The Set of Real Functions are Countable in Applied Mathematics, Algebra of the Functions and Their Classification | Chapter 8 | Current Topics on Mathematics and Computer Science Vol. 7

Because the functions must be computable, that is, there must be a method for computing them, the set of real functions is countable. However, the number of algorithms may be counted. Uncomputable functions are useless in applied mathematics; they don't exist. The set of countable real numbers is also a set of computable real numbers. Numbers that cannot be computed are useless. A classification of subalgebras with one-element bases is established, as well as a definition of algebra of computable real functions. This classification is also a function classification. Multielement bases algebras are fictitious, and thus are useless for classifying functions. Infinite sequences of subalgebra inclusions are built.

Author (S) Details

M. A. Malkov
Russian Research Center for Artificial Intelligence, Russia.

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