What is a starfinite set?

Question

In the book Field Arithmetic by Fried and Jarden, the following definition is given on p. 273:

Consider an enlargement of a higher order structure that contains both $P$ and $K$. Call the elements of $\mathcal{P}^*_{\mathrm{fin}}(P)$ the starfinite subsets of $P^*$.

What is a starfinite set? I can't understand the definition.

Answer 1

A set $A \subseteq \mathbb{N}$ is finite iff it is contained in $\{0, \ldots, n\}$ for some $n \in \mathbb{N}$.

Applying transfer, an internal set $A \subseteq {}^*\mathbb{N}$ is $*$-finite iff it is contained in $\{0, \ldots, \xi\}$ for some $\xi \in {}^* \mathbb{N}$.

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Edit: equivalently, a $*$-finite set is an element of ${}^* \mathcal{P}_\mathrm{fin}(\mathbb{N})$, where $\mathcal{P}_\mathrm{fin}(\mathbb{N})$ is the collection of finite subsets of $\mathbb{N}$.