In this paper, we take up the outstanding problem of axiomatically characterizing what we have referred to in the paper as the additive choice function on the classical domain for choice problems. Apart from an impossibility result for the additive choice function, there is an axiomatic characterization, which as a by-product provides a counter example to a conjecture for the egalitarian choice function. In an appendix, we provide a proof of an axiomatic characterization of the egalitarian choice function using a superadditivity axiom. Further we show several non-rationalizability properties of utilitarian consistent solutions. In this paper, we also provide proofs of axiomatic characterizations of the family of non-symmetric Nash choice functions and the family of weighted hierarchies of choice functions. Our conclusion is that earlier axiomatizations are essentially preserved on the classical domain for choice problems. The proofs are significant in being non-trivial and very dissimilar to existing proofs on other domains.