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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.

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    $\begingroup$ Asking someone to do all the work for you is generally not the best way to get s fruitful response. Perhaps give details of which Morley theorem you're interested in, and write up (as best you can) your point of confusion. If you're really just confused by some slides, it would be best to contact the author directly. $\endgroup$
    – Ryan Budney
    Feb 20, 2015 at 20:23
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    $\begingroup$ @RyanBudney Could you please clarify how your comment applies to this post? The author reports that he was unable to find an independent reference to the article; I believe him. Question 2 looks like a real question to me, and evidently based on some background knowledge. I just don't see how the OP is asking MO users "do all the work for [him]". And (pardon me for asking) is this subject area one that is familiar to you? $\endgroup$
    – Todd Trimble ♦
    Feb 20, 2015 at 20:53
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    $\begingroup$ There appears to be a closely related set of slides here: citeseerx.ist.psu.edu/viewdoc/… Page 34-35 outlines the intended Morley's theorem, and page 37 repeats some of the material you cited from that page 45. $\endgroup$
    – Todd Trimble ♦
    Feb 20, 2015 at 20:57
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    $\begingroup$ @RyanBudney My reading is that the question is rather trying to track down a mysterious reference involving the three names Keisler, Chudnovsky, and Shelah, and asking what might be in that reference. (There may not be such a reference involving all three; see Ed Dean's nice answer.) I really don't believe the OP is asking the community to explain Morley's classical work on categoricity (or results obtained inter alia on omitting types), which context he presumably thought was clear. "Doing all the work for you" really seems an unwarranted conclusion about what the OP wants here. $\endgroup$
    – Todd Trimble ♦
    Feb 21, 2015 at 1:15
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    $\begingroup$ Why are people voting to close this? It's a perfectly appropriate question. $\endgroup$
    – Noah Schweber
    Feb 21, 2015 at 3:40

1 Answer 1

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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.

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    $\begingroup$ @ToddTrimble, my hunch is that Chudnovsky's relevant work can be found in: "Some results in the theory of infinitely long expressions," Math. Dokl. 9, 556-559. Keisler offers no direct citation in the text, but that is one of two Chudnovsky works in his bibliography. (For good measure, the other is given just as Algebra and Logic 9, 80-120, with no title given, but a mention that it's a work in Russian.) I don't have access to the first paper or to a JSL review of it by Lopez-Escobar. Someone who does could (dis)confirm its relevance. $\endgroup$
    – Ed Dean
    Feb 23, 2015 at 15:19
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    $\begingroup$ The JSL review says that the first half of Chudnovsky's paper is about compactness and preservation theorems in $L_{\alpha,\beta}$, and the second half "is devoted to mathematical characterizations of the classes $K$ which are axiomatizable by a set of sentences of $L_{\alpha,\beta}$ of certain prescribed forms... Unfortunately, although the characterizations are of theoretical interest, they are a little too complicated to be completely satisfying." $\endgroup$
    – Alex Kruckman
    Feb 23, 2015 at 18:47

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