Combining Determinism and Indeterminism
This work addresses foundational issues in mathematical logic and computation theory, offering incremental insights into the structure of bi-immune sets and their symmetries.
The paper tackles the problem of combining quantum randomness with computational determinism to preserve non-mechanistic behavior, resulting in the discovery of an uncountable subgroup called the bi-immune symmetric group within the infinite symmetric group on natural numbers, which is shown to be highly transitive and dense.
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.