In set theory (say, ZF) the 'class notation' is introduced. In general one could say that a class is something of the form $\{ x \, | \, \varphi ( x ) \}$ where $\varphi ( x)$ is some formula in the language of set theory, having only $x$ as free variable.

I know that these (i.e. classes) are just a matter of handy notation, and that in principle they can always 'deleted', so they're not really part of the language.

Now I've heard that the big difference between sets on the one hand, and classes on the other hand, is that sets can be element of other sets while classes cannot be element of a set (and also, classes cannot be element of another class). I thought I understood this, however I have the following concern. I will give a concrete example below.

Let $A=\{ x \, | \, \varphi ( x ) \}$ be a class, and let $B$ be the class $\{ x \, | \, \emptyset \in x, \mbox{ and if } y \in x \mbox{ then } \{ y \} \in x \}$. Now, the expression "$A \in B$" does make sense, in this particular case, I think. Namely it would be simply an abbreviation of the sentence $\emptyset \in A \wedge \forall y \, ( y \in A \rightarrow \{ y \} \in A )$. And this means that $\varphi ( \emptyset ) \wedge \forall y \, ( \varphi ( y ) \rightarrow \varphi ( \{ y \} ) )$. (Note that in this example the 'a priori' class $B$ in fact happens to be a set itself, assuming some axioms.)

So my point is: it seems to be the case that in some cases it would make sense. (Of course, the reason why it makes sense in this particular example is because in the sentence "$\emptyset \in x, \mbox{ and if } y \in x \mbox{ then } \{ y \} \in x$" defining the class $B$, there are only terms of the form $\ldots \in x$, and not the other way around; if for example there would have appeared $x \in \emptyset$ we would finally get $A \in \emptyset$ which doesn't make any sense.)

Thus, I think that in general it does not make sense to write things like "$A \in B$" where $A$ and $B$ are classes.

But I'm a bit in doubt about all of this, as it seems pretty confusing. Hence,


1) What I have described above, does that make sense? Is my 'analysis' correct? Or is what I've said essentially false.

2) In cases when $A \in B$ (with $A,B$ classes) does make sense, can we just write it like this? Or maybe write it along with a comment that in this particular case it makes sense, as it is just an abbreviation of this and that formula?


When we write $A\in B$, we invariably require that $A$ is an object of the universe, which means that we require it to be a set. But remember that sets are themselves classes.

Your example can be seen as a "complication" of the following example: $$\{x\mid x=x\}\in\{\{x\mid x=x\}\}$$

But this statement means that $B$ is not even a class. It is a $2$-class, or a conglomerate. If we want $B$ to be a class them its elements must be sets.

So while it may make some sense to apply $A\in B$ for proper classes sometimes, we don't want to have a convention which is too confusing. So classes which are elements of other classes have to be sets.

  • $\begingroup$ Well I don't really see why my example is 'similar' to yours. You perform a set-theoretic operation on a class. I.e. your right-hand side (namely $\{ \{ x \, | \, x = x \} \}$) doesn't even make sense. In my example both $A$ and $B$ make perfectly sense. But I guess I understand your main point. You say "classes which are elements of other classes have to be sets". If I understand correctly, this is a (notational) convention right? $\endgroup$ – Elisheva Mar 28 '13 at 13:46

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