Categoricity of theories in Lϰω, with ϰ a compact cardinal

1 5 A pr 1 99 6 Categoricity of Theories in L κ ∗ ω , when κ ∗ is a measurable cardinal . Part II

We continue the work of [KlSh 362] and prove that for λ successor, a λ-categorical theory T in Lκ∗,ω is μ-categorical for every μ, μ ≤ λ which is above the (2)-beth cardinal. Research supported by the United States-Israel Binational Science Foundation. Done 6-7/88. Publication number 472

Cardinal characteristics at κ in a small u(κ) model

We provide a model where u(κ) < 2κ for a supercompact cardinal κ. [10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that...

Categoricity of computable infinitary theories

A MEASURABLE CARDINAL WITH A CLOSED UNBOUNDED SET OF INACCESSIBLES FROM o(κ) = κ

We prove that o(κ) = κ is sufficient to construct a model V [C] in which κ is measurable and C is a closed and unbounded subset of κ containing only inaccessible cardinals of V . Gitik proved that o(κ) = κ is necessary. We also calculate the consistency strength of the existence of such a set C together with the assumption that κ is Mahlo, weakly compact, or Ramsey. In addition we consider the ...

Categoricity from One Successor Cardinal in Tame Abstract Elementary Classes

We prove that from categoricity in λ we can get categoricity in all cardinals ≥ λ in a χ-tame abstract elementary classe K which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ > LS(K) and λ ≥ χ. For the missing case when λ = LS(K), we prove that K is totally categorical provided that K is categorical in LS(K) and LS(K).