Target of a superstrong embedding

Question

Let $j\colon V \to M$ be a superstrong elementary embedding (i.e. $M$ is transitive, $j\neq id$, and $V_{j(\kappa)} \subseteq M$ where $\kappa$ is the critical point of $j$).

Is $j(\kappa)$ necessarily inaccessible in $V$?

Answer 1

The answer seems to be negative. In fact, more is true:

Claim: Let $j\colon V\to M$ be a superstrong embedding with critical point $\kappa$ and target $\lambda$ (i.e. $j(\kappa) = \lambda$). If $\lambda$ is regular then there is a club $C \subseteq \lambda$ such that for all $\mu \in C$, there is a superstrong embedding with critical point $\kappa$ and target $\mu$. In particular, the first superstrong cardinal has a singular target.

Proof: Let $E = \langle E_\alpha \mid \alpha < \lambda\rangle$ be an extender that witnesses the superstrength of $\kappa$ with target $\lambda$. For every $x\in V_\lambda$ there is a finite sequence $\eta \in \lambda^{<\omega}$ and a function $f\colon \kappa\to V_\kappa$, such that $j_E(f)(\eta) = x$.

Let $C$ be the set of all $\beta < \lambda$, such that:

• For all $\eta \in \beta^{<\omega}$ and $f\colon \kappa \to \kappa$, $j_E(f)(\eta) < \beta$.
• For all $x\in V_\beta$ there is $\eta \in \beta^{<\omega}$ and $f\colon \kappa \to V_\kappa$ such that $j_E(f)(\eta) = x$.

$C$ is a club as the set of closure points of $2^\kappa < \lambda$ many functions.

I claim that for every $\beta\in C$ the elementary embedding using the extender $E \restriction \beta$ is a superstrong embedding with target $\beta$. Indeed, every $x\in V_\beta$ is represented in this extender by a function from $\kappa$ to $V_\kappa$ and a generator $\eta\in\beta^{<\omega}$. Clearly, $j_{E\restriction \beta}(\kappa) \geq \beta$. If $j_{E\restriction \beta}(\kappa) > \beta$ then there is a function $f\colon \kappa \to \kappa$ and $\eta \in \beta^{<\omega}$ such that $j_{E\restriction \beta}(f)(\eta) = \beta$. But $j_E(f)(\eta) \geq j_{E\restriction \beta}(f)(\eta) = \beta$, which contradicts the assumption that $\beta \in C$. Thus, we conclude that $\beta = j_{E\restriction \beta}(\kappa)$, as wanted.

Edit: Apparently, for any superstrong cardinal $\kappa$, the minimal target has cofinality between $\kappa^{+}$ and $2^\kappa$ (so under GCH - it is always $\kappa^{+}$). This follows from Theorem 3.3 in this paper of Perlmutter. The idea is that if $j\colon V \to M$ is a superstrong embedding with critical point $\kappa$ and $\theta = \sup_{f\colon \kappa \to \kappa} j(f)(\kappa)$, then there is a superstrong embedding with critical point $\kappa$ and target $\theta$. If the cofinality of $j(\kappa)$ is below $\kappa$ or above $2^\kappa$, it is possible to deduce that $\theta < j(\kappa)$.