User blog:Sleepy Dragonfly/Recursion Level of Worldly Cardinals

Since there are worldly cardinals below the smallest inaccessible, does this mean that they can are reachable through the recursion of aleph numbers? If so, can they be expressed with collapsing functions, e.g. Taranovsky's C but collapsing inaccessibles to get alephs instead of collapsing omegas to get countables? This is a wild guess but given worldly cardinals are related to ZF, does it mean that the recursion level to get to worldlies from alephs would be similar to the recursion level of the proof-theoretic ordinal of ZF? I don't have a good understand of set theory so please help me understand! I've been trying to find out more about this and I'm surprised that worldly cardinals have never been mentioned on this website before.