Each Feferman–Levy real layer is countable

The ordinal ${\mathrm{\aleph }}_{m+1}^V$ is nonempty. By F2 and F3, fix in $N$ one surjection $f:\omega \twoheadrightarrow {\mathrm{\aleph }}_{m+1}^V$, and form $g_m=e_m\circ f$. Both factors are sets of $N$, and ordinary ordered-pair Separation produces their composition. For each $x\in R_m$, its $e_m$-preimage is nonempty, so take its least ordinal member $\xi$; then the $f$-preimage of $\xi$ is a nonempty set of naturals and has a least member $k$. Thus $g_m(k)=x$, so $g_m:\omega \twoheadrightarrow R_m$. This fixes one witness for one already fixed $m$; it does not choose a family indexed by $\omega$.

The layer is nonempty because it contains the interpretation of the constantly-zero Boolean name. Sending each $x\in R_m$ to its least $g_m$-preimage gives an injection into $\omega$, so $R_m$ is at most countable. The least-preimage clauses are definable and involve one fixed map; no choice function for the family $(R_m{)}_{m<\omega }$ is formed.