tag:blogger.com,1999:blog-8148573551417578681.post4452078199236302303..comments2020-05-26T07:09:09.183-07:00Comments on Dark Buzz: Horgan interviews MaudlinRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-8148573551417578681.post-89083841578020538372018-12-30T04:52:57.902-08:002018-12-30T04:52:57.902-08:00Wow. I make a point about the objective moral prog...Wow. I make a point about the objective moral progress that was made by the civil rights movement and the women's suffrage movement, allowing women and blacks to both vote and hold political office, and you respond with a rant about Donald Trump? Res ipsa loquitor.Anonymoushttps://www.blogger.com/profile/17918668471205376513noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-45844806740085433292018-12-04T17:09:39.143-08:002018-12-04T17:09:39.143-08:00Incompleteness has no impact on mathematics at all...Incompleteness has no impact on mathematics at all. Gödel sentences are grammatical (well-formed formulas) gibberish about totalities of properties. Even Cohen confused trivial incompleteness with Gödel incompleteness.<br /><br />Feferman: As things stand today, these explorations of the set-theoretical stratosphere are clearly irrelevant to the concerns of most working mathematicians. A reason for this was also given by Gödel near the outset of his 1951 Gibbs lecture (posthumously published in 1995), where he said that the “phenomenon of the inexhaustibility of mathematics” follows from the fact that “the very formulation of the axioms [of set theory over the natural numbers] up to a certain stage gives rise to the next axiom. It is true that in the mathematics of today the higher levels of this hierarchy are practically never used. It is safe to say that 99.9% of present-day mathematics is contained in the first three levels of this hierarchy.” In fact, modern logical studies have shown that the system corresponding to the second level of this hierarchy, called second-order arithmetic or analysis and dealing with the theory of sets of natural numbers, already accounts for the bulk of present-day mathematics. Indeed, much weaker systems suffice, as is demonstrated in Simpson (1999). Even more, I have conjectured and verified to a considerable extent that all of current scientifically applicable mathematics can be formalized in a system that is proof-theoretically no stronger than PA (cf. Feferman 1998, Ch. 14). https://math.stanford.edu/~feferman/impact.pdfMD Coryhttps://www.blogger.com/profile/05342743632013663077noreply@blogger.com