The definitions of Morley rank & degree I use are
I understand these definitions, but I am having a hard time to use them concretely in exercises. For example,
Let $L$ be a countable language and $A$ an $\omega$-saturated
$L$-structure.(a) let $p$ be a ranked type over $A$. Show that $p$ has Morley degree
1.(b) Let $\varphi(x_1, \dots, x_n)$ be a ranked $L_A$-formula, show that
$$ \text{dM}(\varphi(\bar{x})) = |\{p \in S_n(\text{Th}_{L_A}(A)) : \varphi \in p \text{ and } \text{RM}(p) = \text{RM}(\varphi) \}| $$
I don't know how to connect $\omega$-saturation with Morley rank & degree. I thought to use that $\text{Th}(A)$ has an $\omega$-saturated model iff its type spaces are countable iff there is no binary tree of consistent formulas, in the hope that $\text{dM}(p) \geq 2$ would allow me to construct such a binary tree, but I cannot get it to work.
Best Answer
I'm assuming your source defines Morley degree to be the $k$ from Lemma 5.10.
For (a), Lemma 5.10 ensures that every $L_A$-formula $\varphi(x)$ can be decomposed into $k$ disjoint pieces of full Morley rank, where $k = \text{dM}(\varphi(x))$. The main observation is that the formulas $\varphi_j(x)$ in this maximal decomposition all have Morley degree $1$. Indeed, if $\varphi_j(x)$ had Morley degree $d>1$, then applying the lemma to this formula, we could split $\varphi_j(x)$ into $d$ disjoint pieces of full Morley rank. Replacing $\varphi_j(x)$ with these formulas in the original decomposition of $\varphi(x)$ would decompose $\varphi(x)$ into $k-1+d > k$ disjoint pieces of full Morley rank.
Now for any complete type $p(x)$, let $\varphi(x)$ be a formula in $p(x)$ of minimal Morley rank and degree. Suppose for contradiction that $\text{dM}(\varphi(x)) = k > 1$. Then $\varphi(x) \leftrightarrow \bigvee_{j=1}^k \varphi_j(x)$, and each $\varphi_j(x)$ has Morley rank equal to $\text{RM}(\varphi(x))$ and Morley degree $1$. Since $p(x)$ is a complete type, $\varphi_j(x)\in p(x)$ for some $1\leq j\leq k$, contradicting minimality of $\varphi(x)$.
For (b), I'll give you a hint. Let $k = \text{dM}(\varphi(x))$, and let $n$ be the number of complete types of full Morley rank containing $\varphi(x)$. You can prove $k\leq n$ by decomposing $\varphi(x)$ by Lemma 5.10 and extending each formula in the decomposition to a complete type of maximum Morley rank. And you can prove $n\leq k$ by picking disjoint formulas containing each of the types of full Morley rank, and applying the bound from Lemma 5.10.