## Properly $$\Sigma_2^0$$ enumeration degrees and the high/low hierarchy.(English)Zbl 1114.03034

In the paper under review the authors prove that every high enumeration degree bounds a noncuppable enumeration degree. Since every noncuppable enumeration degree is downwards properly $$\Sigma_2^0$$, it follows that there are downwards properly $$\Sigma_2^0$$ degrees that are not high. By definition, an enumeration degree $${\mathbf a}\leq_{\text{e}}{\mathbf 0}_{\text{e}}^{\prime}$$ is downwards properly $$\Sigma_2^0$$ if $${\mathbf a}\neq{\mathbf 0}_{\text{e}}$$ and any nontrivial enumeration degree below a is a properly $$\Sigma_2^0$$ enumeration degree.

### MSC:

 03D30 Other degrees and reducibilities in computability and recursion theory 03D55 Hierarchies of computability and definability

### Keywords:

enumeration reducibility; high degrees; low degrees
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### References:

 [1] Annals of Pure and Applied Logic 82 pp 317– (1997) [2] DOI: 10.1002/malq.19880340603 · Zbl 0667.03034 [3] Recursion Theory Week, Oberwolfach 1989 1432 pp 57– (1990) [4] The distribution of properly enumeration degrees 65 pp 19– (2000) [5] Complexity, logic and recursion theory pp 303– (1997) [6] Recursively enumerable sets and degrees (1987) [7] DOI: 10.1090/S0002-9939-1954-0063995-6 [8] DOI: 10.1090/S0002-9947-1963-0155747-3 [9] Structural properties and enumeration degrees 65 pp 285– (2000) [10] On minimal pairs of enumeration degrees 50 pp 983– (1985) · Zbl 0615.03031 [11] Archive for Mathematical Logic 42 (2003) [12] Archive for Mathematical Logic [13] Recursion theory and complexity pp 157– (1999)
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