Combining Determinism and Indeterminism

TL;DR

This paper explores mathematical operations combining quantum indeterminism with classical determinism, focusing on bi-immune sets and the structure of the bi-immune symmetric group, revealing its uncountability, transitivity, and density properties.

Contribution

It introduces the bi-immune symmetric group, an uncountable subgroup of Sym(N), and studies its properties and relation to bi-immune sets and rearrangements.

Findings

Bi-immune symmetric group is uncountable and highly transitive.

Contains the finitary symmetric group on natural numbers.

Dense in the symmetric group with respect to pointwise convergence.

Abstract

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.