User blog comment:Alemagno12/Huge Ordinal Analysis/@comment-1605058-20180203102941/@comment-1605058-20180207173517

ZFC is not the same as FOST. FOST is a language in which ZFC and many other theories (like KP or ZFC + large cardinals) are expressed.

Similarly, SOST is just a language, but just like there is ZFC for FOST, there is a "standard" theory for SOST, namely Morse-Kelley set theory (MK). This theory is stronger than ZFC, but weaker than ZFC + inacessible cardinal. Hence PTO(ZFC)<PTO(MK)<PTO(ZFC + inaccessible).