From: Richard Carr Newsgroups: sci.math Subject: Re: Regular Limit Cardinals Date: Thu, 10 Dec 1998 19:54:32 -0500 On Thu, 10 Dec 1998, Felix Dilke wrote: :Date: Thu, 10 Dec 1998 21:55:16 -0000 :From: Felix Dilke :Newsgroups: sci.math :Subject: Regular Limit Cardinals : :Could somebody confirm that there does exist a limit :cardinals (=aleph(a) for some limit ordinal a) which :is its own cofinality? : :I think these are called 'regular limit cardinals' & have a :""proof"" that there aren't any, which I am fairly sure :has a hole in it. : :Any answers please cc to fdilke@usa.net, as I can't :keep up with this group. : :Thanks very much in advance :Felix : Such cardinals are generally called weakly inaccessible. If you can prove that they do not exist (on the basis of ZFC) then you should, if there's any justice on the committee, win a Fields Medal. This would be an amazing result. ============================================================================== From: Richard Carr Newsgroups: sci.math Subject: Re: Regular Limit Cardinals Date: Sat, 12 Dec 1998 20:58:41 -0500 The argument is somewhat like this. If one can prove that the existence of a weakly inaccessible cardinal is consistent with ZFC (arguing in ZFC), then we can prove in ZFC, by G"odel's Completeness theorem, that there is a model for ZFC+there exists a weakly inaccessible cardinal. Then within this model, construct L. L satisfies GCH and the weakly inaccessible cardinal is still weakly inaccessible in L, but in the presence of GCH weakly inaccessible and strongly inaccessible cardinals are the same. Thus we get the existence of a model of GCH+there is a strongly inaccessible cardinal. So, if ZFC is consistent then so is ZFC+there exists a strongly inaccessible cardinal (+GCH, but that won't matter). Now this model has a strongly inaccessible cardinal, but the existence of a strongly inaccessible cardinal implies the consistency of ZFC. So, if ZFC could prove the existence of an inaccessible cardinal to be consistent, then it would also prove itself consistent and this would contradict G"odel's 2nd Incompleteness theorem. ============================================================================== From: hrubin@b.stat.purdue.edu (Herman Rubin) Newsgroups: sci.math Subject: Re: Regular Limit Cardinals Date: 12 Dec 1998 16:18:02 -0500 In article , Richard Carr wrote: >On Thu, 10 Dec 1998, Felix Dilke wrote: >:Could somebody confirm that there does exist a limit >:cardinals (=aleph(a) for some limit ordinal a) which >:is its own cofinality? .............. >Such cardinals are generally called weakly inaccessible. If you can prove >that they do not exist (on the basis of ZFC) then you should, if there's >any justice on the committee, win a Fields Medal. >This would be an amazing result. It would be more than that; it would mean that the foundations would have to be reworked, at least if GCH holds. It MIGHT be possible to prove than none such exists, as this is consistent with the other axioms. However, many, if not most, working in this area believe the existence of much worse, such as strongly inaccessible and measurable cardinals, is consistent. As before, this CANNOT be proved if mathematics is itself consistent. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558