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?
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1The answer is $2$, because the two elements $a$ and ${a,{a}}$ are different, else we would have $a\in a$. (This is excluded by set axioms.) – dan_fulea Oct 22 '18 at 18:02
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Sam, where did you get the question? Do you need to prove that $a\neq {a,{a}}$? – Ennar Oct 22 '18 at 18:04
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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.
Ennar
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The relation $a\ne b$ needs explanation, this is maybe the only point in the exercise. – dan_fulea Oct 22 '18 at 18:04
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Thanks for adding the reference to the axiom; presumably OP is using ZF or ZFC (although it's possible that OP doesn't know that). – John Hughes Oct 22 '18 at 19:23