[FOM] uncountable structures and core mathematics II

John Baldwin jbaldwin at uic.edu
Mon Feb 20 22:14:16 EST 2006

I discussed connections between model theory and core mathematics and
admitted that the weak continuum hypothesis played a key role at points.

Shipman asked:
> So what happens to these connections in the case that 2^aleph_0 =
> 2^aleph_1?  Do the model-theoretic properties of larger models then
> have anything to say about currently popular mathematics?

This is wide open. Some of the application of weak CH are blatant.
Keisler proved that if there are few (less than 2^{aleph_1} models of 
cardinality aleph_1, then there are only countably many types over the 
empty set.  If you name constants you conclude the result for the number 
of types over countable sets (omega-stability).

However, some of the results hold in ZFC under the hypothesis of omega 
stability:  E.g. an omega stable sentence of L omega-1 omega with at least 
1 but fewer than 2^{aleph_1} models in aleph_1 has a model of cardnality 

Few models implies omega stable is consistently false.  But I know of know 
work in other extensions of ZF.

John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics, 
and Computer Science  M/C 249
jbaldwin at uic.edu
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607

Assistant to the director
Jan Nekola: 312-413-3750

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