I. Abstract
The Interval Sieve Algorithm is a method for generating a list of real numbers on any closed interval of real numbers [ri, rj] where ri < rj. Georg Cantor in his 1891 paper demonstrated a constructive proof that the real numbers are uncountable. Cantor developed a method for showing how a particular objective cannot be accomplished; in this case, establishing a one to one correspondence between the natural numbers and the real numbers, using his diagonal argument.
What Cantor didn’t show is that there are no ways of demonstrating a one to one correspondence between the natural numbers and the real numbers. This is important because even if one can demonstrate one or more ways that something cannot be done, it is only necessary to develop one way that shows how it can be done to invalidate the ways that show it cannot.
The interval sieve algorithm partitions a closed interval of real numbers [ri, rj] where ri < rj to create a complete list, L, of numbers in the interval. We will prove that the list L is complete, and derive the bijective function f : ℕ → [r1, r2].