Definition 1. An uncountable cardinal $\kappa$ is Shelah if for every function $f:\kappa\rightarrow \kappa$ there exists a transitive class $M$ and a non-trivial elementary embedding $j:V\rightarrow M$ such that $^\kappa M\subseteq M$, $crit(j)=\kappa$ and $V_{j(f)(\kappa)}\subseteq M$.

Definition 2. Woodin's fast function forcing on $\kappa$ consists of partial functions $p$ from $\kappa$ to $\kappa$ ordered by inclusion such that:

• The domain of $p$ consists of inaccessible cardinals $\lambda <\kappa$ which are closed under $p$, namely for every $\lambda, \theta\in dom(p)$ if $\theta<\lambda$ then $p(\theta)<\lambda$.

• For every $\lambda\in dom(p)$ we have $|dom(p)\cap \lambda|<\lambda$

On one hand, preserving Shelah cardinals through lifting arguments often needs dominating the corresponding functions $f:\kappa\rightarrow \kappa$ by functions in the ground model.

On the other hand, we know that the fast function forcing $\mathbb{P}_{\kappa}$ adds a very fast (and so non-dominatable) function of this type into the universe (i.e. the fast function) and simultaneously fails to satisfy $\kappa$-cc property which is a usual condition for providing dominating functions in the ground model.

Thus it is somehow natural to expect that Woodin's fast function forcing can kill Shelah cardinals. Is it true? Any example?