Can each number be specified by a finite text?

TL;DR

This paper explores the complex hierarchy of definable numbers, clarifying misconceptions and distinguishing them from other well-known classes like natural or computable numbers, emphasizing the infinite levels involved.

Contribution

It provides an explanatory overview of the hierarchy of definable numbers, highlighting the distinctions from other number classes and addressing misconceptions.

Findings

Defines the hierarchy of definable numbers

Clarifies misconceptions about specifying numbers

Highlights the infinite levels involved in definability

Abstract

Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels involved distinguishes the concept of "definable number" from such notions as "natural number", "rational number", "algebraic number", "computable number" etc. (Explanatory essay for non-experts.)