## 3.3 Classes

Informally we use the notion of a class. Given a formula $\phi (x, p_1, \ldots , p_ n)$, we call

$C = \{ x : \phi (x, p_1, \ldots , p_ n)\}$

a class. A class is easier to manipulate than the formula that defines it, but it is not strictly speaking a mathematical object. For example, if $R$ is a ring, then we may consider the class of all $R$-modules (since after all we may translate the sentence “$M$ is an $R$-module” into a formula in set theory, which then defines a class). A proper class is a class which is not a set.

In this way we may consider the category of $R$-modules, which is a “big” category—in other words, it has a proper class of objects. Similarly, we may consider the “big” category of schemes, the “big” category of rings, etc.

Comment #1481 by verisimilidude on

You say in section 3.3 (tag 000A) that "A proper class is a class which is not a set." Yet in the previous section, 3.2, the declaration was made that everything is a set. So is the set of proper classes an empty set? And if so then what is the use of defining a proper class. If this is not true then I think we readers are owed a little more explanation of the term "proper class".

Comment #1482 by on

Note the word "informally" at the beginning of the paragraph.

Comment #4157 by 蒋云江 on

It would be nice to outline why the class of R modules is proper, or at least mention which ZFC axiom is violated.

Comment #4366 by on

Hmm. Well, given any set $S$ the set of maps from $S$ to $R$ is an $R$-module of cardinality greater than or equal to $|S|$. Thus we have $R$-modules of arbitrarily large cardinality and hence these cannot all be elements of a single set.

Of course no axioms are being violated! Not sure what you meant by that statement.

I'm going to leave this alone for now unless other people speak up or make the edits themselves.

Comment #7879 by parsec on

@verisimidude

(I am not a graduate student; tell me if this is inaccurate)

It's not true that everything is a set. The axioms of ZFC are actually very specific rules for how a set can be constructed. This is because of stuff like Russel's Paradox (does the "set" of all sets that do not contain themselves contain itself). A class is a way to talk about some sets without placing them in a set: by definition, classes can't contain themselves, which avoids Russels's paradox. The "set" of all proper classes would actually not be a class or a set: it would be the next step up in the hierarchy.

Comment #7905 by chris on

Everything is a set but a class is not a thing. Actually, most notions are not, like art, justice, wisdom or beauty. There is no way to know whether a set of proper classes exists; we can safely assume it does not. If it existed, it would not be empty.
Of course, there always is a "minimal model" of the set of all proper classes: just enumerate every possible formula in the formalism and associate a class with each one. But this set will not be a true thing, it will be a metaobject that depends on the particular formalism.

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