Weakly extendible cardinals and compactness of extended logics

TL;DR

This paper introduces weakly extendible cardinals, characterizes their relation to weak compactness in second order logic, and explores their consistency strength and connections to extended logics.

Contribution

It defines weakly extendible cardinals, relates them to weak compactness of extended logics, and situates their consistency strength between known large cardinal notions.

Findings

Weakly extendible cardinals are characterized by weak compactness in second order logic.

Their consistency strength lies between strongly unfoldable and strongly uplifting cardinals.

Under V=L, the weak compactness number of certain extended logics coincides with weakly extendible cardinals.

Abstract

We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located strictly between that of strongly unfoldable (i.e. shrewd) cardinals, and strongly uplifting cardinals. Weak compactness of many other logics can be connected to certain variants of the notion of weakly extendible cardinals. We also show that, under V=L, a cardinal $\kappa$ is the weak compactness number of ${\cal L}^{\aleph_0,II}_{stat,\kappa,\omega}$ if and only if it is the weak compactness number of ${\cal L}^{II}_{\kappa,\omega}$. The latter condition is equivalent to the condition that $\kappa$ is weakly extendible by the characterization mentioned above (this equivalence holds without the assumption of V=L).