Is the Halting probability a Dedekind real number?

In a historical overview, Cristian S. Calude, Elena Calude, and Solomon Marcus identify eight stages in the development of the concept of a mathematical proof in support of an ambitious conjecture: we can express classical mathematical concepts adequately only in a mathematical language in which both truth and provability are essentially unverifiable. In this essay we show, first, that the concepts underlying their thesis can, however, be interpreted constructively; and, second, that an implicit thesis in the authors’ arguments implies that the Halting problem is solvable, but that, despite this, the probability of a given Turing machine halting on a random input cannot be assumed to define a Dedekind real number.