  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.

Logic;Computational Complexity;Computation and Language;Group Theory