Using non-well-founded models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.
12.
Quine atoms cannot exist in systems of set theory that include the axiom of regularity, but they can exist in non-well-founded set theory.
13.
Note that saying that the membership relation of some model of ZF is well-founded is stronger than saying that the axiom of regularity is true in the model.
14.
It can be carried out in Zermelo & ndash; Fraenkel set theory, without using the axiom of choice, but making essential use of the axiom of regularity.
15.
This principle, sometimes called the "'axiom of induction "'( in set theory ), is equivalent to the axiom of regularity given the other well-founded induction.
16.
Appending this schema, as well as the axiom of regularity ( first proposed by Dimitry Mirimanoff in 1917 ), to Zermelo set theory yields the theory denoted by "'ZF " '.
17.
So " M " satisfies the axiom of regularity ( it is " internally " well-founded ) but it is not well-founded and the collapse lemma does not apply to it.
18.
The axiom of induction in KP is stronger than the usual axiom of regularity ( which amounts to applying induction to the complement of a set ( the class of all sets not in the given set ) ).
19.
Given the other axioms of Zermelo Fraenkel set theory, the axiom of regularity is equivalent to the intuitionistic theories ( ones that do not accept the law of the excluded middle ), where the two axioms are not equivalent.
20.
ZF set theory with the axiom of regularity removed cannot prove that any non-well-founded sets exist ( or rather, this would mean ZF is inconsistent ), but it is compatible with the existence of Quine atoms.
