 NSFPAR ID:
 10325699
 Date Published:
 Journal Name:
 Science Advances
 Volume:
 8
 Issue:
 12
 ISSN:
 23752548
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Brattka, Vasco ; Greenberg, Noam ; Kalimullin, Iskander ; Soskova, Mariya (Ed.)In her 1990 thesis, Ahmad showed that there is a socalled “Ahmad pair”, i.e., there are incomparable Σ 2 0 enumeration degrees a 0 and a 1 such that every enumeration degree x < a 0 is ⩽ a 1 . At the same time, she also showed that there is no “symmetric Ahmad pair”, i.e., there are no incomparable Σ 2 0 enumeration degrees a 0 and a 1 such that every enumeration degree x 0 < a 0 is ⩽ a 1 and such that every enumeration degree x 1 < a 1 is ⩽ a 0 . In this paper, we first present a direct proof of Ahmad’s second result. We then show that her first result cannot be extended to an “Ahmad triple”, i.e., there are no Σ 2 0 enumeration degrees a 0 , a 1 and a 2 such that both ( a 0 , a 1 ) and ( a 1 , a 2 ) are an Ahmad pair. On the other hand, there is a “weak Ahmad triple”, i.e., there are pairwise incomparable Σ 2 0 enumeration degrees a 0 , a 1 and a 2 such that every enumeration degree x < a 0 is also ⩽ a 1 or ⩽ a 2 ; however neither ( a 0 , a 1 ) nor ( a 0 , a 2 ) is an Ahmad pair.more » « less

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