# Proof of theorem 2.1 and conclusion 2.2 in Sh:31 (Categoricity of Uncountable theories)

Shelah in his paper stated the following two results (directly quoting from the paper):

Theorem 2.1. For $$\lambda > |T|$$, $$T$$ has a model $$M$$ of cardinality $$\lambda$$, such that for all $$m < \omega$$, $$A \subseteq |M|$$, $$|\{p\ |\ p \in S^m(A), p \text{ realized in } M\}| \leq |A| + |T|$$.

Proof. Take $$M$$ as an Ehrenfeucht-Mostowski model, which is the closure of a well-ordered set (See [Mo 1]).

And,

Conclusion 2.2. If $$T$$ is categorical in $$\lambda$$, $$|T| \leq \mu < \lambda$$, then $$T$$ is stable in $$\mu$$.

Both of these references Morley's Categoricity in Power paper, which I have tried to go over, but did not get very far.

My question is essentially, for theorem 2.1, $$M = EM(I)$$ for some well-ordered set $$I$$, how is this $$I$$ defined?

For conclusion 2.2, this follows directly from theorem 2.1 and Upwards Lowenheim-Skolem, correct?

• The point of Sh:31 is to generalize Morley's Categoricity Theorem to uncountable languages. The proof in this case requires significantly more technical work. Just a piece of advice: If you haven't thoroughly internalized the proof of Morley's theorem, I highly recommend that you work through a textbook presentation of that proof (e.g. from Marker or Hodges or Tent-Ziegler or Chang-Keisler) before reading Shelah's paper. May 30 at 18:50

You may find a proof of of Theorem 2.1 (with $$T$$ countable, but it does not really matter) in the book Model Theory by Chang and Keisler. In the 3rd edition (1990) it is Corollary 3.3.14
The key point is that the well-ordered set that generates $$M$$ has to be an indiscernible sequence. Also, the language needs to have Skolem functions (which we can add without increasing the cardinaity of the language).