Is the Halting probability a Dedekind real number?

In a recent 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 paper we show, firstly, that the concepts underlying their thesis can, however, be interpreted constructively; and, secondly, that an implicit thesis in the authors' arguments implies that the probability of a given Turing machine halting on a given input cannot be expressed as a Dedekind real number.