I have some questions regarding the proof that if M is a group and Th(M) is $\omega$-stable then there is no infinite, strictly decreasing sequence of definable subgroups, $R_0\subsetneq R_1\subsetneq R_2\subsetneq ...$ I am reading it from Martin Ziegler, Model theory of modules. I will copy the text:
Let $\varphi_i$(M) be a proper descending sequence of pp-definable subgroups of M. Choose $a_i \in R_i $ \ $ R_{i+1}$. The types $$ p_\eta (x)=\{x \in b_{i,\eta} + \varphi_i (M) | i \in \omega \} \text{ , } \eta : \omega \rightarrow 2$$ where $b_{i,\eta}=\sum_{j<i}\eta(j) a(j) $ are (in M) consistent and pairwisely contradictory. Whence $$ |S_{M|R_0}(\{a_0,a_1,...\})| \ge 2^{א_0}$$ if $L_{R_0}$ contains the $\varphi_i$.
Questions:
We represent the definable subgroups of the sequence with the formulas <$\varphi_i$(M) | $i<\omega$>. What are the formulas defining the types $p_{\eta}(x)$?
I understand they are pairwisely contradictory because the subgroups are proper one to another and $a_i \in R_i $ \ $ R_{i+1}$. But why they are consistent in M?
"What formulas are defining these types"?
I suppose you are asking how to rewrite the given formula $x\in b_i^\eta+\varphi_i(M)$ in the syntax of the first-order language of groups? Like so: $$\exists y\, (\varphi_i(y)\land x = b_i^\eta+y)$$
Or equivalently (and more efficiently): $$\varphi_i(x-b_i^\eta)$$
"Why are they consistent in M?"
By compactness!
Ok, I'll say a bit more, but you should really try to work it out for yourself before reading the solution below.
Fixing $\eta\in 2^\omega$, let's write $\theta_i(x)$ for the formula $\varphi_i(x-b_i^\eta)$. I claim that $\theta_{i+1}(x)$ implies $\theta_i(x)$. Note that $b_{i+1}^\eta = \sum_{j<i+1}\eta(j)a_j = b_i^\eta + \eta(i)a_i$. Now if $x-b_{i+1}^\eta\in \varphi_{i+1}(M)$, then $x-b_i^\eta = x-b_{i+1}^\eta+\eta(i)a_i\in \varphi_i(M)$, since both $x-b_{i+1}^\eta\in \varphi_{i+1}(M)\subseteq \varphi_i(M)$ and $a_i\in \varphi_i(M)$. It follows that $\theta_j(x)$ implies $\theta_i(x)$ whenever $i\leq j$. So if we take a finite subset of $p_\eta(x)$, say $\theta_{i_1}(x),\dots,\theta_{i_k}(x)$ with $i_1<\dots<i_k$, then it suffices to show that $\theta_{i_k}(x)$ is consistent. Taking $x = b^\eta_{i_k}$ works.