Let us fix some mathematical theory T=T(0), such as ZFC. The aim is to develop algorithm A, which takes any statement S independent of T(0) as an input, and outputs axiom a(S) such that T(1)=T(0)+a(S) is consistent if T(0) is consistent, and either S or negation of S can be proved in T(1). Then algorithm can take another statement S' independent of T(1), and proceed iteratively, forming infinite sequence of theories, such that theory T(n) decides the first n statements given as inputs.
One simple example of A is a randomised algorithm such that a(S) is either S or negation of S with probabilities 0.5.
My question is whether there exists a version of algorithm A witch is (a) deterministic, and (b) the way how it decides every statement S does not depend on the order in which statements are arriving.
An example how the answer may be positive is a restricted setting when T(0)=ZFC, and all statements S are in the form "This particular Diophantine equation has a solution". Then the algorithm always returning a(S)="negation of S" does the job.
A positive answer to this question in general would imply a way to decide all provably independent statements in set theory one way or another in a consistent way.