[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
aleph_2.
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
312-413-2149
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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