FOM: a query about cardinals

[Robert Black]

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