Dear friends, We can look back at Goedel's work as one early attempt of fibring of formal systems. Thus, the problem of determining the consistency and the completeness of the system seems to be like the same problem for a given logic L which is obtained by the fibring of two logics L* and L**. First order arithmetics is thus the fibring of first order logic and Peano's arithmetics, which yields the known results presented by Goedel.
We have no a priori intuitive reason to assume that a certain fibring will produce a consistent and complete system. This has to be checked as the systems are combined. If in combining two systems one produces results like those of Goedel's theorems, it must still be possible to investigate the resulting system by investigating other features of the same, namely checking whether it has different, new, unusual or unexpected properties. In sum, I don't expect to see novelties that can or will overshadow Goedel's contributions, except in consequence of more advances in the field of fibring. Em 1 de outubro de 2011 14:21, Walter Carnielli <walter.carnie...@gmail.com>escreveu: > Caros colegas: > > como vimos, o Ed Nelson retirou seu "claim" sobre a inconsistência de > P. > Isso, contudo, não significa o fim das discussões, nem menos a > garantia da consistência de P! > > Quero aqui parabenizar ao nosso colega Daniel Tausk do IME USP > (http://www.ime.usp.br/~tausk/) > pela excelente observação que convenceu o Nelson. Imagino que o > Rodrigo Freire e outros coelgas da USP tenham participado > ativamente da atmosfera de discussão que levou o Daniel a observar o > erro na pretensa prova do Ed Nelson. > > E finalmente parabenizo o Ed Nelosn pela coragem em propor sua > alegação, e pela humildade em reconhecer publicamente o erro. Acho > que é assim que se faz ciência, e acho que esse tipo de atitude é > que devemos ensinar aos nossos estudantes, e aprendermos nós > mesmos. Ed Nelson merece uma homegagem. > > Abraços, > > Walter > > ---------- Forwarded message ---------- > From: Edward Nelson <nel...@math.princeton.edu> > Date: 2011/10/1 > Subject: [FOM] inconsistency of P > To: f...@cs.nyu.edu > > > Terrence Tao, at > http://golem.ph.utexas.edu/category/2011/09/ > and independently Daniel Tausk (private communication) > have found an irreparable error in my outline. > In the Kritchman-Raz proof, there is a low complexity > proof of K(\bar\xi)>\ell if we assume \mu=1, but the > Chaitin machine may find a shorter proof of high > complexity, with no control over how high. > > My thanks to Tao and Tausk for spotting this. > I withdraw my claim. > > The consistency of P remains an open problem. > > Ed Nelson > _______________________________________________ > FOM mailing list > f...@cs.nyu.edu > http://www.cs.nyu.edu/mailman/listinfo/fom > > > > -- > ----------------------------------------------- > Prof. Dr. Walter Carnielli > Director > Centre for Logic, Epistemology and the History of Science – CLE > State University of Campinas –UNICAMP > 13083-859 Campinas -SP, Brazil > Phone: (+55) (19) 3521-6517 > Fax: (+55) (19) 3289-3269 > Institutional e-mail: walter.carnie...@cle.unicamp.br > Website: http://www.cle.unicamp.br/prof/carnielli > _______________________________________________ > Logica-l mailing list > Logica-l@dimap.ufrn.br > http://www.dimap.ufrn.br/cgi-bin/mailman/listinfo/logica-l > _______________________________________________ Logica-l mailing list Logica-l@dimap.ufrn.br http://www.dimap.ufrn.br/cgi-bin/mailman/listinfo/logica-l
