Generic I0 at ${\aleph }_{\omega }$

Abstract

In this paper it is introduced a generic large cardinal akin to I0, and its consequences are analyzed in the case that ${\aleph }_{\omega }$ is such a generic large cardinal. In this case ${\aleph }_{\omega }$ is Jónsson, and in a choiceless inner model many properties hold that are in contrast with PCF in ZFC.