Generic I0 at $\aleph_\omega$
  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 J\'{o}nsson, and in a choiceless inner model many properties hold that are in contrast with PCF in ZFC. 
Vincenzo Dimonte
math.LO