Frege's definition of the empty set was not a raw extensional one: he did not simply write the partial string {} and say, "That's it: that's the empty set." His account was more intensional: "There is a set that contains all and only true contradictions," but since per his LNC-compliant logic he would say that there are no true contradictions, it follows that the set in question is empty (has no elements that we actually quantify over, here).
Frege's definition has been generalized (or at least adapted) to other qualifications: one that I saw was, "The empty set is the set of all ill-founded sets," but since modulo ZFC there is a "fact" that V = WF (for WF = the class of well-founded sets), such a set would be empty. So another could be, "There is a set of all disordered sets," or, "There is a set of all sets that don't have powersets," etc. Inside a world where the axioms say that all sets have property f, that world's empty set will be the set of all sets that don't have f.
But so from outside of any world as such, things might appear quite different. In the literature, they usually frame transworld witness so that the emphasis is on how elements of V can vary from world to world: some versions of V have more or different elements than other versions, V can be extended by forcing, etc. and it is variation of content on this relatively maximal level that is the concern of theorists. However, per the Fregean option, we can also look at how each world's empty set (as opposed to its universal set/proper class) has more or different elements between worlds. If V = L, for example, then 0♯ "doesn't exist," neither do zero dagger, zero pistol, or any zero section that might be proposed. So in the constructible universe, the empty set = zero = the set of all the zero sections, which is empty only in constructible or sufficiently similar worlds but is teeming with elements from the perspective of other worlds.
The question: but suppose we were trying to be inclusive of finitism or even ultrafinitism, or at least respectful of their degree of caution with respect to infinitary reasoning, and so we wanted to introduce infinitesimals in such a way that they did not have to be reciprocals of actually-infinite numbers. Bassler[16] uses the perspicuous term parafinite to navigate these waters, so let us speak of parafinitesimals instead. Now, the relative elements of a Frege-flavored empty set could often by styled as parts/aspects/etc. of emptiness or the number zero, which despite its elementarity is fraught with nontrivial effects (e.g. the indeterminacy and undefinability of divison by zero, or the divergence between the Conway argument for 00 = 1 and the quasi-model where the expression is undefined). Would describing these "parts"/"aspects" as parafinitesimally small, as in parafinitesimally close to zero without necessarily being equal/identical to it, be a way to introduce "logical infinitesimals" without properly offending finitistic sensibilities? After all, even by the strict Fregean introduction, we have that infinitesimals as examples of "actual infinity" are then nonexistent elements of an empty set in a finitistic world where the closest counterpart to "actual infinity" is the class of all sets (which following the resolutions to Russell's paradox, is usually not taken for a set, even by non-finitist set theorists). Without claiming that there are infinitely many axioms/models buttressing the appeal to the multiversal standpoint, can a finitist allow for a finite number of universes (perhaps even some where actual infinity is countenanced), such that as transworld witnesses they can see into parafinitesimal or even "properly" infinitesimal elements of an empty set?