M 3:30 - 6:00 PM

Wean 6423

THEOREM Let T be a first-order theory. If T is categorical in a cardinal greater than |T|+\aleph_0 then it is categorical in all cardinals greater than |T|+\aleph_0.

This paper is fundamental to model theory as it introduces several important concepts. Baldwin in his detailed review of this paper for MR wrote "The proof is a miracle of localization".

While Shelah's paper is not self contained, the lectures will assume familiarity of only an introductory course in model theory as last year's 21-603.

THEOREM (Keisler). If an L_\omega_1,\omega theory has a model realizing uncountable many types over a fixed countable fragment then the theory has 2^{\aleph_1}-many pairwise non-isomorphic models of cardinality \aleph_1.

The proof require an omitting-types theorem and a theorem about end-extensions of countable models. Both are results of independent importance and interest.

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Last modified:
April 6^{th}, 2005 |