[Date Index] [Thread Index] [FOM Postings] [FOM Home]

*To*: fom@math.psu.edu*Subject*: FOM: a query about cardinals*From*: Robert Black <Robert.Black@nottingham.ac.uk>*Date*: Sat, 17 Jul 1999 14:59:54 +0100*Sender*: owner-fom@math.psu.edu

Could someone out there who knows more set theory than I do help me with the following query? Suppose you start with a set of Urelemente and then pile on it a structure of simple types (sets of Urelemente, sets of sets of Urelemente etc.). If you have kappa Urelemente you will have 2^kappa items at level 1, 2^2^kappa at level 2 etc. Go to all finite levels. Suppose you now want the additional constraint that for each set which occurs somewhere in this hierarchy you want its 'cardinal number' to be one of the Urelemente, so there must be (at least) as many Urelemente as there are cardinalities occuring in the whole structure. If GCH holds, this is obviously possible. e.g. if there are countably many Urelemente, the cardinalities which occur in the structure will be all those less than beth_omega, but if GCH holds beth_omega is aleph_omega and there are only countably many cardinals less than aleph_omega. But what if GCH doesn't hold, what if the 'continuum function' kappa |-> 2^kappa grows a lot faster (in aleph terms) than GCH says? (ZFC cerainly permits it to grow a lot faster.) Consider the claim: There is a cardinal kappa such that the structure of finite types over kappa contains at most kappa cardinalities. Two questions: (1) Am I right to guess that the claim is not provable in ZFC? (2) If I am right, is there any mathematically salient condition weaker than GCH from which the claim does follow? Or which is equivalent (in ZFC) to the claim? And what about the (presumably stronger) claim that *every* infinite kappa is such that the structure of finite types over kappa contains at most kappa cardinalities? Robert Black Dept of Philosophy University of Nottingham Nottingham NG7 2RD tel. 0115-951 5845

[Date Prev] [Date Next] [Thread Prev] [Thread Next]

[Date Index] [Thread Index] [FOM Postings] [FOM Home]