[1][2][3] The theory of non-well-founded sets has been applied in the logical modelling of non-terminating computational processes in computer science (process algebra and final semantics), linguistics and natural language semantics (situation theory), philosophy (work on the Liar Paradox), and in a different setting, non-standard analysis.
After Zermelo adopted Foundation into his own system in 1930 (from previous work of von Neumann 1925–1929) interest in non-well-founded sets waned for decades.
Then in 1957 Rieger's theorem was published, which gave a general method for such proof to be carried out, rekindling some interest in non-well-founded axiomatic systems.
A more recent approach to non-well-founded set theory, pioneered by M. Forti and F. Honsell in the 1980s, borrows from computer science the concept of a bisimulation.
Four mutually independent anti-foundation axioms are well-known, sometimes abbreviated by the first letter in the following list: They essentially correspond to four different notions of equality for non-well-founded sets.
In published research, non-well-founded sets are also called hypersets, in parallel to the hyperreal numbers of nonstandard analysis.