Alternate proof of Morley's theorem?

Question

I'm trying to understand the result given in the first box at slide 45 of this talk. Specifically:

1) What is the source cited? I have not been able to find any article by Keisler, Chudnovsky and/or Shelah corresponding to the situation.

2) Is this an alternate proof of the classical Morley's theorem (using $\mathcal{L}_{\omega_1\omega}$ as a tool, I guess like how you can use cut-elimination in $\omega$-logic to prove consistency of ordinary first-order PA) or an approach to proving the $\mathcal{L}_{\omega_1\omega}$ version of Morley's theorem?

Thanks.

Answer 1

Regarding (2), some evidence that Baldwin refers to some sort of $L_{\omega_1 \omega}$ version of Morley's theorem, rather than just an alternate proof making use of $L_{\omega_1 \omega}$ machinery, comes from a 1970 survey by Keisler himself. He mentions that "various forms" of Morley's theorem were extended to $L_{\omega_1 \omega}$ by "Choodnovsky [sic], Keisler, and Shelah, 1969" (p.149) though no citation is included in the references. And a look through the Shelah archive seems to turn up no relevant joint work with either of the other two.

I don't have a copy on hand, but one promising source for clarification (beyond inquiring with Baldwin about the content of his slides) is Keisler's 1971 book Model Theory for Infinitary Logic, which likely covers the result(s) in question such as they are; and though perhaps only a coincidence, that does match the year Baldwin's slides assign to the matter.

ETA: Baldwin's Categoricity book confirms both the nature of the result and his direct source: "Keisler [Kei71] generalized Morley’s categoricity theorem to sentences in $L_{\omega_1 \omega}$, assuming that the categoricity model was $\aleph_1$-homogeneous" (p.22). Though Baldwin points to Keisler's book as the basis for transferring Morely's theorem to infintary logic, he also attributes most of the machinery to Shelah (p.xi).

Having now gotten ahold of Keisler's book, the main generalization of Morley's theorem there (see Section 23) is as Baldwin describes:

Theorem. Let $T$ be a set of sentences from a countable fragment $L$ of $L_{\omega_1 \omega}$, and let $\kappa,\lambda > \omega$. Assume:

1. $T$ is $\kappa$-categorical.
2. For every countable model $M$ of $T$ there are models $N$ of $T$ of arbitrarily large size such that $M \prec_{L} N$.
3. Every model $M$ of $T$ of size $\kappa$ is $(\omega_1,L)$-homogeneous.

Then $T$ is $\lambda$-categorical, and every model of $T$ of cardinality $\lambda$ is $L$-homogeneous.

When $L$ is first-order logic, (2) is just upward Lowenheim-Skolem and (1) implies (3), so Morley's theorem really is a special case. Keisler explicitly notes that the special cases where either $\kappa=\omega_1$ or $\lambda=\omega_1 \alpha$ for $\alpha\ge 1$ follow from results due independently to Chudnovsky, Shelah and himself, so that seems to clarify everything.