Is the set of all real numbers whose decimal expansions are computed by a machine not countable?
Assume a digital machine not capable of computing infinite decimal places.
2026-03-30 06:09:20.1774850960
Countability of computable real numbers
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For any programming language (Turing-complete or not) where programs consist of a finite (but unbounded) number of symbols taken from a finite alphabet, the number of possible programs is countable.
Any computable number can be defined by a program P that calculates the number accurate to within n decimal (or binary) places, where n is an integer provided by the user. Since P and n are both part of countable sets, the possible outputs of $(P, n)$ pairs also forms a countable set.
Therefore, the number of computable numbers is countable.