tag:blogger.com,1999:blog-11807812.post111789674289989168..comments2023-05-29T08:58:13.381-04:00Comments on Recycled Knowledge: Who knows? Maybe nonstandard arithmetic is just unavoidableJohn Cowanhttp://www.blogger.com/profile/11452247999156925669noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-11807812.post-56042364744977282852007-02-17T20:01:00.000-05:002007-02-17T20:01:00.000-05:00I know this is an old entry for you (I just came a...I know this is an old entry for you (I just came across it in a Google search), but I was quite confused by your claim here. You say, "But although Gödel showed by a metamathematical argument that there can be no proof of G within Peano arithmetic, he did <I>not</I> show that there is no finite proof of ~G. Which means that in principle there might be one."<BR/><BR/>Assume for a moment that PA is indeed consistent. If PA proves ~G, then PA+G cannot possibly be consistent (since ~G and G would both be true in PA+G). This contradicts Gödel's First Incompleteness Theorem, which shows that the consistency of PA implies the consistency of PA+G.<BR/><BR/>So PA cannot prove ~G. Am I missing something?Anonymousnoreply@blogger.com