How can I express an infinitely nested unit set?

Question

A unit set S can be expressed by $\{e\}$ where $e$ is a given element. Similarly, a nested unit set can be expressed by $\{\{e\}\}$. Would the following notation be promptly understood as representing an infinitely nested unit set: $\{\cdots\{\{e\}\}\cdots\}$?

Answer 1

Yes I think that is reasonably clear, although I think it's always a good practice to clarify notation which has the potential for being confused. So for example you could use that notation and add a line with something like "... denoting the infinite nested singleton set" or something like this. As Brian Moehring noted however, the axiom of regularity (https://en.wikipedia.org/wiki/Axiom_of_regularity) which is one of the ZF(C) axioms dictates that not set is an element of itself, which your infinitely nested singleton set is. That doesn't mean what you're doing is wrong, just that you can not construct such an object in ZF(C).