An axiomatic approach to higher order set theory

Higher order set theory has been a topic of interest for some time, with recent efforts focused on the strength of second order set theories [KW16]. In this paper we strive to present one 'theory of collections' that allows for a formal consideration of 'countable higher order set theory'. We will see that this theory is equiconsistent with $ZFC$ plus the existence of a countable collection of inaccessible cardinals. We will also see that this theory serves as a canonical foundation for some parts of mathematics not covered by standard set/class theories (e.g. $ZFC$ or $MK$), such as category theory.