What is the cardinality of the set {a, {a, {a}}}?

Question

I think that the answer is 2, but I'm not 100% sure. If the answer isn't 2, could someone help lead me to the correct answer?

Answer 1

Let $b = \{a,\{a\}\}$. Then your set is $\{a,b\}$ which has $2$ elements, since $a\neq b$. The thing that $b$ is a set with two elements itself doesn't change the number of elements of $\{a,b\}$.

To prove $a\neq b$, assume the contrary. Then, $a\in a$, which is in contradiction with axiom of regularity.