Combining Determinism and Indeterminism

Our goal is to construct mathematical operations that combine indeterminism measured from quantum randomness with computational determinism so that non-mechanistic behavior is preserved in the computation. Formally, some results about operations applied to computably enumerable (c.e.) and bi-immune sets are proven here, where the objective is for the operations to preserve bi-immunity. While developing rearrangement operations on the natural numbers, we discovered that the bi-immune rearrangements generate an uncountable subgroup of the infinite symmetric group (Sym$(\mathbb{N})$) on the natural numbers $\mathbb{N}$. This new uncountable subgroup is called the bi-immune symmetric group. We show that the bi-immune symmetric group contains the finitary symmetric group on the natural numbers, and consequently is highly transitive. Furthermore, the bi-immune symmetric group is dense in Sym$(\mathbb{N})$ with respect to the pointwise convergence topology. The complete structure of the bi-immune symmetric group and its subgroups generated by one or more bi-immune rearrangements is unknown.