tag:blogger.com,1999:blog-7068528325708136131.post6657181336778652062..comments2017-02-22T19:49:07.412-08:00Comments on Structural insight: Computation and truthJohn Shutthttp://www.blogger.com/profile/00041398073010099077noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-7068528325708136131.post-20477339774971290502015-05-15T14:16:23.749-07:002015-05-15T14:16:23.749-07:00This description of Godel's incompleteness the...This description of Godel's incompleteness theorem is similar to other traditional accounts in that it misses an important fact: according to Godel's *completeness* theorem (a result much more fundamental than the celebrated incompleteness one) a sentence is provable iff it is valid in all models. This means that the "Godel's sentence", which is unprovable in PA, is necessary invalid in some versions of natural numbers.<br /><br />Of course, up to this day, there is no shortage of people who believe that there is only one true set of natural numbers (sanctioned by the Pope and Congress, presumably) and that their Turing machines compute functions on this sacred object.furia-kruchahttp://furia-krucha.livejournal.com/noreply@blogger.com