TITLE: Which large cardinals are upward reflecting?
QUESTION [7 upvotes]: Let the first order formulas $p(x)$ and $wi(x)$ assert "$x$ is a large cardinal of type $p$" and "$x$ is weakly inaccessible" respectively.
The large cardinal type $p$ is upward reflecting if $ZFC\vdash \forall \kappa~~(p(\kappa)\longrightarrow\exists \lambda>\kappa~~wi(\lambda))$. 
In the other words existence of a large cardinal of type $p$ like $\kappa$ implies existence of a large cardinal (at least weakly inaccessible) above $\kappa$.
Which large cardinals are upward reflecting? 
In particular, are Shelah cardinals upward reflecting?

REPLY [3 votes]: As you are mainly interested in Shelah cardinals, let me add a few more remarks about them. Given a Shelah cardinal $\kappa$, let $wt(\kappa),$ the witnessing number of $\kappa,$ be the least $\lambda$ such that for each $f:\kappa\to\kappa,$ there is an extender $E$ in $V_\lambda,$ witnessing the Shelahness of $\kappa$ with respect to $f$.
Then we can show that:
1) for all $\xi<wt(\kappa), \kappa$ is $\xi-$strong,
2) Measurable Woodin cardinals are unbounded in $wt(\kappa).$
For more information and additional results see "Witnessing Numbers of Shelah Cardinals" by Toshio Suzuki, Math. Log. Quart. 39 (1993), 62 - 66.