tag:blogger.com,1999:blog-8148573551417578681.post7763572186288074600..comments2020-06-02T16:10:03.590-07:00Comments on Dark Buzz: Logicism did not failRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-8148573551417578681.post-83726752662532770612015-09-29T10:30:19.814-07:002015-09-29T10:30:19.814-07:00Thanks. You can download a free copy of the first ...Thanks. You can download a free copy of the first paper <a href="http://opus.ipfw.edu/philos_facpubs/298/" rel="nofollow">here</a> or <a href="http://philpapers.org/rec/BULTSO-4" rel="nofollow">here</a>.<br /><br />Yes, there are some consistency proofs of Peano arithmetic and other systems, but they are not very satisfying.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-1639681257095337702015-09-29T07:37:40.468-07:002015-09-29T07:37:40.468-07:00Your views are strongly supportive of a reversed n...Your views are strongly supportive of a reversed narrative about Cantor's set theory. Set theory came after most useful mathematics and the use of the term "finite" is quite abused. Hilbert had trouble explaining what he meant by it. Edward Nelson said “[f]initism is the last refuge of the Platonist.” The argument Godel made was the same type of diagonalization that Cantor performed and Turing after Godel. It has nothing to do with productive mathematics. By the way, people who think Godel and Turing spawned the computer revolution are plainly mistaken. It's a myth of mathematics departments.<br /><br />"No compelling evidence has yet been presented that G1 affects, or future refinements of it will affect, mainstream mathematics."<br />http://link.springer.com/article/10.1007/s11787-014-0107-3<br /><br />Self-referencing formulas and impredicative sets are not about mathematics proper. I can't stress how badly people don't understand this! By the way, people who are trying to connect this to the continuum hypothesis are crackpots. They similar to the extent they are both about nonsense.<br /><br />Chaitin finds that systems have a specific complexity bound beyond which provability falls to zero. Ron Maimon: "Gödel's theorem is a limitation on understanding the eventual behavior of a computer program, in the limit of infinite running time." (http://physics.stackexchange.com/a/14944)<br /><br />Roger: "Supposedly Hilbert thought that an axiomatization of math should first prove the consistency of its axioms. If so, that was a stupid belief, because inconsistent axioms allow proof of anything."<br /><br />Precisely! Poincare knew that assuming induction to prove it is circular anyways. I won't bother mentioning the weak retort about infinitely axiomitized systems because it's a fanciful and stupid reply to the absurdity of incompleteness.<br /><br />"Generalizing a bit, this suggests that logicism may be safe from Gödel's first incompleteness theorem and even the second, as well. If we do not and cannot know the consistency of a putative logicist system, then we are probably no better off with regard to mathematical knowledge than it is (assuming it is otherwise powerful enough). And, if we can know of a system that it is consistent, then, for the benefit of logicism, we should regard the system as too weak to be the relevant one for logicism"<br />http://www.jstor.org/stable/2214847Anonymousnoreply@blogger.com