TITLE: What is the strength of chains of 1-extendibles?
QUESTION [12 upvotes]: Let $X$ be a collection of cardinals such that if $\kappa,\lambda\in X$ and $\kappa<\lambda$, then there is a non-trivial elementary embedding $j:V_{\kappa+1} \to V_{\lambda+1}$ with $crit(j) = \kappa$. 
What is the consistency strength of the claim that there are such $X$ for various sizes of $X$? In particular, what is the consistency strength when $X$ is a proper class of cardinals?

REPLY [8 votes]: I think for the Theorem one can get by with a subcompact cardinal, which is weaker than a $\kappa^+$-supercompact.
( $\kappa$ is subcompact if for any $A\subseteq H_{\kappa^+}$ there are $j,\mu < \kappa, B\subseteq H_{\mu^+}$ with $j:(H_{\mu^+},\in, B) \longrightarrow_e (H_{\kappa^+},\in, A)$. ( where $ \longrightarrow_e$ means ``is an elementary map''. ) )
Let $\kappa$ be subcompact as above, and let $A = \mathop{Sat} =
{Sat}_{\kappa^+}$  Then $\mathop{Sat} \subseteq H_{\kappa^+}$, and
applying the definition of subcompactness there are $\mu < \kappa$, $j$,
and $\overline{\mathop{Sat}}$ with
$j : (H_{\mu^+}, \overline{\mathop{Sat}}) \longrightarrow_e (H_{\kappa^+},
\mathop{Sat})$.
Then $\overline{\mathop{Sat}}= {Sat}_{\mu^+}$ can be checked easily. $j$ is easily in $H_{\kappa^+}$ but the point of having the Sat predicate is that we can express in a first order way (in the extended language) that there is a $j$ with $ran(j)$ an elementary substructure of $(H_{\kappa^+},\in)$. 
Thus $(H_{\kappa^+}, {Sat}) \models$``${ran} (j) \prec_e (V, \in)$'' indeed:
$(H_{\kappa^+}, {Sat}) \models$``There is $k, \kappa_0,$ with $k
: (H_{\kappa_0^+}, \in) \longrightarrow (V, \in)$ and ${ran} (k)
\prec_e (V, \in) ,$ and thus $k$ is an elementary map.''
This goes down to  $(H_{\mu^+}, {Sat}_{\mu^+})$. This starts out links in the chain, using the restriction of the map(s) $k$ to the domain $V_{\kappa_0 +1}$ etc. and the rest of the argument is as before.