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Classes are larger than sets.

What's larger than classes ?

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What types of things do you have in mind? The question has no context to tell what type of answer you are looking for. –  Carl Mummert Aug 6 '10 at 18:16
    
There are things that are too big to be sets. Are there things that are too big to be too big to be sets ? –  user882 Aug 6 '10 at 18:18
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in ZFC there are only sets and no classes, but yes you get set theories which have classes and there are also type theories which have types, types of types, types of types of types of types... in an infinite hierarchy. The actual structure of the hierarchy doesn't seem to be very interesting, it's just necessary to avoid paradoxes. –  anon Aug 6 '10 at 18:33
    
Is there a Cantor-diagonal-style argument or something to show there are different sizes of classes, as there is for sets ? –  user882 Aug 6 '10 at 18:38
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No, if you have Cantor-diagonal-style machinery available you are dealing with sets. Classes are really there as a catching net for the paradoxes of set theory (so Russell's Paradox escapes being a paradox because the class of all sets where $x \notin x$ is not itself a set). –  Jason Dyer Aug 6 '10 at 19:44

4 Answers 4

Sets are first-order objects in a system like ZFC, although there are systems like second-order arithmetic where the first-order objects are non-set individuals and sets of individuals are the second-order objects.

Bernays-Gödel set theory (BG set theory), which I see both WP and Wolfram prefer to call by the unwieldy term Von Neumann–Bernays–Gödel set theory, adds second-order quantifiers over classes. The result is conservative over ZFC (i.e., its first-order theorems are the same), but allows much shorter proofs of theorems. You can formalise talk of classes in first-order set theory by using class terms, where you treat one-place predicates as defining a set by extension, but this is weaker than using second-order quantifiers, and an important issue in second-order set theories is what classes a particular set theory is committed to. This question can be addressed axiomatically in third-order set theory.

I've seen the usage "second-order set" and "third-order set" to talk about classes and their third-order generalisation.

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From one possible point of view, conglomerate come next.

Take a model of set theory $\mathcal{U}$ and assume you have an inaccesible cardinal; let $S \in \mathcal{U}$ be a set of that cardinality. Then $S$ itself is another model of set theory. So you can (re)-define:

1) Sets as elements of $S$. That is, $A \in \mathcal{U}$ is a set if and only if $A \in S$.

2) Classes as subsets of $S$. That is, $A \in \mathcal{U}$ is a class if and only if $A \subset S$.

3) Conglomerates as elements of $\mathcal{U}$.

In this way sets and classes will satisfy the usual axioms, and you have conglomerates as even bigger collections.

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Wikipedia doesn't even have this word (as a mathematical concept) o_O. –  KennyTM Aug 8 '10 at 20:56
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Is it a new criterion for deciding what exists in mathematics? :-D –  Andrea Ferretti Aug 8 '10 at 22:35
    
For what it's worth it does not have "stability condition" either... –  Andrea Ferretti Aug 8 '10 at 22:40

Classes are defined differently in systems, but essentially--

Sets are defined from bottom-up (adding elements starting from nothing) while classes are defined from top-down (starting with the universal class V and then deciding from there which classes are sets). So it doesn't make sense to talk about things larger than the universal class, and if notions of comparing sizes are possible, you're simply dealing with classes that are also sets. (V itself is not a set.)

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It is actually possible to talk of things larger than classes, as I explain in my answer. –  Andrea Ferretti Aug 6 '10 at 23:57

A class is a collection of things. Some classes are sets (in some sense, only "smaller" classes can be sets). Since any collection is a class, there can't be anything larger.

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That is incorrect - there are collections of classes which are not classes. –  Jacques Carette Aug 8 '10 at 13:30

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