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.