Page "Constructivism (mathematics)" Paragraph 19

And in fact, Cantor's diagonal argument is constructive, in the sense that given a bijection between the real numbers and natural numbers, one constructs a real number which doesn't fit, and thereby proves a contradiction.

We can indeed enumerate algorithms to construct a function T, about which we initially assume that it is a function from the natural numbers onto the reals.

But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating ( T is a partial function ), so this fails to produce the required bijection.

In short, one who takes the view that real numbers are effectively computable interprets Cantor's result as showing that the real numbers are not recursively enumerable.