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 $\textrm{Ord}$.


1 Answer 1


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.


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