The Löwenheim–Skolem–Mal'tsev Theorem for $\mathbb{HF}$-Structures

We deal with the problem asking whether hereditarily finite superstructures have elementary extensions of the form $\mathbb{HF}(\mathfrak M)$. In so doing, we settle the question whether a theory for some hereditarily finite superstructure have $\mathbb{HF}(\mathfrak M)$ models of arbitrarily large cardinality. A Hanf number is shown to exist, and we provide an exact bound for the countable case.