 Here \(A(n)\) is a decidable property for which \(\forall n A(n)\) is not known to be true or false. Decidability means that at present for any given \(n\) there exists (can be constructed) a proof of \(A(n)\) or of \(\neg A(n)\). At the time of this writing, we could for example let \(A(n)\) express that \(n\), if greater than 2, is the sum of three primes; \(\forall n A(n)\) then expresses the (original) Goldbach conjecture that every number greater than 2 is the sum of three primes. The sequence \(\langle r_n \rangle\) defines a real number \(r\) for which the statement \(r=0\) is equivalent to the statement \(\forall n A(n)\). It follows that the statement \((r = 0 \vee r \neq 0)\) does not hold, and therefore that the law of trichotomy \(\forall x(x \lt y \vee x=y \vee x \gt y)\) is not true on the intuitionistic continuum.