The paradox of the infinity

TL;DR

This paper investigates a paradoxical situation where a subset with a property has a density of at least 1/2 in an infinite set, despite each partition having only finitely many such elements, challenging intuitive notions of rarity.

Contribution

The paper presents examples demonstrating that the density of elements satisfying a property can be high even when each partition contains only finitely many such elements, revealing a paradoxical phenomenon.

Findings

Density can be at least 1/2 despite finite counts in partitions

Partitioning an infinite set can obscure the rarity of certain elements

The phenomenon resembles the Simpson paradox in probability

Abstract

\textit{Let $E$ be an infinite set on which a property $(\bf P)$ is defined. Suppose that $E=\cup_{i\in I} E_i$ is a partition, where each $E_i$ is infinite. Suppose also that, in each $E_i$, the number of elements satisfying $(\bf P)$ is finite. Then, clearly the density of the elements satisfying $(\bf P)$ is 0 in every $E_i$. Is it possible that the density of the subset of $E$ containing all the elements satisfying $(\bf P)$ will be at least equal to $ 1/2$?} We were first confronted with this situation while reading the paper of Arno et al. [1]. In fact, it is in the paper [1] where it is shown that the density of certain algebraic numbers in $\overline{\mathbb{Q}}$, which we will call Arno et al. numbers in section 5, is equal to $1/\zeta(3)$. We have partitioned $\overline{\mathbb{Q}}$ in a way that suggests these Arno et al. numbers are rare. This phenomenom struck us as contradictory, which lead us to consider the situation in greater detail. We will show in the sequel, through two examples, that the answer to the above question may be positive. At first glance, this problem resembles to the so called Simpson paradox in probability and statistics. In this paper, when we say the density, we mean the natural density.