tag:blogger.com,1999:blog-4987609114415205593.post1794725895144520277..comments2021-06-19T18:51:04.369+01:00Comments on M-Phi: Voevodsky: "The consistency of PA is an open problem"Jeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-4987609114415205593.post-62274635650591339492011-10-12T08:12:34.929+01:002011-10-12T08:12:34.929+01:00By the way, I've been feeling sorry that I cam...By the way, I've been feeling sorry that I came across as a bit fierce in my comment above. When I saw the phrase "he seems to display a complete lack of understanding of Goedel's incompleteness results," it made me see red, despite all the qualifiers it was couched in, such as "at least according to some". I got the feeling that "some" - I'm not sure who - were criticizing minutia while ignoring the amazing developments occuring right before their eyes.<br /><br />This of course is common in academia, but also elsewhere. I just watched a great video documentary about Thelonious Monk on YouTube. It's blisteringly clear that he was smarter than all the journalists commenting on him, who seemed to see him mainly as an eccentric who wore funny hats. <br /><br />Your June 6 post makes me much happier. <br /><br />There are huge and fascinating philosophical questions involved in the homotopification of logic, and I hope philosophers get in there and explore them.John Baezhttp://math.ucr.edu/home/baez/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-37872852899671090872011-10-07T11:15:40.302+01:002011-10-07T11:15:40.302+01:00@John,
Indeed, in the end my conclusion was that ...@John,<br /><br />Indeed, in the end my conclusion was that Voevodsky's comment concerning PA was for the most part tangential within the bigger picture of the homotopy project. My third post on the 'affair' elaborates on this point:<br /><br />http://m-phi.blogspot.com/2011/06/latest-news-on-inconsistency-of-pa.htmlCatarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-709860212615931142011-10-07T04:18:19.717+01:002011-10-07T04:18:19.717+01:00I find it bizarre that Voevodksy would be expected...I find it bizarre that Voevodksy would be expected to give a precise formal statement of a well-known result like Goedel's 2nd incompleteness theorem or risk people thinking he "displayed a complete lack of understanding" of the subject, or is "dismissing Gentzen's proof".<br /><br />Voevodsky is a great mathematician, and if you go to talks by great mathematicians you'll find they often run roughshod over well-known details in their pursuit of challenging new ideas. They don't feel the need to prove their ability to parrot standard results. They've got bigger things in mind.<br /><br />Voevodsky is working on integrating ideas from homotopy theory into mathematical logic. This gives rise to a new subject that some people are now starting to call "homotopy type theory". It's part of the switch from set theory towards infinity-category theory, since homotopy types are essentially the same as infinity-groupoids. <br /><br />Anyone who wants to understand where mathematical logic is heading in the 21st century needs to pay attention to this stuff.John Baezhttp://math.ucr.edu/home/baez/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-16258700596512183322011-06-01T08:04:12.460+01:002011-06-01T08:04:12.460+01:00Franklin,
Good question! And I'd add: is PRA+i...Franklin,<br />Good question! And I'd add: is PRA+induction up to epsilon_0 *sound*? (It may be consistent but not sound.) Ultimately, if Voevodsky wants to dismiss Gentzen's proof, he would have to have arguments to question the soundness of PRA+induction up to epsilon_0. From what I gather, mathematicians by and large trust the soundness of the system, but as philosophers we are allowed to ask such questions... (speaking for myself, I don't know whether you are a philosopher).<br /><br />Anonymous, thanks for mentioning the connection with debates in phil mind regarding Godel's theorem. Certainly relevant in this context too.Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-16078618452347952552011-05-30T14:11:41.324+01:002011-05-30T14:11:41.324+01:00Corrections:
"Cognitive Sciences hold comput...Corrections:<br /><br />"Cognitive Sciences hold computabilism which implies that there's not any MORE POWERFUL THAN Turing Computability going on IN THE brain and mind. Turing Computability imposes certain restrictions on what infinities can be legitimately presupposed to hold any mathematical truth ACCEPTABLE EVEN TO INTUITIONISTS."Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-63237977474682814272011-05-30T14:04:58.217+01:002011-05-30T14:04:58.217+01:00Interesting discussion. In an adjacent problem ste...Interesting discussion. In an adjacent problem stepping on the fields of Metaphysics (Philosophy of Mind, specially) known as 'gödelian arguments' computabilists tend to echo Voevodsky's spirit by denying that we can even know PA's consistency. This gives them a shot declaring Gödelian Arguments invalid formally. Most detractors of Gödelian Arguments use Gödel's Incompleteness results as their only evidence to the impossibility of PA's consistency being true. (Hayes, LaForte and Ford, clearly do this, but it is a commonplace even between those who hold at least some version of a Gödelian Argument, but criticize others such as Selmer Bringsjord, H. Putnam, ...)<br /><br />Well, anyway, it is a delicate thing to state that it is a metaphysical fact and maybe a natural fact, if not a physical one, that THERE IS actually a phenomenon (the cognitive features of the mind doing/contemplating Arithmetic) that is not or has some access to non-finitary resources. Cognitive Sciences hold computabilism which implies that there's not any higher Turing Computability going on the in brain and mind. Turing Computability imposes certain restrictions on what infinities can be legitimately presupposed.<br /><br />So, by a long trip, I think that there's a case for those thinking that PA CAN be inconsistent.<br /><br />[vperaltadelrriego@gmail.com]Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-56515848715224528782011-05-30T00:58:40.403+01:002011-05-30T00:58:40.403+01:00Is PRA+induction up to eptsilon_0 consistent?Is PRA+induction up to eptsilon_0 consistent?Franklinhttps://www.blogger.com/profile/16273024510749563188noreply@blogger.com