I have been reading Kanamori's Higher Infinite and I am trying to understand that a cardinal $\kappa$ is $\Pi^1_1$-indescribable iff it has the extension property.
We say that $\kappa$ has the extension property iff for every $R \subseteq V_\kappa$ there are sets $X$, $S$ such that $X \neq V_\kappa$ and $S \subseteq X$, such that $\langle V_\kappa, \in, R \rangle \prec \langle X, \in, S\rangle$.
This sounds like a rather easy property to satisfy, can someone help me acquire some intuitive insight why this is not satisfied for $\kappa$s smaller than the $\Pi^1_1$-indescribabile cardinal?
EDIT: A cardinal $\kappa$ is $\Pi^1_1$-indescribable iff for every $\Pi^1_1$ formula $\varphi$ and a set $R \subseteq V_\kappa$ with $\langle V_{\kappa+1}, \in, R \rangle \models \varphi$, there is an ordinal $\alpha$ such that $\langle V_{\alpha+1}, \in, R \cap V_\alpha \rangle \models \varphi$.
End-extensions are not at all trivial to obtain.
First of all, note that if $X$ is an end-extension of $V_\kappa$, then $X$ is a transitive model of $\sf ZFC$ of height $>\kappa$. This is not to be taken lightly.
Suppose $V=L$ and there exists a single inaccessible cardinal $\kappa$. Moreover assume that $\sf\operatorname{Con}(ZFC+I)$ is false in our universe. So there is no model of $\sf ZFC+I$ (where $\sf I$ is the existence of a strongly inaccessible cardinals). Now I claim that there is no end-extension of $V_\kappa$, since any end-extension of $V_\kappa$ will have to satisfy that $\kappa$ is a strongly inaccessible cardinal. And then you will have a model of $\sf ZFC+I$ (in fact a well-founded model!) which is impossible in our universe.
You might want to argue that the same would work for weak-compactness, but this is not the case. Weak-compactness is a second-order property over $V_\kappa$, and the end-extension only guarantees that there is a first-order end-extension.
So what does this mean? It means that if you have a weakly compact $\kappa$, then there are transitive models which have height greater than $\kappa$. This is not a trivial fact anymore.
Now you can easily prove that the end-extension property implies that $\kappa$ is at least strongly inaccessible, this is the first thing Kanamori proves (in the relevant direction). The idea is that a function $F\colon X\to\kappa$ which is cofinal satisfies that $\forall y\exists z(z\in X\land F(z)\notin y)$, as a first-order sentence over $V_\kappa$ (with $F$). But if you take an end-extension, this is not true anymore since now $\kappa$ is an element of the end-extension. This implies both regularity and being a strong limit.
So we see that not only the end-extension property implies strong inaccessibility. It is in fact much stronger than just inaccessibility.