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We study abstract notations for natural numbers, which may differ from

standard notations, so that the functions and relations classically

uncomputable become computable, and vice versa. This question was

raised by a philosopher Stewart Shapiro in course of the debate on

Church-Turing thesis.

We discover that in some notations the standard ordering of natural

numbers is computable, but successor is not; or multiplication is

computable, whereas addition and exponentiation are not. Apart from

surprising counter-examples, we search for a general theory,

explaining how computability of some functions and relations may

induce computability of some others.

How computability depends on notation?