nLab neutral non-well-founded set theory

Idea

Neutral non-well-founded set theory is set theory without any of the foundation-like axioms, including

• membership induction

• the axiom of regularity

• transitive closure

• Mostowski's principle

• the axiom of well-founded materialization

but also without any axioms which contradict the axioms above, such as the axiom of anti-foundation, or any weaker versions of the above axioms.

The axiom of choice is not in general accepted here because it implies the axiom of well-founded materialization in general, as well as Mostowski's principle for material set theories.

Important examples of neutral non-well-founded set theory include

• $ZF^{\circlearrowleft}$ and its structural counterpart SEAR,

• $IBZ^{\circlearrowleft}$ and its structural counterparts intuitionistic bounded SEAR and intuitionistic ETCS.

Here the superscript “$\scriptsize{\circlearrowleft}$” probably means “omit the axiom of foundation”, and “$ZF$” refers to Zermelo-Fraenkel set theory.

See also

• neutral constructive mathematics (for the equivalent with excluded middle)

 References

The foundation-like axioms are given in section 2 of

• Michael Shulman (2018). Comparing material and structural set theories. arXiv:1808.05204.

The set theory in this textbook does not have the axiom of regularity, and is thus equivalent to $ZFC^{\circlearrowleft}$. However, it isn’t fully neutral, since Mostowski’s principle still holds.

• Paul Halmos, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960.