The iterative conception of set. A (bi-)modal axiomatisation (Q380997)

The author offers a modal set theory. The modal logic language refers to the rank hierarchy structure of the universe of all sets as its accessibility relation, and has two necessity operators similar to the temporal modalities G and H, i.e.\ one ``forward-looking'' and one ``backward-looking''. The extension of this setting to a modal stage theory \(\mathsf{MST}\) comes via axioms of extensionality, priority (all elements of a set \(A\) are formed prior to \(A\)), and plenitude (a kind of unrestricted comprehension at any stage). This setting allows for a natural interpretation of Zermelo set theory \(\mathsf{Z}^-\) without infinity. And an extension with an axiom concerning the extent of the modal frame even allows the author to interpret \(\mathsf{ZF}\). Reviewer's remark: The author's discussion of the axiomatization of the rank relation in his modal context seems to indicate that he is unaware of the 1957 axiomatization of this rank relation by \textit{D. Klaua} [Z. Math. Logik Grundlagen Math. 3, 303--316 (1957; Zbl 0080.03701)].