Let's call a cardinal $\delta$ an $\text{I1}$-tower cardinal if for each $A\subseteq V\_{\delta}$, there exists a $\kappa<\delta$ such that whenever $\kappa<\alpha<\delta$ there is some $\lambda<\delta$ and a $j:V\_{\lambda+1}\rightarrow V\_{\lambda+1}$ such that $\text{crit}(j)=\kappa$ and $j(\kappa)>\alpha$ and $j(V\_{\lambda}\cap A)=V\_{\lambda}\cap A$.


Is the existence of an $\text{I1}$-tower cardinal consistent? If so, then where do the $\text{I1}$-tower cardinals stand on the large cardinal hierarchy? Is the existence of an $\text{I1}$-cardinal implied by the existence of an $\text{I0}$-cardinal? Does $\text{Con(I0)}$ imply $\text{Con}(\text{there is an I1-tower cardinal})$? Does $\text{Con}(\text{there is an I1-tower cardinal})$ imply $\text{Con(I0)}$?



Where does this strengthening of I1 stand?
Suppose $\delta$ is an ordinal. We first note that $\delta$ is an $I\_1$-tower cardinal if and only if it has the following superficially weaker property: for all $X\subseteq V\_\delta$ there is some $\kappa < \delta$ such that for arbitrarily large $\lambda < \delta$, there is an elementary embedding $j:V\_{\lambda+1}\to V\_{\lambda+1}$ with critical point $\kappa$ fixing $X\cap V\_\lambda$. (This is because if there is an embedding $j:V\_{\lambda+1}\to V\_{\lambda+1}$ with critical point $\kappa$, there is an embedding $j':V\_{\lambda+1}\to V\_{\lambda+1}$ with critical point $\kappa$ that sends $\kappa$ arbitrarily high below $\lambda$ and fixes anything that $j$ fixes: just let $j' = j\circ j \circ \cdots\circ j$.)


Suppose $\delta$ is the critical point of an embedding $j: L\_1(V\_{\lambda+1})\to L\_1(V\_{\lambda+1})$. We show $\delta$ is an $I\_1$-tower cardinal. Fix $X\subseteq V\_\delta$. Let $j\_{0\omega}: V\_{\lambda+1}\to M\_\omega$ denote the $\omega$-th iterate of $j$. Let $X\_\* = j\_{0\omega}(X)$. A standard fact is that $j$ fixes $X\_\*$: $$j(X\_\*) = j(j\_{0\omega}(X)) = j\_{1\omega}(j(X)) = j\_{0\omega}(X) = X\_\*$$ We make the following claim: there exist arbitrarily large $\bar \lambda < \lambda$ such that there is an elementary embedding $\bar j : V\_{\bar \lambda+1}\to V\_{\bar \lambda+1}$ with critical point $\delta$ fixing $X\_\*\cap V\_{\bar \lambda}$. This is proved by Laver's inverse limit reflection technique, but assume it is true for the moment and let us conclude the proof. Since $V\_\lambda \subseteq M\_\omega$, the claim is absolute to $M\_\omega$. Pulling the claim back from $M\_\omega$ to $V\_{\lambda+1}$ through $j\_{0\omega}$, and noting that $j\_{0\omega}(\delta) = \lambda$ and $j\_{0\omega}(X) = X\_\*$, there is some $\kappa < \delta$ such that for arbitrarily large $\bar \lambda < \delta$, there is an elementary embedding $\bar j: V\_{\bar \lambda +1 }\to V\_{\bar \lambda +1 }$ with critical point $\kappa$ fixing $X\cap V\_{\bar \lambda}$. This shows that $\delta$ has the superficially weakened $I\_1$-tower property with respect to $X$, as desired.


We now prove the claim. Fix an arbitrary $\alpha < \lambda$, and assume for ease of notation that $\alpha > \delta$. We must find an ordinal $\bar \lambda\in (\alpha,\lambda]$ and an elementary embedding $\bar j : V\_{\bar \lambda+1}\to V\_{\bar \lambda+1}$ fixing $X\_\*\cap V\_{\bar \lambda}$ with critical point $\delta$. Inverse limit reflection yields an ordinal $\bar \lambda\in (\alpha, \lambda]$ and an elementary embedding $J:V\_{\bar \lambda + 1}\to V\_{\lambda+1}$ with critical point above $\alpha$ sending $X\_\*\cap V\_{\bar \lambda}$ to $X\_\*$ with $j\restriction V\_\lambda$ in the range of $J$. The only part of this that is not a clause in the standard inverse limit lemma is that we can find $J$ sending $X\_\*\cap V\_{\bar \lambda}$ to $X\_\*$, so we indicate how this part is achieved. We first iterate $j$ finitely many times to obtain an embedding $j\_n : L\_1(V\_{\lambda+1})\to L\_1(V\_{\lambda+1})$ with critical point above $\alpha$, in order to ensure that the critical point of $J$ is above $\alpha$, though we will ignore this part of the construction. Of course $j\_n$ fixes $X\_\*$ since $j$ does. We build $J$ as an inverse limit of embeddings $\langle k\_i: i < \omega\rangle$, choosing the component embeddings $k\_i:V\_{\lambda+1}\to V\_{\lambda+1}$ to be square roots of $j\_n$ that also fix $X\_\*$, which is possible by the square root lemma. (To obtain square roots, we use the assumption that $j$, and hence $j\_n$, is stronger than just an $I\_1$-embedding.) Letting $J = k\_0\circ k\_1\circ k\_2\circ \cdots$, we have $k\_0\circ\cdots \circ k\_n(X\_\*\cap V\_{\text{crt}(k\_n)}) = X\_\*\cap V\_{k\_0\circ\cdots \circ k\_n(\text{crt}(k\_n))}$. Hence $J(X\_\*\cap V\_{\bar \lambda}) = X\_\*$ by the definition of an inverse limit. (For the rest of the details of this construction, see Laver's paper "Implications between strong large cardinal axioms.") Now let $\bar j = J^{-1}(j\restriction V\_\lambda)$. Since $J:V\_{\bar \lambda+1}\to V\_{\lambda+1}$ is elementary, $\bar j$ extends to an elementary embedding $V\_{\bar \lambda + 1} \to V\_{\bar \lambda + 1}$ with critical point $J^{-1}(\delta) = \delta$ fixing $J^{-1}(X\_\*) = X\_\*\cap V\_{\bar \lambda}$. This proves the claim.


