Meanwhile, of course, many set theorists, including almost every large cardinal set theorist (but not quite all, like Silver), believe not only that ZFC is consistent, but that consistency rides much higher in the large cardinal hierarchy. Thus, Silver's world view will be as consistent as his opponent, and there will never be an argument against the view that ZFC is inconsistent that does not beg the question by assuming that this theory is consistent or something stronger.
We cannot show that Silver's ideas about inconsistency are incoherent.īecause of the incompleteness theorem, we know that even if ZFC is consistent, then it is consistent with ZFC to hold that they are inconsistent. What an amazing and valuable contribution to the subject he had made that way, even though he had never achieved his initial goal of refuting measurable cardinals.Īnd although I have heard many people speak against Silver's inconsistency research program, my view has always been: His attempt at refuting measurable cardinals led directly to the theory of ordinal indiscernibles over $L$ and $0^\sharp$, which are now a core part of our understanding of the nature of the constructible universe in the presence of large cardinals. When I was a graduate student Jack Silver was famous for trying to refute first, the existence of measurable cardinals, and second, the consistency of ZFC.
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