The structure of the ordinals and the interpretation of ZF in double extension set theory

Andrzej Kisielewicz has proposed three systems of "double extension set theory" of which we have shown two to be inconsistent in an earlier paper. Kisielewicz presented an argument that the remaining system interprets ZF, which is defective: it actually shows that the surviving possibly consistent system of double extension set theory interprets ZF with Separation and Comprehension restricted to Δ₀ formulas. We show that this system does interpret ZF, using an analysis of the structure of the ordinals.