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.

Comments (4)

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.

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 000A. Beware of the difference between the letter 'O' and the digit '0'.