Can a set containing non-standard elements only, have a legal set formation? Is it possible to form a set that contains only non-standard elements legally? A legal set formation means that only internal formulas can be used to form the set. Here's what I thought: A set containing non-standard elements must be a non-standard set. Otherwise, by transfer principle, it has to be empty. Thus it can't be uniquely described by internal mathematics. It can't take the form as $\\{x : A(x)\\}$.

A set containing a single non-standard element is perfectly legal.

If the "standard" predicate cannot be used, then translated into NSA you can only define sets that are natural extensions of real sets $X$. The natural extension of the empty set $X=\emptyset$ is still empty. If $X$ is inhabited, then its natural extension, which contains all the elements of $X$, will necessarily contain standard elements.