Dana Scott and Peter Aczel name statements called “anti-[this axiom]” that replace this axiom in some alternate set theories. For 10 points each:
[10h] Name this axiom of ZF set theory, which states that for every non-empty set A, there exists an element of A that is disjoint from A. It implies that there does not exist infinite descending chains.
ANSWER: axiom of regularity [accept axiom of foundation]
[10e] Regularity applied to the singleton set containing only A implies that A does not contain itself, thus helping avoid this paradox named after a British philosopher.
ANSWER: Russell’s paradox [accept Bertrand Russell]
[10m] Regularity is equivalent to every set appearing in the Von Neumann universe, which is constructed inductively by applying unions and this operation. By Cantor’s theorem, applying this operation on a set strictly increases cardinality.
ANSWER: powerset
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