Abstract
A real number is computable if it is the limit of an effectively converging computable sequence of rational numbers, and left (right) computable if it is the supremum (infimum) of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable (multiple) sequences of rational numbers we introduce a non-collapsing hierarchy Σ n , Π n , Δ n : n ∈ ℕ of real numbers. We characterize the classes Σ 2 Π 2 and Δ 2 in various ways and give several interesting examples.
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References
K. Ambos-Spies. A note on recursively approximatable real numbers, Forschungsberichte Mathematische Logik, Mathematisches Institut, Universität Heidelberg. Nr. 38, 1998.
O. Demuth and P. Filipec. Differentiation of constructive functions of a real variable and relative computability, in Mathematical Logic and Its Applications, D. G. Skordev ed., Plenum Press, New York and Londin, 1987, pp 81–106.
H. S. Gaskill and P. P. Narayanaswami Elementary of Real Analysis, Prentice-Hall, Upper Saddle River, 1997.
A. Grzegorczyk. On the definitions of computable real continuous functions, Fund. Math. 44(1957), 61–71.
Chun-Kuen Ho. Relatively recursive real numbers and real functions, Technical Report of Department of Computer Science, University of Chicago 94-02. 1994.
L. Hudes. Hyperarithmetical real numbers and hyperarithmetical analysis, Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, Mass. 1962.
S. K. Kleene. Recursive predicates and quantifiers, Trans. Amer. Math. Soc. 53(1943), 41–73.
S. Mazur. Computable Analysis. PWN: Warsaw, 1963.
P. Odifreddi. Classical Recursion Theory, vol. 129 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdan, 1989.
M. Pour-El and J. Richards. Computability in Analysis and Physics. Springer-Verlag, Berlin, Heidelberg, 1989.
J. R. Shoenfiled. On degrees of unsolvability, Ann. Math. 69(1959) 644–653.
R. Soare. Recursively Enumerable Sets and Degrees, Springer-Verlag, Berlin, Heidelberg, 1987.
E. Specker. Nicht konstruktiv beweisbare Sätze der Analysis, J. of Symbolic Logic 14(1949), 145–158.
K. Weihrauch. Computability. EATCS Monographs on Theoretical Computer Science Vol. 9, Springer-Verlag, Berlin, Heidelberg, 1987.
K. Weihrauch. An Introduction to Computable Analysis. (In Preparation.)
K. Weihrauch and X. Zheng. A finite hierarchy of recursively enumerable real numbers, MFCS’98, Brno, Czech Republic, August 24–28, 1998.
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Zheng, X., Weihrauch, K. (1999). The Arithmetical Hierarchy of Real Numbers. In: Kutyłowski, M., Pacholski, L., Wierzbicki, T. (eds) Mathematical Foundations of Computer Science 1999. MFCS 1999. Lecture Notes in Computer Science, vol 1672. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48340-3_3
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DOI: https://doi.org/10.1007/3-540-48340-3_3
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