tag:blogger.com,1999:blog-7225373.post496709812879284775..comments2024-02-29T03:34:23.190-05:00Comments on Who Were the Sea Peoples?: Cantor set IIIgcallahhttp://www.blogger.com/profile/10065877215969589482noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-7225373.post-2667233607311449782008-05-27T03:43:00.000-04:002008-05-27T03:43:00.000-04:00Does anybody know the proof of this obvious(?) fac...Does anybody know the proof of this obvious(?) fact: "every recursively enumerable set is recursive"?Yehonathanhttps://www.blogger.com/profile/12200119323595401195noreply@blogger.comtag:blogger.com,1999:blog-7225373.post-40875423886447145752008-03-25T17:48:00.000-04:002008-03-25T17:48:00.000-04:00shonk: fascinating--thanks for the tip; I shall in...shonk: fascinating--thanks for the tip; I shall investigate. There is a catch here that I left out of everything, partly because I wasn't sure how to express it concisely, partly because I thought it woud help the ratiocination of you guys out there if I left it for you. Classically, recursive, creative, and other related properties of sets are defined in countable sets. Clearly ~C is r.e. because--even though it is uncountably infinite, we are defining it in a countably infinite number of uncountably infinite chunks. But my take on all this is highly unconventional: for example, it is no longer obvious that a recursive set is r.e.--the classical argument no longer applies when the set in question is uncountably infinite! Good luck, and let me hear from you again.Wabulonhttps://www.blogger.com/profile/16838347174718251102noreply@blogger.comtag:blogger.com,1999:blog-7225373.post-84039156265338294952008-03-23T02:24:00.000-04:002008-03-23T02:24:00.000-04:00If C were recursive, then of course it would be re...If C were recursive, then of course it would be recursively enumerable. I haven't read it, but supposedly somewhere in <A HREF="http://dx.doi.org/10.1090/S0273-0979-1989-15750-9 " REL="nofollow" TITLE="Bulletin of the American Mathematical Society">this paper</A> the authors note that any recursively enumerable set over the reals must have integral Hausdorff dimension (I say "supposedly" because I'm just taking the author of <A HREF="http://dx.doi.org/10.2307/2048094" REL="nofollow" TITLE="JSTOR - Proceedings of the American Mathematical Society - Vol. 110, No. 2 (Oct., 1990), pp. 495-497">this paper</A> at his word). <BR/><BR/>Since C has Hausdorff dimension log 2/log 3, this would mean C is not recursively enumerable and, hence, cannot be recursive.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7225373.post-31410121283669283062008-03-22T17:29:00.000-04:002008-03-22T17:29:00.000-04:00Quite right--but *is* C recursive? This is the for...Quite right--but *is* C recursive? This is the formal way to ask the question that I asked in CsII, and is why I went through the stuff in CsIII. I admit that it looks like no, but maybe this is just a way of saying that the question is hard. (I do not know the answer, but I suspect that it is indeed No.)Wabulonhttps://www.blogger.com/profile/16838347174718251102noreply@blogger.comtag:blogger.com,1999:blog-7225373.post-17992931604771619092008-03-21T22:12:00.000-04:002008-03-21T22:12:00.000-04:00Well, C isn't recursively enumerable, so C (and, t...Well, C isn't recursively enumerable, so C (and, therefore, ~C) isn't recursive, if that's what you're getting at.Anonymousnoreply@blogger.com