In "Compositionality in Montague Grammar" (In "The Oxford Handbook of Compositionality"), Markus Kracht writes that "In a standard model (where we allow quantifying over all subsets) there is a biunique correspondence between the individuals of the domain and the set of all subsets of the domain containing that individual (such sets are also called principal ultrafilters)" (p.58).
Why is the qualification "where we allow quantifying over all subsets" necessary here? Is this to imply that correspondence he alludes holds only in a full model of second order logic?
If the bijection only holds in a full second order model why would that be the case? Why would it fail in first order logic?
What is the proof of the biunique correspondence the Kracht mentions (he provides no proof himself)?