Every finite ground aleph is countable in the Feferman–Levy model

Define $f_n(i)=\alpha$ exactly when some $p\in G$ contains the triple $(n,i,\alpha )$. Functionality follows because two conditions in the filter are compatible and conditions are functional at $(n,i)$. For each $i<\omega$, conditions assigning a value at $(n,i)$ are dense; for each $\alpha <{\mathrm{\aleph }}_n^V$, conditions putting $\alpha$ at some fresh $(n,i)$ are dense. Therefore genericity, equivalently F2 and F3, makes $f_n:\omega \twoheadrightarrow {\mathrm{\aleph }}_n^V$.

The canonical name for $f_n$ uses only Boolean values from layer $n$. Every member of $H_{n+1}$ fixes all layers below $n+1$, hence fixes this name and its canonical ordinal subnames by F1. It is hereditarily symmetric, so $f_n\in N$.

The ordinal ${\mathrm{\aleph }}_n^V$ is nonempty. In ZF a surjection $f:\omega \twoheadrightarrow A$ onto a nonempty set gives an injection $A\to \omega$ by sending $a$ to the least $i$ with $f(i)=a$; hence $A$ is at most countable. Applying this inside $N$ to $f_n$ proves the claim. The construction is for one specified $n$ and does not assert that the sequence $⟨f_n:n<\omega ⟩$ belongs to $N$.