5
CONSTRUCTIVE CONTINUITY
1.4
of a weaker type. A Brouwerian counterexample shows that a given
proposition implies some omniscience principle whose nonconstructivity has
been firmly established. The following result, Theorem 16.5, shows that
LCP is equivalent to the Almost Separating Principle (ASP). This
principle seems quite nonconstructive, and is somewhat related to other
omniscience principles, but its nonconstructivity has not yet been
definitely established.
THEOREM. The following are equivalent.
(a) LCP: Every monotone real-valued function on the closed unit
interval which approximates intermediate values is continuous.
(b) ASP: If c is a real number such that for any real number x,
either x 0 is contradictory or x c is contradictory, then c 0.
The structure of ASP suggests that it is nonconstructive. From a mere
dichotomy of contradictories, it purports to derive quite affirmative
information, the construction of specific integers which demonstrate that
c is positive (see paragraph 3.2). And thus LCP appears nonconstructive.
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