  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.

Logic;Category Theory