1. Note that one could also choose to leave out foundation (and set-induction) altogether. One may then wish to add an axiom, called antifoundation, which allows for the formation of non-well-founded sets, also called hypersets (see Aczel 1988 or Barwise and Moss 1996). In fact, a constructive treatment of non-well-founded sets can be given which mirrors the classical one (see for example Rathjen 2004). Further, by using a construction by Hällnas and Lindström of non-well-founded sets as limits of finite approximations (Lindström 1989), one can see that in constructive contexts the antifoundation axiom comes as proof-theoretically “inexpensive” if compared with set induction (Rathjen 2003).
2. Inaccessible sets in CZF corresponds in ZFC to Vk for k a strongly inaccessible cardinal, where Vk denotes the k-th level of the von Neumann hierarchy (Crosilla and Rathjen 2001).