On the hierarchies of ${\Delta }_{2}^{0}$-real numbers
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 1, pp. 3-25.

A real number $x$ is called ${\Delta }_{2}^{0}$ if its binary expansion corresponds to a ${\Delta }_{2}^{0}$-set of natural numbers. Such reals are just the limits of computable sequences of rational numbers and hence also called computably approximable. Depending on how fast the sequences converge, ${\Delta }_{2}^{0}$-reals have different levels of effectiveness. This leads to various hierarchies of ${\Delta }_{2}^{0}$ reals. In this survey paper we summarize several recent developments related to such kind of hierarchies shown by the author and his collaborators.

DOI : https://doi.org/10.1051/ita:2007008
Classification : 03D55,  26E40,  68Q15
Mots clés : computably approximable reals, ${\Delta }_{2}^{0}$-reals, hierarchy
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Zheng, Xizhong. On the hierarchies of $\Delta ^0_2$-real numbers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 1, pp. 3-25. doi : 10.1051/ita:2007008. http://www.numdam.org/articles/10.1051/ita:2007008/

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