tag:blogger.com,1999:blog-4987609114415205593.post7358423765926094175..comments2024-03-28T07:29:53.593+00:00Comments on M-Phi: Counting InfinitiesJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-4987609114415205593.post-58441410269363877272014-04-03T03:33:12.979+01:002014-04-03T03:33:12.979+01:00Correction, "this property" seems to ref...Correction, "this property" seems to refer to an attribute of their measure (if I can call it that) and not to the entire idea. It's not clear whether they have extended their ideas to uncountable sets or not. Catarina, can you give a summary of the technical ideas? That was missing from your article.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-23743806289747312292014-04-03T03:06:35.983+01:002014-04-03T03:06:35.983+01:00Interesting. I Googled the subject and found this ...Interesting. I Googled the subject and found this quote from one of the key papers:<br /><br />"Extending this property to uncountable sets seems to be a difficult problem"<br /><br />http://www.ams.org/journals/tran/2010-362-10/S0002-9947-2010-04919-0/<br /><br />So let's not hold that wake for Cantor's theory just yet.<br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-92116059502966714322014-03-31T12:48:14.428+01:002014-03-31T12:48:14.428+01:00Sigh, so now that NF has turned out (or is very li...Sigh, so now that NF has turned out (or is very likely) to be a "vanilla" logical system, one needs to come up with new shiny 'alternatives' somehow?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-6185592068539391662014-03-28T13:36:47.885+00:002014-03-28T13:36:47.885+00:00Nice ideas. Maybe they could also find an applicat...Nice ideas. Maybe they could also find an application in formal semantics, where most of the models are restricted to finite universes? It is usually justified but sometimes we also want to talk about natural numbers, e.g., say "There are more non-primes than primes among integers" with an intention that the sentence is actually true. There is not much literature treating that problem, that I'm aware of. One existing proposal is to treat quantifiers in such sentences as measure quantifiers (so-called ambiguous quantifiers): in infinite universes we can compare quantities by a proper measure functions. For instance, a proper measure function would make the above sentence true as primes become less common as the numbers become larger (asymptotic distribution from Prime Number Theorem). However, that is not a very satisfactory solution as measures are non-logical and context dependent. Anonymoushttps://www.blogger.com/profile/01786156551779108959noreply@blogger.com