Abstract
Sacks [14] showed that every computably enumerable (c.e.) degree ≥ 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ 2 splitting above each proper predecessor (by ’splitting’ we understand ’nontrivial splitting’). Arslanov [1] showed that 0’ has a d.c.e. splitting above each c.e. a < 0’. On the other hand, Lachlan [9] proved the existence of a c.e. a > 0 which has no c.e. splitting above some proper c.e. predecessor, and Harrington [8] showed that one could take a = 0’. Splitting and nonsplitting techniques have had a number of consequences for definability and elementary equivalence in the degrees below 0’.
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Soskova, M.I., Cooper, S.B. (2007). The Strongest Nonsplitting Theorem. In: Cai, JY., Cooper, S.B., Zhu, H. (eds) Theory and Applications of Models of Computation. TAMC 2007. Lecture Notes in Computer Science, vol 4484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72504-6_18
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