What is the difference between class and set?

[chingkui]

What is the difference between class and set? I have seen statements like "Let M be a class of subsets of X...", and it seems to me we can still do everything we like as though M is a set, and we are just avoiding the word "set" and replace it with "class". I am aware of the paradox "the set of all sets..." lead to, my question is what exactly is the difference between class and set? What sort of operations is prohibited in class while we can do in sets? How should we judge whether "class" rather than "set" is appropriate when describing a collection of sets? Thanks.

 

[matt grime]

You need to learn what a set is first. Simply put something may or may not be a set. Really when we say X is a set we mean X is a set in some model of some axiomatic set theory. Confusing? Try this for starters:

www.maths.bris.ac.uk/~maxmg/maths/philosophy/sets.html[/URL]

Loosele, a Class is a collection of objects. Something is a set *in some model*, it's just an extra label that we can apply to a class.

For instance we agree that the rational numbers exist and have the properties we think they do? They also are a group. That is we can add the label "group" to them. But we had the rationals before the label group was invented...

 

[honestrosewater]

In my book, all the objects in the universe are sets. Classes are not necessarily objects. (Whether something is an object is the whole point.) Classes are extensions of properties (the class of all objects having some definite property). You basically start with a class and look at your axioms to determine whether the class is a set or not. So all sets are classes, but not all classes are sets. Those classes that aren't sets are called proper classes and cannot be members of any class or do anything that would make them a set, like being a subset of a set or whatever your rules say. For instance, the universe is usually a proper class. The empty class is usually a set.
So in the beginning, everything is a class. You then add some axioms, allowing some classes to be sets. At least, that's how I'd sum it up. If you want a more formal grounding, you can read a little about models, like matt says.

 

[neurocomp2003]

http://mathworld.wolfram.com/SetClass.html
mathworld.com is awesome.