# How do chains of elementary extensions compare to shrewdness?

I thought of the following large cardinal axiom, extending the notion of $$\theta$$-upliftingness:

Let $$\eta$$ be be an ordinal, and $$X$$ be a class of ordinals. $$\kappa$$ is called $$\eta$$-iteratively uplifting onto $$X$$ iff, for every ordinal $$\theta$$, there is a monotonically increasing sequence $$(\gamma_i)_{0 \leq i \leq 1 + \eta}$$ such that: (1) $$\gamma_0 = \kappa$$. (2) $$\gamma_1 > \theta$$ (3) For every $$0 \leq i \leq 1 + \eta$$, $$\gamma_i \in X$$ (3) For every $$0 \leq i < j \leq 1 + \eta$$, $$(V_{\gamma_i}, \in) \prec (V_{\gamma_j}, \in)$$ is a proper elementary extension.

It's easy to see that $$\kappa$$ is pseudo-uplifting iff it is $$0$$-iteratively uplifting onto $$\textrm{Ord}$$. Now, I'm wondering, how does $$\eta$$-iterative upliftingness (onto $$\textrm{Ord}$$) compare to $$\eta$$-shrewdness in the hierarchy of large cardinal consistency strength? Recall that $$\eta$$-shrewdness is defined like so:

Let $$\eta$$ be an ordinal. $$\kappa$$ is called $$\eta$$-shrewd iff, for every formula $$\varphi$$ and set $$A \subseteq V_\kappa$$ such that $$(V_{\kappa+\eta}, \in, A) \vDash \varphi$$, there exist $$\bar{\kappa}, \bar{\eta} < \kappa$$ such that $$(V_{\bar{\kappa}+\bar{\eta}}, \in, A \cap V_{\bar{\kappa}}) \vDash \varphi$$.

EDIT: This problem has been solved. It turns out that, for every $$\eta$$, the existence of a 1-shrewd cardinal above $$\eta$$ implies the consistency of a cardinal $$\kappa$$ which is $$\eta$$-iteratively uplifting on $$\textrm{Ord}$$.

As Cantor's Attic explains, if $$\kappa$$ is 0-uplifting, that is, there is a cardinal $$\lambda \gt \kappa$$ such that $$V_\kappa \prec V_\lambda$$ and $$\lambda$$ is inaccesible, $$\lambda$$ has a club subset $$C$$ of cardinals such that $$V_\kappa \prec V_{\gamma_1} \prec V_{\gamma_2} \prec... \prec V_\lambda$$ for all $$\gamma_1, \gamma_2... \in C$$. Thus $$\langle V_\lambda, C \rangle \vDash \text{\kappa is Ord-iteratively uplifting onto Ord}$$. As every Mahlo cardinal is a limit of 0-uplifting cardinals, the strength of these properties are below the bottom of the shrewdness hierarchy.