Question
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
<Other Science>
Summary
| 2024 ACF Regionals @ Nebraska | 01/27/2024 | Y | 5 | 10.00 | 80% | 20% | 0% |
| 2024 ACF Regionals @ Ohio State | 01/27/2024 | Y | 2 | 5.00 | 50% | 0% | 0% |
| 2024 ACF Regionals @ Imperial | 01/27/2024 | Y | 8 | 15.00 | 88% | 63% | 0% |
| 2024 ACF Regionals @ Vanderbilt | 01/27/2024 | Y | 1 | 10.00 | 100% | 0% | 0% |
| 2024 ACF Regionals @ MIT | 01/27/2024 | Y | 3 | 13.33 | 100% | 33% | 0% |