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Categoricity properties for computable algebraic fields

“…Lemma 3.3 shows (4) =⇒ (1), and its proof also explained how (4) yields (3), which in turn clearly implies (2). So we now prove (2) =⇒ (5). Fix any computable chain with union F , and fix a Σ 0 1 Scott family S for F .…”

“…On the other hand, although relative computable categoricity clearly implies computable categoricity, it was established independently by Khoussainov and Shore in [19] and by Kudinov in [21] that the two concepts are not equivalent. More recently, Downey, Hirschfeldt, and Khoussainov showed in [5] that relative computable categoricity can be viewed as a kind of uniform version of computable categoricity, although this fact was already implicit in work of Ventsov [40].…”