Already after sending the first two parts of this paper ([5], [6]) to the editor, two new results on the subject have appeared — namely the results of G. Wilmers and Z. Ratajczyk. So for the sake of completeness let us review them here.

We consider expansions of models of Peano arithmetic to models ofA 2 s -¦Δ 1 1 +Σ 1 1 −AC which consist of families of sets definable by nonstandard formulas.

We study cofinal extensions of models of arithmetic, in particular we show that some properties near to expandability are preserved under cofinal extensions.