Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? A professor said it to me a long time ago, but I don't have any references. Thanks (and sorry for my english).
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3$\begingroup$ Maybe a dim recollection of the second incompleteness theorem? $\endgroup$– Nik WeaverMay 30, 2016 at 15:17
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$\begingroup$ I don't think a proof that inconsistency of ZFC is unachievable could exist: that would be a proof of consistency, contradicting Goedel, right? $\endgroup$– QfwfqMay 30, 2016 at 19:25
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$\begingroup$ @Qfwfq: the comment is that (informally speaking) if ZFC is consistent then it cannot prove its own consistency. In that sense a proof of consistency could be "unachievable". $\endgroup$– Nik WeaverMay 30, 2016 at 20:16
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There are six mentions of incompleteness and eight of sets on Professor Feferman’s publications page, https://math.stanford.edu/~feferman/papers.html, but I’m not sure that any of them are what you’re asking for. Could you instead be thinking of a result by a different Stanford logician, that the “C” in ZFC is independent of ZF: https://en.wikipedia.org/wiki/Paul_Cohen#Contributions_to_mathematics?
