# I am learning about non-well-founded sets and I wanted to know what books everyone recommends.

I am currently studying "A Course in Model Theory" by Bruno Poizat and I am doing research with altering the set of set theory's axioms. I want to further learn about the effect of the axiom of regularity and non-well-founded sets. I have already had someone recommend Peter Aczel's "Non-Well-Founded Sets" and I was curious what books on non-well-founded set theory anyone would recommend.

• Does Poizat actually cover the axioms of set theory in A Course in Model Theory? I promise you that the choice of whether to include regularity among the axioms of set theory has no impact whatsoever on the theorems of model theory covered in Poizat's book. Commented Aug 15 at 1:16
• @AlexKruckman I'm sorry, I worded this poorly and edited the question to better reflect what I am trying to say. I am studying model theory as part of my own research at school and I wanted book recommendations for non-well-founded sets. Commented Aug 15 at 4:00

Another book on non-well-founded set theory is Vicious Circles by Barwise and Moss. It also includes material on applications of non-well-founded sets in computer science and philosophy.

Some entry-points:

1. Start with Lawrence S. Moss, ‘Non-wellfounded set theory’, The Stanford Encyclopedia of Philosophy, tinyurl.com/sep-zfa.

2. Keith Devlin, The Joy of Sets (Springer, 2nd edn. 1993), Ch. 7. The last chapter of Devlin’s book, added in the second edition, starts with a very lucid introduction, and develops some of the theory.

3. Peter Aczel, Non-well-founded Sets (CSLI Lecture Notes 1988). This is, as no doubt others have mentioned, a very readable short classic book. It is available at tinyurl.com/aczel.

4. Luca Incurvati, ‘The graph conception of set’ Journal of Philosophical Logic (2014) pp. 181-208, or his Conceptions of Set and the Foundations of Mathematics (CUP, 2020), Ch. 7, very illuminatingly explores the motivation for such set theories.

• Fantastic. Thank you. Commented Aug 15 at 13:54