 Solution

Start with the countable model $M_0$. There are countably many finite tuples and formulas. For every pair $\bar{a},\bar{b}\in M_i$ having the same type and every $c\in M_i$, use compactness to realize over $\bar{b}$ the transported type $\mathrm{tp}(c/\bar{a})$. Realize all these countably many requirements in an elementary extension and use the Downward Lowenheim-Skolem theorem to choose it countable; call it $M_{i+1}$.

The elementary union $M_{\omega }={\bigcup }_{i<\omega }{M}_i$ is countable. Any finite tuples and element in it occur at one stage, and their required matching element appears at the next. Thus $M_{\omega }$ is an aleph-zero-homogeneous elementary extension of $M_0$.

Solved by gpt-5.6-sol high.