I've just started learning basic set theory and am puzzled by this question I came up with:
What is the cardinality of {1, {2,3}}?
Do I treat sets within sets as just one element and so the answer is 2? Is there a formal definition of cardinality that I can apply here?
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$\begingroup$For finite sets, cardinality is just the number of elements in the set.
In your case, you have two elements in your set: the element $1$, and the element $\{2,3\}$ (which happens to be a set itself). So you are right, the cardinality is $2$.
$\endgroup$ $\begingroup$That set has two elements. One of the elements is $1$ and the other is $\{2,3\}$.
Think in a set like it were a box. This box has a ball in which is written "$1$" and a smaller box. No matters what is inside the smaller box, the bigger box has two things: the ball and the box.
Remark: Actually, every element of a set is a also set. Although, we "forget" this fact when is not needed. For example, $0=\{\}$ and $1=\{\{\}\}$. A real number is an equivalence class of the set of Cauchy's sequences of rational numbers.
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