicate.

but I wanted to first ask here if it looks ok. I may have used wrong terminology and/or misinterpreted the idea, or maybe it represents a controversial view needing more clarification, etc. It's inspired by a recent Reference Desk post of Trovatore that I can't seem to find right now. Please feel free to use anything from it if you want to edit the article directly. Thanks. 66.127.55.192 (talk) 18:57, 17 February 2010 (UTC)

I think that is very clear, so I've incorporated it into the article. I didn't know what was meant by "logical formulas for predicates" so I skipped that part. Sam (talk) 16:19, 18 February 2010 (UTC)
Hmm, should it say "proper classes"? The above description conflicts with the idea that a class is just any collection of objects in the structure (including collections that don't satisfy the set axioms). In particular, any set should also be a class, and there are many sets that can't be described by any formula. (Edit: hmm, but "any set should also be a class" conflicts with "classes exist only in the metalanguage", since most sets don't have any distinct existence in the metalanguage.) Trovatore, if you're reading this, can you help? 66.127.55.192 (talk) 22:56, 18 February 2010 (UTC)
You want to use a metalanguage with a constant symbol for every set to ensure that every set is definable in the metalanguage. This is a typical technique in model theory when considering some given model: start by expanding the language.
Personally, I think "In ZF set theory, classes exist only in the metalanguage" is not ideal wording. In ZF set theory, classes don't exist, period. Whether we can use them in the metalanguage is a separate issue. Of course there are well-known ways of extending the language of ZF to include terms for certain classes, as explained in thorough detail by Levy's Basic set theory. — Carl (CBM · talk) 03:21, 19 February 2010 (UTC)
Thanks, I tried to adjust the phrasing in the article slightly. 75.62.109.146 (talk) (new address) 08:42, 24 February 2010 (UTC)

Proper class = no powerset?[edit]

User:Tkuvho added a comment that being a proper class means that it has no powerset; the same sentence has been added to powerset and to Cantor's paradox. But "class" is often just an informal term, and moreover different people use it to mean different things. I don't think the sentence necessarily makes sense in the generality that it is stated, let alone being true. For instance one very common way of setting things up is to assume an inaccessible cardinal; call anything smaller a "set", and anything bigger a "class"; then there is a reasonable notion of "powerset" on classes.

I've removed the sentence from this page -- of course, feel free to reinstate it with more context and/or a reference. I'll hold off removing it from the other pages until I've checked for any consensus or disagreement. ComputScientist (talk) 16:27, 14 April 2011 (UTC)

The informal meaning of "class" that you referred to is going to be different from the technical meaning under any circumstances. The issue of the informal meaning is orthogonal to the issue of the technical definition. Saying that a proper class does not have a powerset is a better explanation of the difference between a proper class and a set than what we currently have in the lead. Tkuvho (talk) 18:14, 14 April 2011 (UTC)
Furthermore, Von Neumann–Bernays–Gödel set theory, a conservative extension of ZF, makes it clear that the power set operation applies only to sets, not to classes. Tkuvho (talk) 18:22, 14 April 2011 (UTC)
There isn't really a powerset operation in ZF(C) or in NBG set theory. There are no term-forming operations at all in the basic language, just one binary relation symbol. It's true that in ZFC, for any proper class A there is no set b that contains every subset of A. But there may well be a class C that contains every subset of A. Levy's Basic Set Theory defines powerclasses in this way on p. 19; for every class A he has an associated powerclass P(A) = { x : x ⊆ A }. Levy's book has a very detailed treatment of proper classes compared to other common books like Kunen's. — Carl (CBM · talk) 19:04, 14 April 2011 (UTC)
The reader may find it helpful to know that a basic difference between a set and a class is that all sets have a power set, but the power operation does not (always) apply to a class. Is there any reason to obfuscate this point? Tkuvho (talk) 05:07, 15 April 2011 (UTC)
By "informal meaning" I didn't mean "layman's meaning", I just meant the fact that classes don't have any formal status in the language of ZF, which makes it difficult to say what "poweroperation" would even mean. But Carl has cleared this up. I am still not sure that Tkuvho's sentence about powersets is helpful in understanding what proper classes are. We already have a sentence "the axioms of ZF do not immediately apply to classes". The axiom of power set just one of the axioms of ZF. We could change the sentence to "the axioms of ZF (e.g. the Axiom of Power Set) do not immediately apply to classes"... ComputScientist (talk) 10:16, 15 April 2011 (UTC)
I was trying to say that all classes do have powerclasses; the powerclass of A is the class of all sets that are subsets of A. — Carl (CBM · talk) 10:58, 15 April 2011 (UTC)
If the meaning of "power operation" is ambiguous, perhaps the point that could be mentioned to help the reader is that unlike a set, a class is not a member of anything. Usually the power operation applied to C is understood as creating a collection where the outcome contains C as an element. I still think such remarks can be helpful for the unitiated. Classes after all were historically introduced so as to create weaker objects to which the power operation does not apply. To someone who believes set theory is foundation of reality it may be disturbing to hear "classes" described as objects introduced to fix a bug in the theory; but the average reader deserves to know this nontheless. Tkuvho (talk) 12:08, 15 April 2011 (UTC)
The article doesn't seem to mention powerclasses at all. I think you're saying we should include them, only so that we can say that if A is a class then A is not a member of the powerclass of A. I don't really see the benefit in mentioning powerclasses at all. The point of proper classes is that they are not members of anything, so how could they be members of their own powerclass? It seems very tangential to me.
Historically, classes were not "introduced"; the distinction between sets and classes was introduced by redefining the word "set" to mean something different than it originally did. The study of classes has always been present in set theory, for example Cantor studied the class of ordinals and the class of cardinals. — Carl (CBM · talk)

I undid the edits to Power set and Cantor's paradox as well. The power operation can be applied to a proper class; it just gives back another proper class. So just claiming that it can't be applied is, at face value, wrong. Moreover, in the set theory most people use (ZF or ZFC) there is no power "operation" at all, just an axiom that for any set there is a corresponding powerset. So in ZFC you can't "apply the power operation" to anything, set or class. The definitional extension that adds a powerset operation works just as well for classes as for sets, as in Levy's book. I hope we can keep the discussion here, instead of on three talk pages. — Carl (CBM · talk) 13:06, 15 April 2011 (UTC)

Just to add, in case it is helpful: the idea of having power operations for all classes also plays a central role in the Algebraic Set Theory of Joyal, Moerdijk and others. ComputScientist (talk) 17:48, 16 April 2011 (UTC)
That's all fabulous but this notion of power operation is a refinement of the usual one, and not the one people naively think of. The naive version does not exist for classes. At any rate, the real issue appears to be whether a class is an element of another entity. This point should be explained to the reader. This would be a way of clarifying the difference between set and class at all three pages. Currently our explanations of class are mainly "negative". Tkuvho (talk) 18:17, 16 April 2011 (UTC)
Carl, thanks for your comment. I defer to your judgment in matters set-theoretic, though I am a bit puzzled. Saying that the power operation still exists for classes is like saying that every function f has a derivative f': simply define the domain of f' to be the collection of points where the appropriate limit exists. Be that as it may, it is not clear to me what your intention is here. You deleted my edits at the other two pages, but left in my edit here, concerning class not being a member of another entity. Do I surmise that this is a legitimate thing to say about classes? Can it be added at the other pages? I do think it clarifies the nature of a "class" more than evasive descriptions such as "a collection defined by a predicate", etc., which don't really explain why it is different from a set. Tkuvho (talk) 07:07, 18 April 2011 (UTC)
Hi Tkuvho. As we have discussed, precisely what a class is depends on the foundation. You have written "e.g., as entities that are not members of another entity". This is reasonable for NBG, and it's explained further down the page, so it's not really necessary at the top, though I'm willing to accept it. Other foundations do not support this intuition. For example, I am happy with the class of all order types as a definition of the ordinals. An order type is itself a proper class, so you will not like this concept. ComputScientist (talk) 11:35, 18 April 2011 (UTC)
Thanks for filling me in on nuances of set theory, which is not my field of expertise. Did I understand you correctly above that some authors even define anything at epsilon_0 level and above, as "class", so that my intuition completely breaks down? I like that, actually. This illustrates that there are many variations on the theme of "class", but again I come back to my metaphor of the derivative. One should be able to say something to the effect that a first "naive" notion of class differs from a set in that it is not a member of another entity. Is that fair enough? If so, this may be useful to a reader at the Cantor page, as well. Tkuvho (talk) 12:03, 18 April 2011 (UTC)
Hi. Epsilon_0 is probably too small, but that's the idea. Anyway, I don't think that there is a "first naive notion" of proper class. My understanding is that set theory came about as a fix, a way of carving out a well-behaved collection of classes that avoid the paradoxes. I understand that the primary thing is that the set theories fix the paradoxes, and that any intuition about sets/proper classes comes afterwards (if indeed there is any intuition to be had). ComputScientist (talk) 16:38, 20 April 2011 (UTC)

@Tkuvho: the note that you added here is exactly the definition of a proper class in the usual set theories that have proper classes, such as NBG and MK set theory. So I think it's certainly something we should say here. I don't think that we should try too hard to explain proper classes in other articles, particularly because what they "are" depends on the set theory that is used, and sometimes even then different people handle them somewhat differently (as with ZFC). — Carl (CBM · talk) 00:41, 21 April 2011 (UTC)

At NBG set theory this point is mentioned but somewhat indirectly, perhaps this should be clarified. Tkuvho (talk) 05:12, 21 April 2011 (UTC)
I'll need to look up a published axiomatization of NBG to double-check, but I will try to clarify that article. — Carl (CBM · talk) 10:44, 21 April 2011 (UTC)

Classes "introduced" by von Neumann?[edit]

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)

A couple questions[edit]

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)

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)

proper classes can't be memebers of a set??[edit]

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)

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)