# Many $(\kappa+\alpha)$-strong cardinals below a $(\kappa+\beta)$-strong one for $\alpha < \beta < \kappa$

Let $\lambda$ be an ordinal. A cardinal $\kappa$ is $\lambda$-strong iff there is some inner model $M$ and an elementary embedding $$j \colon V \rightarrow M$$ s.t. $crit(j) = \kappa$ and $V_\lambda \subseteq M$.

I am given the following exercise:

Let $\alpha < \beta < \kappa$ be ordinals, where $\kappa$ is $(\kappa+\beta)$-strong. Then $$\{ \mu < \kappa \mid \mu \text{ is } (\kappa+\alpha) \text{-strong} \}$$ has size $\kappa$.

I'd like to show that $$M \models \kappa \text{ is } (j(\kappa)+\alpha) \text{-strong},$$ but I'm not sure where to begin...

(It seems that I have to "stretch" a $(\kappa, \kappa+\alpha)$-extender $E \in V$ to a $(\kappa, j(\kappa)+\alpha)$-extender $\tilde E \in M$, but how? Is this even the right approach?)

edit:

Consider the case $\alpha =0$ and recall that a cardinal $\mu$ is $\mu$-strong iff it is measurable.

Now, if $\kappa$ is $\kappa+2$-strong, we may fix a witnessing measure $U$ on $\kappa$. Then $U \in V_{\kappa+2} \subseteq M$ and some basic observations yield $$M \models \kappa \text{ is measurable}$$ Thus for any fixed $\xi < \kappa$ we get $$M \models \exists \mu \colon j(\xi) < \mu < j(\kappa) \wedge \mu \text{ is measurable}$$ (take $\mu = \kappa$) and elementarity yields that there is some measurable $\mu \in (\xi, \kappa)$. This proves that there are $\kappa$ many $\mu$-strong cardinals $\mu < \kappa$ and also suggest that we should be more careful and require something like $\alpha+n < \beta < \kappa$ for some $0 < n < \omega$. (Given the kind of proof I have for the edited question below, the exact value of $n$ is then easily calculated.)

I think that this exercise contains a typo and should read $$\{ \mu < \kappa \mid \mu \text{ is } (\mu+\alpha) \text{-strong} \}$$ instead. I will ask my professor and in the case that this is what he was asking for, I will tidy up my post and provide an answer for future reference.

• At which level $V_\alpha$ can we find an extender witnessing that there is an embedding $j:V\to M$ with critical point $\kappa$ and such that $V_{\kappa+\tau}\subset M$? – Andrés E. Caicedo Jul 4 '15 at 19:23
• @Andres Such an extender is an element of $V_{\kappa + \tau +1}$ and an argument similar to the one above yields that $\{ \mu < \kappa \mid \mu \text{ is } (\mu+\alpha) \text{-strong} \}$ has size $\kappa$. I still fail to see how I can improve this result to $\kappa$ many $(\kappa+\alpha)$-strong cardinals below $\kappa$. – Stefan Mesken Jul 4 '15 at 19:38
• Actually, it may be $V_{\kappa+\tau+3}$... (using flat pairing functions it seems that we could get away with $+2$ at the end)? – Stefan Mesken Jul 4 '15 at 19:42
• This is false, notice that when $\kappa$ is measurable, then it is $(\kappa+1)$-strong as we can get a transitive class $M$ and an elementary embedding $j:V\rightarrow M$ with $crit(j)=\kappa$ and $M^\kappa\subseteq M$. If what you're trying to prove was true, you would always get $\kappa$ many measurables below any measurable $\kappa$. – Camilo Arosemena-Serrato Jul 5 '15 at 14:29
• @Camilo Yeah, I already figured that, but I somehow managed to delete the relevant part of a previous edit where I said that for this reason we should be more carefully and require $\alpha + 1 < \beta < \kappa$ (or maybe even a little more, but this will fall out of a careful observation of a proof). – Stefan Mesken Jul 5 '15 at 14:36