1. The problem asks for the cardinalities (sizes) of the following sets involving the empty set $\varphi$: 2. Recall that the empty set $\varphi$ has no elements, so $|\varphi| = 0$. 3. For the set $\{\varphi\}$, it contains exactly one element, which is the empty set itself, so $|\{\varphi\}| = 1$. 4. For the set $\{\varphi, \{\varphi\}\}$, it contains two distinct elements: the empty set $\varphi$ and the set containing the empty set $\{\varphi\}$. Thus, $|\{\varphi, \{\varphi\}\}| = 2$. 5. For the set $\{\varphi, \{\varphi\}, \{\varphi, \{\varphi\}\}\}$, it contains three distinct elements: the empty set $\varphi$, the set $\{\varphi\}$, and the set $\{\varphi, \{\varphi\}\}$. So, $|\{\varphi, \{\varphi\}, \{\varphi, \{\varphi\}\}\}| = 3$. Final answers: $$|\varphi| = 0$$ $$|\{\varphi\}| = 1$$ $$|\{\varphi, \{\varphi\}\}| = 2$$ $$|\{\varphi, \{\varphi\}, \{\varphi, \{\varphi\}\}\}| = 3$$