[FOM] a simple constructive example when AC is not true

[Kreinovich, Vladik]

A simple example re recent question:

In purely constructive approach, when existence is algorithmic and functions are computable, by computing a computable real number x with accuracy 0.1, we can efficiently check whether x > 0 or x < 1. Thus, we have Ex Av ((v = 0 & x > v) \/ (v = 1 & x < v)). However, it is impossible to have a computable function v(x) for which Ax ((v(x) = 0 & x > v(x)) \/ (v(x) = 1 & x < v(x)) since computable functions are continuous and thus, cannot jump from 0 to 1.
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