1. Singletons are sets with just one member. For example, \(\{\, a \,\}\) is the singleton set containing just the one member \(a\). \(\{\, \{\, \,\} \,\}\) is the singleton set containing just the one member \(\{\, \,\}\), that is, the empty set. Since another notation for \(\{\, \,\}\) is \(\varnothing \), we could also write \(\{\, \{\, \,\} \,\}\) as \(\{\, \varnothing \,\}\).

  2. If \(\Gamma\) is a subset of \(\Delta\) (that is, \(\Gamma \subseteq \Delta\)), then we can also say that \(\Delta\) is a superset of \(\Gamma\).

  3. Pure sets are sets like \(\{\, \,\}, \{\, \{\, \,\} \,\}, \{\, \{\, \,\},\{\, \{\, \,\} \,\} \,\}\), and so on, where no matter how deeply nested, none of the sets involved have any members that aren't themselves sets.

  4. One kind of set is a set of real numbers between two endpoints. This is known as an interval. If the set includes the upper endpoint, it's called a closed interval (on the top). If it doesn't, it's called an open interval (on the top). So for example:

    \(\{\, x\in\mathbb{R}\mid 0\le x\texttt{ and }x\le 1 \,\}\) is closed on both sides. This interval is commonly written as \([0, 1]\).

    \(\{\, x\in\mathbb{R}\mid 0\le x\texttt{ and }x\lt 1 \,\}\) is closed on the bottom, and open on the top. It's described as "half-open". This interval is commonly written as \([0, 1)\).

    \(\{\, x\in\mathbb{R}\mid 0\lt x\texttt{ and }x\lt 1 \,\}\) is open on both sides. This interval is commonly written as \((0, 1)\).

    Note the unfortunate coincidence between the notation for the open interval \((0, 1)\) and for the ordered pair \((0, 1)\). Those are very different things. Even those who want to reduce ordered pairs to sets don't want to reduce the ordered pair \((0, 1)\) to the set of real numbers which is the interval \((0, 1)\). The notation \([0, 1]\) also has some uses other than the one described above.

    We won't need to work with intervals much in this class, so I will just avoid the notations \([0, 1]\) and \((0, 1)\). I only mention these here for your reference: in case you come across a discussion of intervals you should be prepared to know what's being discussed.

  5. Another notion you may come across is the question of whether a set is convex. This is primarily applied to sets of points on a line, or in a plane, or in a 3-space, and so on. Suppose \(x\) and \(z\) belong to the set \(\mathrm{A}\). We could then ask whether the point directly halfway between them is also in \(\mathrm{A}\), and whether the point \(\frac13\) of the way from \(x\) to \(z\) is in \(\mathrm{A}\), and so on. In general, if \(\Upsilon\) is the open interval between \(0\) and \(1\), we can ask whether \(\forall y\in\Upsilon (\)the point \(y\) of the way between \(x\) and \(z\) is in \(\mathrm{A})\). If so, then the set \(\mathrm{A}\) counts as convex. See Wikipedia for some diagrams.

    This notion has some importance in some discussions of probability, but it won't play a role in what we're going on to study. Again, I mention it only for your reference. If you come across it, hopefully you will recognize this as a term you've been introduced to before and will either remember what it means, or you'll know to look it up.

  6. We will discuss issues about the size or cardinality of sets later.

  7. You may encounter talk of the rank of a set. The way this talk is used is this. At rank 0, we have all the sets whose members are only urelements. This includes at least the empty set. Potentially it could also include the set containing you and me, but standardly the rank-talk is used in theoretical settings where we assume everything being talked about is set-like --- which you and I are not. So in that setting, there is only one set of rank 0, the empty set. A set belongs to rank \(k+1\) when all of its members are urelements or sets of rank \({} \le k\), and at least some of its members are from rank \(k\).

    You may encounter talk of the cumulative hierarchy of sets. This is another way of talking about rank --- but the sets of rank \(k\) belong to level \(k+1\) of the cumulative hierarchy. So urelements belong to level 0, the empty set \(\varnothing \) has rank 0 and belongs to level 1, the set \(\{\, \varnothing \,\}\) has rank 1 and belongs to level 2, and so on.

    You may encounter talk of classes. One can begin by thinking of these as collections like sets --- and indeed every set is a class. However there are also thought to be some classes that are too complex to be sets. If there is some rank such that all the members of the class are \(\le\) that rank, then the class is allowed to be a set. If not, it's not a set but a proper class. A distinguishing property of these is that sets can be members of other classes (sets or proper classes). However, proper classes can't be members of anything.

    The reason for making this distinction is that it enables us to avoid some paradoxes, like Russell's set of all sets that aren't members of themselves. In today's mainstream set theory, every set fails to be a member of itself --- but there is no set that contains them. (There is a class that contains them, but it's not a set. It's a proper class.) Neither is there any set that contains all the singleton sets (of any rank). In both cases, this is because there is no bound on the maximum rank of the things we're talking about.

    You might wonder: well, if there can be classes of such things but not sets, why don't we just repeat Russell's paradox at the level of classes? That is, what about the class of all classes that aren't members of themselves? Well, among the classes that aren't members of themselves are some proper classes, and these can't be the members of anything. So there can be no class whose members are all classes that aren't members of themselves, any more than there can be a set of all sets that aren't members of themselves.

    It may be helpful to think of all the class-talk as really just a covert way of talking about predicates. When we say that some classes are sets, we mean that some predicates define a set. When we say that other classes aren't sets, we mean that those predicates don't. In the latter cases, the classes can't be members of anything because really there is no object that contains exactly the objects satisfying the predicate. Some of our talk about classes makes it look like they are a special kind of object, that somehow magically resists being a member of anything. But perhaps its best to think of that talk as superficial and misleading.

    If you study set theory, you will learn more about this. We are not going to discuss it further here. I mention the notions of rank and class just to give you some orientation, so that if you encounter talk about these, you know where it should go in your mental map.