TITLE: Set theory and alternative foundations
QUESTION [12 upvotes]: Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set theory? Is there any alternative foundation which is not a set-theory?

REPLY [3 votes]: This may not satisfy the request for something that is "not, in any sense, a set theory" but Oliver Deiser has worked out two versions of foundations, one based on lists and one on multisets.  This is in his book "Orte, Listen, Aggregate" (and his Habilitationsschrift with the same title).