I have a question about a corollary in Jech's set theory text which states:
Corollary 17.19. Every Weakly Compact cardinal $ \kappa $ is a Mahlo cardinal, and the set of Mahlo cardinals below $ \kappa $ is stationary.
I have a question about the proof of the first part . In particular,
Proof : Let $ C \subset \kappa $ be a closed unbounded set. Since $ \kappa $ is inaccessible , $ ( V_{ \kappa} , \in , C)$ satisfies the $ \Pi_{1}^{1} $ sentence:$ \not\exists F ( F \text{ is a function from some } \lambda < \kappa \text{ cofinally into } \kappa )$ and $C$ is unbounded in $ \kappa$.
I am confused as to why this last line is true. In particular , doesn't $ \kappa $ having rank $ \kappa $ imply that $V_{\kappa}$ cannot even interpret the sentence? Am I misunderstanding what is meant by this line in the first place? Thanks
In $V_\kappa$ it is very easy to understand what is $\kappa$. It's the class of all the ordinals. So the sentence is really "For every relation over $V_\kappa$ which is a function with domain being an ordinal, the range is bounded".
For every relation over $V_\kappa$, since if $\kappa$ were singular, then a function witnessing that is not an element of $V_\kappa$, but rather a subset of $V_\kappa$.
The domain is an ordinal simply means that for some $\lambda<\kappa$, the domain is this $\lambda$.
The range is bounded means that the function is not cofinal.
Alternatively, you could say that for every subset of $V_\kappa$ which is a function with domain being an element of $V_\kappa$, there is an element of $V_\kappa$ which realizes this function entirely. I'll leave you to think about why this alternative sentence works too.