# Saturated Ehrenfeucht-Mostowski models

Inspired by this question on MSE I tried to prove the following:

Let $T$ be a complete theory in a first order language and $\kappa$ a cardinal. If $T$ is $\kappa$-stable, then there exists a $\kappa$-saturated model $\mathcal M$ of $T$ such that there is some sequence of indiscernibles $(a_i)_{i<\gamma}$, with $\gamma$ an ordinal of size less than $\kappa$, such that $\langle (a_i)_{i<\gamma}\rangle^{\mathcal M}=\mathcal M$, where $\langle (a_i)_{i<\gamma}\rangle^{\mathcal M}$ is the Skolem hull of $(a_i)_{i<\gamma}$.

As the following argument shows this holds provided $\kappa$ is regular:

By induction on $\alpha$, let us construct for each $\alpha<\kappa$ a model $\mathcal M_{\alpha}$ of $T$ of size $\kappa$ and a sequence of indiscernibles $I_{\alpha}\subseteq M_{\alpha}$ whose order-type is an ordinal less than $\kappa^+$, and for given $\alpha<\beta$, $\mathcal M_{\alpha}\prec \mathcal M_{\beta}$, $I_{\alpha}\subseteq I_{\beta}$ and each element of $I_{\beta}\setminus I_{\alpha}$ greater than all elements of $I_{\alpha}$ (1).

Let $\mathcal M$ be a model of $T$, and consider $I_0=\{a_{\alpha}:\alpha<\kappa\}$ an enumeration of $\mathcal M$, then using the compactness theorem and Ramsey's theorem we get $\mathcal M_0$ a model of $T$ of size $\kappa$ containing $\mathcal M$ with a sequence $I_0'$ of indiscernibles of order type $\kappa$ which realizes $EM(I_0)$, WLOG we may assume $I_0=I_0'$.

For $\alpha$ limit we simply put $\mathcal M_{\alpha}=\bigcup_{\beta<\alpha}\mathcal M_{\beta}$ and let $I_{\alpha}$ be the concatenation of the $I_{\beta}'$s $(\beta<\alpha)$.

Now suppose $\alpha$ is not limit.

If $\alpha$ is an even ordinal , let $I_{\alpha}=\{a_{\gamma}:\gamma<\mu\}$, where $(a_{\gamma})_{\gamma<\mu}$ is an enumeration of $\mathcal M_{\alpha-1}$ such that for any $\beta<\alpha$ we have that $I_{\beta}$ is bounded in $I_{\alpha}$, $\mu<\kappa^+$, let $\mathcal M_{\alpha}$ be an elementary extension of $\mathcal M_{\alpha-1}$ of size $\kappa$ that realizes $EM(I_{\alpha})$.

If $\alpha$ is an odd ordinal let $\mathcal M_{\alpha}$ be a model of $T$ of size $\kappa$ realizing all types over subsets of $I_{\alpha-1}$ of size less than $\kappa$ in $\mathcal M_{\alpha-1}$, then let $I_{\alpha}$ be the union of all the previous $I_{\beta}'$s and the set of all this realizations, with such a enumeration satisfying (1); $|I_{\alpha}|=\kappa$ as $T$ is $\kappa$-stable.

When we finish at step $\kappa$, let $I$ be the concatenation of all the $I_{\alpha}'$s and put $\mathcal N=\bigcup_{\alpha<\kappa}\mathcal M_{\alpha}$, then by the construction we have that $I$ is a sequence of indiscernibles in $\mathcal N$ of order type less than $\kappa^+$, $\langle I\rangle^{\mathcal N}=\mathcal N$, and thus $\mathcal N$ is $\kappa$-saturated because of the construction.

This solves the MSE question for $\kappa$ regular, as elementary equivalent saturated structures of the same size are isomorphic, and the above model has $2^{\kappa}$ automorphisms.

My question is, can we construct such model for $\kappa$ singular, under the above hypothesis?

Thanks

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What do you mean by $\langle (a_i)_{i<\gamma}^{\cal M}\rangle = {\cal M}$? Are you assuming $T$ has built-in Skolem functions? – Dave Marker Jul 15 '13 at 12:29
If $T$ is stable with built in Skolem functions (or has a stable Skolemization) I think there is a problem. Morely showed that if you take an EM model with a well ordered set of indiscernibles you only realize countably many types over any countable set (Thm 5.2.9 in my book). But if T is stable, the order type of the indiscernibles is irrelevant. This would be a problem if $T$ is stable but not $\omega$-stable. – Dave Marker Jul 15 '13 at 12:36
Hello David. I meant, and forgot to writte, $\langle (a_i)_{i<\gamma}\rangle^{\mathcal M}$ is the Skolem hull of $(a_i)_{i<\gamma}$ – Camilo Arosemena Jul 15 '13 at 17:56
I don't see what is the problem between the theorem in your book and what I'm asking, could you please be more clear? – Camilo Arosemena Jul 15 '13 at 18:11
I think the following is right--If $T$ is a stable theory (non $\omega$-stable) theory with Skolem functions, then in an any model of $T$ that is the Skolem hull of an infinite set of indiscernibles, we can only realize countably many types over any countable set. Thus the model will not be saturated. – Dave Marker Jul 15 '13 at 18:23