I am given that $M$, a subspace of $\mathbb{R}^n$ is invariant with respect to \begin{equation} \dot x=f(x,t)\hspace{3cm}\cdots\cdots(1) \end{equation} if $x(0)\in M\Rightarrow f(x(t)\in M,t)\in M\forall t $, in otherwords if the trajectory starts at some point $x(0)\in M$ then it remains in $M...

Here (in the context of Abstract Elementary Classes) on the page 43 at the bottom,-6th line, what does it technically mean $$\leq_{\frak K_\lambda}-\text{increasing continuous}$$ ? I think that this should be a condition on limit ordinals, but in his text, Shelah uses $\alpha$ for both, limit an...

In the first paragraph of p.41 of Introduction to: classification theory for abstract elementary class, Shelah gives the following definition of (Galois) type. For $M\preceq N_\ell$ and $a_\ell\in N_\ell \setminus M$, $\mathbf {tp}(a_1,M,N_1)=\mathbf{tp}(a_2,M,N_2)$ iff for some $\preceq$-ext...