I've been trying to do some forcing arguments in intuitionistic ZF using Heyting valued models where the Heyting algebra I'm using is actually a bi-Heyting algebra (both a Heyting algebra and a co-Heyting algebra) and I've found that some interesting questions arise if I use the co-Heyting negations as well as/instead of the Heyting negation. I'm just wondering if anyone knows if any work has been done in constructing some kind of co-Heyting valued models of any kind of set theory? I guess such a theory would have to be paraconsistent. If this hasn't been done, could anybody give me some info/references about the current situation with respect to paraconsistent set theory and its model theory. Is there a dominant axiomatization?

  • $\begingroup$ Cohen's original forcing wasn't closed under double negation, if I recall correctly (this was later fixed by defining "weak forcing"). I wonder if the original forcing relation is really what is in play when forcing in $\sf IZF$. $\endgroup$ – Asaf Karagila Jun 2 '14 at 16:37

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