Abstract
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.
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References
ACKERMANN, W., ‘Zur axiomatik der Mengenlehre’, Mathematische Annalen 131: 336–345, 1956.
HOLMES, M. RANDALL, ‘Paradoxes in double extension set theories’, Studia Logica 77 (2004), 41–57.
KISIELEWICZ, ANDRZEJ, ‘Double extension set theory’, Reports on Mathematical Logic 23: 81–89, 1989.
KISIELEWICZ, ANDRZEJ, ‘A very strong set theory?’, Studia Logica 61: 171–178, 1998.
REINHARDT, W. N., ‘Ackermann’s Set Theory equals ZF’, Annals of Mathematical Logic 2: 149–249, 1970.
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Holmes, M.R. The Structure of the Ordinals and the Interpretation of ZF in Double Extension Set Theory. Stud Logica 79, 357–372 (2005). https://doi.org/10.1007/s11225-005-3611-x
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DOI: https://doi.org/10.1007/s11225-005-3611-x