Lectures on Constructive Mathematics. - Mathematics, Algorithms ...

The upper order locatedness cannot be dropped from the hypotheses of the
constructive least-upper-bound principle.
Consider any statement P . The set
S f0g [ fx 2 R : x = 1 ^ (P _ :P )g
is inhabited by 0 and bounded above by 1. Suppose that sup S exists.
Then 6 1. If < 1, then : (P _ :P ), which is absurd. Hence = 1 and
there exists s 2 S with s > 1=2. It follows that
so P _ :P .
s 2 fx 2 R : x = 1 ^ (P _ :P )g ;
From now on, we drop boldface notation for real numbers.