Is $\{\{\emptyset\}\}$ actually a set? I think this question is not asked here. I apologize in advance if I am wrong.
I have the following two definitions (Joaquín Olivert, Estructuras de álgebra multilineal, 1996):
Class.- A class is an abstract object $C$ which permits to decide if their elements belong to it or not.
Set.- A class $C$ is said to be a set if there exists a class $D$ such that $C\in D$. A class which is not a set is called a proper class.
With this in mind, I'm trying to understand if $A=\{\{\emptyset\}\}$ is a set or not and why. It seems to me the answer should be not but with these definitions, I think $\{ A \}$ is a class and then $A$ is a set. On the other hand, with this reasoning, every class would be a set and that is false (e.g. Russell's set).
Any help please?
Thanks
 A: In ZF(C) set theory, the Axiom of Pairing states that if $x$ and $y$ are sets, then $\{x,y\}$ is a set too.  Applying this with $x=y$ gives that whenever $x$ is a set, $\{x\}$ is a set too (because it has the same elements as $\{x,x\}$ and is therefore equal to it).
Thus,


*

*$\varnothing$ is a set because of the Axiom of the Empty Set. (Sometimes this is not considered an axiom, in which case $\varnothing$ is still a set because the Axiom of Infinity states that there exists some set, and then the Axiom of Separation can be used to remove all of its elements).

*$\{\varnothing\}$ is a set because of Pairing applied to $\varnothing$.

*$\{\{\varnothing\}\}$ is a set because of Pairing applied to $\{\varnothing\}$.


Your definition of "set" sounds like it doesn't belong to pure ZF(C), but more likely to NBG or MK set theory. However, these theories have similar axioms of Empty Set and Pairing, so the argument (at least at the non-formal level I'm writing at) is the same.
A: $A=\{\{\varnothing\}\}$ is a set, not a proper class, because we have $A=\{\{\varnothing\}\}\in\{A\}=\{\{\{\varnothing\}\}\}.$ And $\{A\}$ is not a class either, because we have $\{A\}\in\{\{A\}\}.$
Axiomatically, one usually assumes that $\varnothing$ is a set, and for any set, the singleton containing that set is a sett (for example via the asxiom of pairing), so $\{\varnothing\},$ $\{\{\varnothing\}\}$, etc are all sets, and not proper classes.
In the usual universes of axiomatic set theory, for something to be a proper class, and not a set, it must be very big. For example the class of all sets, the class of all cardinals, or the class of all ordinals. These classes are not contained in any other classes.
