
## 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.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).