Abstract
Three systems of “double extension set theory” have been proposed by Andrzej Kisielewicz in two papers. In this paper, it is shown that the two stronger systems are inconsistent, and that the third, weakest system does not admit extensionality for general sets or the use of general sets as parameters in its comprehension scheme. The parameter-free version of the comprehension principle of double extension set theory is also shown to be inconsistent with extensionality. The definitions of the systems and a self-contained exposition of their properties is given, sufficient to develop the inconsistency proofs.
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References
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.
Quine, W. V. O, ‘On ordered pairs’, Journal of Symbolic Logic 10:95-96, 1945.
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Holmes, M.R. Paradoxes in Double Extension Set Theories. Studia Logica 77, 41–57 (2004). https://doi.org/10.1023/B:STUD.0000034184.03045.7d
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DOI: https://doi.org/10.1023/B:STUD.0000034184.03045.7d