In recent years, since a great deal of circular phenomenon, there has been a furry of interest in them. To explain various circular phenomenon, the study of set theory extended well -founded sets to non-well-founded set. Based on this basis, the paper discusses the logical theoretical basis of circular phenomena. Non-well-founded set theory ZFA allows primitive existence. Primitive is an object that has no elements and is not a class in itself. It is based on the set theory ZFC after the axiom

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... f foundation FA is removed, and the anti-basic axiom AFA is added to ZFC. ZFC here refers to ZF set theory with axiom of choice. According to axiomatic set theory, for ZFC's regular axioms, the set in its universe is a well set. If the regular axiom is removed, and the infinite decline is allowed to belong to the relational chain, then the non-well-founded set can be introduced. Firstly, this paper introduces the basic concept of non-well-founded set, the foundation axiom and the anti-founded axioms. Secondly, we dicusses the limit of the foundation axiom. Thirdly, we exhibit the history and present situation of the research on non-well-founded sets are briefly reviewed. Finally, the applications of non-well-founded sets in philosophy, linguistics, computer science, economics and many other fields is discussed. Because non-well-founded set theory will provide a better tool for dealing with circular phenomena naturally, it can be argued that circle is not vicious.