The axiom of regularity is of a restrictive nature as well.
2.
Sets are well-founded if the axiom of Regularity is assumed.
3.
The axioms do not include the Axiom of regularity and Axiom of replacement.
4.
:See our article at Axiom of regularity.
5.
The axiom of regularity as it is normally stated implies LEM, whereas the form of set induction does not.
6.
The axiom of regularity was introduced by; it was adopted in a formulation closer to the one found in contemporary textbooks by.
7.
The axiom of regularity was also shown to be independent from the other axioms of ZF ( C ), assuming they are consistent.
8.
The axiom of regularity, which is one of the axioms of Zermelo Fraenkel set theory, asserts that all sets are well-founded.
9.
In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.
10.
The hereditarily finite sets, V ?, satisfy the axiom of regularity ( and all other axioms of ZFC except the axiom of infinity ).
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