  It is well known that the R, the set of real numbers, is an abstract set,
where almost all its elements cannot be described in any finite language.
  We investigate possible approaches to what might be called an
epi-constructionist approach to mathematics. While most constructive
mathematics is concerned with constructive proofs, the agenda here is that the
objects that we study, specifically the class of numbers that we study, should
be an enumerable set of finite symbol strings. These might also be called
decidable constructive real numbers, that is our class of numbers should be a
computable sets of explicitly represented computable numbers.
  There have been various investigations of the computable numbers going back
to Turing. Most are however not expressed constructively, rather computable is
a property assigned to some of the abstract real numbers. Other definitions
define constructive real numbers without reference to the abstract R, but the
construction is undecidable, i.e., we cannot determine if a given construction
represents a computable real number or not. For example, we may define a real
as a computable convergent sequence of rationals, but cannot in general decide
if a given computable sequence is convergent.
  This paper explores several specific classes of decidable constructive real
numbers that could form foundational objects for what we might call an
epi-constructionist mathematics.
