Confusion: Classes and Sets

I'm reading the wikipedia article on 'well-founded relation', and this is the beginning part of it:

In mathematics, a binary relation, R, is well-founded on a class X if and only if every non-empty subset of X has a minimal element with respect to R

I am very, very new to set-theory (as in, I started reading about it an hour ago), so I'm reading articles on wikipedia and 'backtracking' if I don't know what a certain definition means, until I can finally understand an article properly.

Here I didn't exactly know what a 'class' is, and I found out that it is a collection of sets which can be defined by a property all of its members share.

But the wikipedia article talks about a subset of X? What does this mean? I'm guessing it means either:

• A bunch of sets

• A subset of every set of the class

• – Maesumi Sep 21 '13 at 14:19

It might be wiser to start reading about set theory from a book, or something, before jumping into slightly more advanced topics like that.

The point in set theory is that all the objects are sets. Classes are collections of objects which may or may not be sets themselves. But in set theory talking about classes is the same as talking about "the set of all the even numbers greater than $53$" when talking about the natural numbers with addition.

When we say that $A$ is a subset of the class $X$ we just mean that $A$ is a set, and all the members of $A$ are members of $X$. For example, the class of all sets is a class, and every set is a subset of this class, because the elements of sets (in set theory) are other sets, which are elements of that class.

• That seems straight-forward, thanks. Any recommendations for books on elementary set theory? As my previous question indicates, my goal is to understand the proof to Goodstein's theorem, because I find the theorem itself to be quite cool. – MacropusRufus Sep 21 '13 at 15:00
• Hm. That's a difficult question actually. I'd probably start with some books about naive set theory, with a reasonable axiomatic background so they talk about classes. I have nothing from the top of my mind, though. – Asaf Karagila Sep 21 '13 at 16:03

In general, the word "class" is often used to mean a collection of objects that is, in some way, "one level up" from the level that we are normally considering. This explanation is heuristic, not formal or historical, but it illustrates the way the terminology is used.

• In computability theory, we talk about "sets of natural numbers", which are usually just called "sets". But then what do we call a set of sets of natural numbers? It would be confusing to also call it just a "set". Instead, we call it a "class". This notion shows up, for example, in the terminology "$\Pi^0_1$ class", which is a certain kind of set of sets of natural numbers. In computablity, the main objects of study are sets of natural numbers, and sets of sets of naturals are "one level up" from that.

• A more general phenomenon happens in descriptive set theory. There, they have a space $X$ and they study the properties of various subsets of $X$. But then, what do they call a set of subsets of $X$? They call it a "pointclass". Again, the key objects of study are sets of points, and so pointclasses are "one level up" from that. The "classes" in computability theory are a special case of this, actually.

• In set theory, the key objects of study are (usually) pure sets. So, just as above, set theorists use the word "class" to denote an arbitrary collection of pure sets. Some classes of pure sets are actually sets themselves, e.g. the class of natural numbers is a set, which is normally called the set of natural numbers. But, in set theories such as ZFC, not every class of pure sets is a set, for example the class of all pure sets is not a set in ZFC. There is no standard word for a "class of classes".

In set theory, there are many interesting issues with classes that are not sets. These issues are intertwined with choices that are made in the definition of specific formal theories in order to avoid the classical set-theoretic paradoxes. For example, ZFC avoids the issue by being uanble to refer to classes at all (the head-in-the-sand approach). Other set theories, such as Morse-Kelley, are able to prove various facts about classes. Other set theories, such as variations of Quine's "New Foundations", have other ways of (presumably) handling the paradoxes.

• This is vaguely analogous to the way that functional analysts use the word "operator" for a function from a Banach space to another Banach space, and use the word "function" with a more limited meaning, even though they would agree that, in the end, every operator is a kind of function. The distinction of terminology helps make it clear what type of object is being described. – Carl Mummert Sep 21 '13 at 15:10

For now, in a statement like this one replace the word class by set and understand that statement. Since all sets are classes, the statement will still be true, just less general.

Later you may learn why there is a distinction between classes and sets (see Russell's paradox for motivation).