Unlike the axiom of foundation, this statement admits no non-standard interpretations.
2.
As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation.
3.
This 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy.
4.
In Zermelo Fraenkel set theory the collection of all sets is not a set; this follows from the axiom of foundation.
5.
By the axiom of foundation, the restriction of the membership relation to the transitive closure of " A " is a well-founded relation.
6.
Although Mirimanoff also introduced a notion of isomorphism between possibly non-well-founded sets, he considered neither an axiom of foundation nor of anti-foundation.
7.
The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel.
8.
Zermelo himself incorporated Fraenkel's axiom in his revised system he published in 1930, which also included as a new axiom von Neumann's axiom of foundation.
9.
*PM : well-foundedness and axiom of foundation, id = 9800 new !-- WP guess : well-foundedness and axiom of foundation-- Status:
10.
*PM : well-foundedness and axiom of foundation, id = 9800 new !-- WP guess : well-foundedness and axiom of foundation-- Status:
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