Axiomatic set theory: hierarchy, cardinal arithmetic and L

How many elements does $V_5$ have? (Recall $V_0 = \varnothing$ and $V_{n+1} = \mathcal{P}(V_n)$.)

Assuming ZFC, which of these cardinals are singular? Choose all that apply.

Which of these values of $2^{\aleph_0}$ does ZFC rule out? Choose all that apply.

In the proof that every set lies in some $V_\alpha$, we form $\alpha = \sup\{\alpha_z + 1 : z \in y\}$. Which axiom guarantees that this supremum exists?

What exactly did Gödel's work on $L$ establish?

Which axioms hold in $(V_{\omega+\omega}, \in)$? Choose all that apply.

Which Lean statement correctly formalises the generalised continuum hypothesis?

The Lean proof of two_power_aleph0_ne_aleph_omega rewrites the inequality $\aleph_0 \lt \operatorname{cf}(2^{\aleph_0})$ using ord_aleph, cof_omega isSuccLimit_omega0 and cof_omega0. Which mathematical fact does this chain of rewrites establish?

In Ackermann's coding, the natural number $x$ codes the set whose members are the numbers $y$ with bit $y$ of $x$ equal to $1$. Write rank(x), computing $\operatorname{rank}(x) = \sup\{\operatorname{rank}(y) + 1 : y \in x\}$ (and $0$ for $x = 0$).