before_revision: 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 operations 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  } on the natural numbers . } We show that this new subgroup contains the bounded symmetric group on the natural numbers, and consequently is highly transitive.  } The complete structure of this new subgroup and its subgroups generated by one or more bi-immune rearrangements is unknown.
after_revision: 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.
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