In question No.134351 it is asserted that there exists a Paraconsistent Set Theory in which the Continuum Hypothesis (CH) is disproved. I would like to know more about this theory, which I will call PST. Is it true that PST is formalizable in the language of first order ZF (even though its rules of inference and some of its logical axioms are different from those of ZF)? Is it also true that "not CH" is a theorem of PST but that "CH" is not a theory of PST?....If the answer to both of these questions is "yes" (and if PST has no special "ad hoc" axioms designed to disprove CH), then this seems to me to be a very noteworthy result which I had never heard about until now.
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asked Jun 21, 2013 at 19:24
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1$\begingroup$ As an aside, I believe the paper in which PST is shown to have these properties is "Transfinite cardinals in paraconsistent set theory" by Zach Weber (journals.cambridge.org/action/…), in the 2012 Review of Symbolic Logic. $\endgroup$– Noah SchweberCommented Jun 21, 2013 at 20:15
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$\begingroup$ Thanks alot for this information. I will try to get hold of that article and hope I will be able to understand it. $\endgroup$– Garabed GulbenkianCommented Jun 22, 2013 at 20:20
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1$\begingroup$ I am currently reading Weber's paper and the gist of his proof that not-CH holds is that |omega|<|P(omega)|<|P(omega)| holds in his system of paraconsistent set theory. $\endgroup$– Thomas BenjaminCommented Jul 12, 2013 at 8:46
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1$\begingroup$ I was interested in this question too and found a PDF of the article on Weber’s website: sites.google.com/site/doctorzachweber/Publications. $\endgroup$– user76284Commented Jun 9, 2019 at 6:20
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