One definition of a finite set is that it can be injected into an initial segment of $ \Bbb N$, thus any $n$ in $\Bbb N$ is finite.
Accordingly, if it's legitmate to define every element in $^* \Bbb N$ as its own intial segment, then every element $^* \Bbb N$ is finite.
What about Dedekind-infinity? Can we define an Dedekind-infinite element in $^* \Bbb N$ by constructing an injection into its proper subset?
Added: I'm not sure whether the construction of $^* \Bbb N$ is compatible with $\bf ZFC$ in the same way as $\Bbb N$ is, which is probably not, I guess. But how?