 In research on category theory from the 1960s onward, several axiomatizations of specific categories have been proposed as alternative foundations for mathematics.  This includes the axiom systems describing the category of sets and functions, on the one hand, and the category of categories, on the other hand, as first presented in Lawvere (1964, 1966). Both were explicitly introduced as foundational systems, and thus, as alternatives to Zermelo-Fraenkel set theory. More recently, elementary topos theory has been developed as a form of categorical set theory that can serve as a foundational system in the above sense (cf.  Landry &amp; Marquis 2005, Marquis 2013).