User blog comment:MachineGunSuper/A world without the infinite/@comment-36957202-20181208204238/@comment-34422464-20181209151650

Actually, most googology is ZFC plus a few other axioms. https://cdn.discordapp.com/attachments/429744905661841419/512393868830965760/table-usual-theories.pdf Here is a few. This table was made by Julian#2338. The second page ranks the set theories in terms of consistency strength. The consistency strength is really amazing. Notice that at the last 3 rows, it is ZF rather than ZFC. That means that at that stage, the axiom of choice becomes inconsistent.