Bounded quantifiers in set theory are represented as $(\forall x \in S)$ or $(\exists x \in S)$. But since the modifier "bounded" brings up an association with "bounded variable", I prefer the term "restricted quantifier" used in Kit Fine. Relatively unrestricted quantification where "four broad grounds upon which the intelligibility of quantification over absolutely everything has been questioned".
My view upon the axiomatization of a theory of multiverses is that a multiverse is a multitude of universes (like Grothendieck universes or the von Neumann universe), and restricted quantifiers are needed to quantify the variables running within each universe, while the unrestricted quantifiers are needed to quantify the variables running within this multitude. But the last ones can be presented as quantifiers restricted by the "multitude" which can be treated as a universe (of universes). Thus, to my mind, this area of research must strongly employ restricted quantifiers.
This is a wide area of research, but my questions refer to a more restricted area:
- Are there results on axiomatizing with restricted quantifiers only, in concrete set theories - ZF, NGB, NF (New Foundations), NFU (New Foundations with urelements) or other?
- Is there any correlation between decidability of a fragment of set theory and restricted quantification?
The question 2 is motivated by the link
which refers to a link about a result which sounds to me strange:
but this last link does not work for me, so that I can check what the author meant.