Categoricity may fail late

We build an example that generalizes [HS90] to uncountable cases. In particular, our example yields a sentence ψ ∈ L(2λ)+,ω that is categorical in λ, λ, . . . , λ but not in ik+1(λ) . This is connected with the Loś Conjecture and with Shelah’s own conjecture and construction of excellent classes for the ψ ∈ Lω1,ω case. 1 The Loś Conjecture, without excellence Early results on the Categoricity Spectrum launched the development of Stability Theory and Classification Theory for first order logic. In the natural quest for generalizing the powerful results of those theories to Nonelementary Classes, some questions on Categoricity – specifically, the status of the Loś Conjecture (=Morley Theorem in First Order) – became a crucial test question. Among the specific issues studied, the following is central. Question 1.1 Old folklore question: what is the status of the Loś Conjecture (= Morley Theorem in the first order case) for ψ ∈ Lω1,ω? ∗ The first author’s research was partially supported by ‘BSF’ (USA-Israel); Publication no 648 in the first author’s publication list. The second author was sponsored by Colciencias and Semilleros de Investigación, Universidad Nacional de Colombia.