Is there a distinction that we make between what we call "elements" and what we call "sets" in Von Neumann type set theories? Two textbooks I have used for an introductory study of NBG both made distinctions between sets and proper classes (the prinipal primitive undefined notion being that of "class"). The distinction was ofcourse made to avoid various logical paradoxes, in particular, Russell's Paradox. However, what wasn't clearly stated in either textbook is whether the concepts of "set" and "element" are identical. Both did state that a "class" is considered an element if and only if it belongs to another class. Would it be safe for me to identify the notions of "set" and "element" completely interchangeably? I don't see an elementary distinction between the two. Then the only two distinctions we should make are of "sets" (elements) and of "proper classes" with the general notion of "classes" referring to either one of the two distinctions. Thank you in advance.
1 Answer
Class theories, to distinguish them from set theories (and also to confuse social philosophers and let's admit it, too many things are named after von Neumann anyway), are really how we used to think about sets back "in the old days".
In other words, naively, from a Fregean point of view, any collection of mathematical objects would be a set. This was observed to be difficult very quickly, with the most famous nail in the coffin provided by Russell in his eponymous paradox, showing that not all collections of mathematical objects form sets.
Elements are simply the objects which reside inside a set, so in this case, and elementhood is something relative to a set. Namely, $0$ is an element of $\{0\}$, but it is not an element of $\{1\}$. So, being an element is like being a neighbour. It is relative to additional information. I am a neighbour of someone if we live nearby. Now, you could argue that you're a neighbour if there's someone to whom you're a neighbour, and therefore you're an element if there is some set to which you're an element. That is a valid approach, and we'll get back to it later.
So, after all the paradoxes, we refined the notion of a set by positing the axioms of Zermelo and later added Replacement and other functions (in part this due to von Neumann) to obtain $\sf ZF$ or $\sf ZFC$ as our de facto theories of sets.
In these theories, the only objects are sets and we are able to interpret any other mathematical structure as a set inside them. This is what it means to be a foundations of mathematics. So in this case, all the objects are sets, and all the objects are elements of larger sets (e.g., singletons), so being an element is the same as being a set is the same as being a mathematical object.
But, as we knew from the beginning, what about those pesky collections that are not sets? What about the collection of all sets? Of all sets which do not contain themselves? Of all the ordinals? These are collections, now called to as "proper classes", and we might want to refer to them and use them, mathematically speaking, but since they are not objects per se, this is difficult. The first-order logic solution is to accept that a collection is merely a formula which sometimes is interpreted as a set (e.g., $x\neq x$ gives rise to the empty set) and sometimes doesn't. But even if it doesn't, we can still use it in proofs and think of it as a collection.
This "hack", while completely reasonable, workable, and to an extent, not at all a hack, is still somewhat dissatisfying. So we turn to the idea of a class theory, where the objects are classes, proper or otherwise. In order for it to be a compatible foundation, or something which extends $\sf ZFC$, we first need to understand how does that extend the notion of a set. In other words, how do class theories interpret set theory?
It turned out to be very very simple, pretty much by design since we chose class theories to extend the idea that a class is just a collection of sets defined by a property. Much like how some classes do define a set (recall $x\neq x$ up there), it turned out that the classes that define sets are exactly those which are elements of some other class, most notable, singletons, but why restrict to that? So, now we can say that a set is simply a class that happens to be an element of another class. So the class $\varnothing$ is a set, since it is an element of $\{\varnothing\}$, whereas the class $V=\{x\mid x\notin x\}$ is not a set.
So, going by our "existential interpretation of relative definitions", we can think of an element as something which is an element of some class, which then means that it coincides with the notion of a set, in the case of these class theories.
Epilogue. There are other ways to defeat the Russell paradox which give interesting rise to very different notions of sets, for example Quine's New Foundation or Positive Set Theory; but in all of these we still have definable collections which are not sets. On the other hand, if we look at the family of von Neumann–Gödel–Bernays style class theories, any collection of sets is still a class.
