I am studying Morley's theorem, which says that a complete theory which is countable and categorical in an uncountable cardinality $\kappa$ is categorical in every uncountable cardinality. In order to prove this we claim that if a theory is categorical in an uncountable cardinality, then it is totally transcendental. But I am stuck here, and I can not prove it.

I tried to show instead it is $\omega$-stable by contadiction: Let $M$ be a model of $T$ with $|M|= \aleph_0 $ but $|S_1(M)| \geq \aleph_1$. By the Löwenheim-Skolem theorem, there exists an elementary extension $M\preceq N$ with $|N|=\kappa$, but I can not find a way to contradict with $\kappa$-categoricity...

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    $\begingroup$ I cleaned up your post, fixing lots of spelling, grammar, and formatting errors. Please try to pay more attention to these things in the future. $\endgroup$ – Alex Kruckman Mar 8 '19 at 14:41
  • $\begingroup$ Also, I changed "there exists an initial expansion of $m$ like $n$" to "there exists and elementary extension $M\preceq N$"... I have no idea what an initial expansion is, but I guessed from context that this is what you meant. I hope I guessed correctly! $\endgroup$ – Alex Kruckman Mar 8 '19 at 14:47
  • $\begingroup$ Yes,thanks a lot...I really have to try to improve my english..... $\endgroup$ – user297564 Mar 8 '19 at 15:13

You have the right idea to prove that the theory is totally transcendental by proving that it is $\omega$-stable. But this is a non-trivial component of Morley's theorem, and one that you're not going to be able to prove just by playing around with Löwenheim-Skolem.

It's also a step that will be included in any proof of Morley's theorem (e.g. it's Corollary 5.2.10 in Marker, Theorem 5.2.4 in Tent and Ziegler, and Theorem 3.8 in Morley's original paper). Maybe "I am studying Morley's theorem" means that you're trying to reprove the whole thing on your own! But if not, you're probably reading from some reference that includes this proof.

In any case, the idea is this: You show that for every every countable theory $T$ with infinite models and every infinite cardinal $\kappa$, there is a model $M\models T$ which only realizes countably many types over any countable set of parameters. How? You build an Ehrenfeucht-Mostowski model generated by a sequence of indiscernibles ordered like the ordinal $\kappa$ (this is the tricky part of the proof, which would take far too long to explain in this answer). But on the other hand, if there is a countable model $N$ of $T$ such that $|S_1(N)|\geq \aleph_1$, then you can realize a set of $\aleph_1$-many types over $N$ in an elementary extension $N\preceq N'$ of size $\kappa$. These two models $M$ and $N'$ aren't isomorphic, so $T$ isn't $\kappa$-categorical.

  • $\begingroup$ Thanks got it.I have written just an abstract of this proof ,but I totally coundn't remember what should I do to prove it... $\endgroup$ – user297564 Mar 8 '19 at 15:21

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