RECURSIVE SELF-INDEXING ENUMERATION OF ALL POSITIVE NUMBERS IN EVERY INTEGER BASE b ≥ 2

Recursive Self-Indexing Enumeration of Numbers in Positional Numeral Systems Author: Szalontay Anna VeronikaDocumented research period: July 2020 – November 2026Earliest documented research: 1–2 July 2020 Research Objective This research presents Szalontay Anna Veronika’s recursive self-indexing enumeration system. The aim of the construction is to assign a natural number, that is, a unique index, to every point on the positive part of the number line, in an arbitrary numeral system, and in the reverse direction also determine which point of the number line corresponds to a given natural number. The correspondence is based on a symmetric recursive pattern built on the separation of even- and odd-exponent place values, together with the use of quotient and remainder operations. Mathematical Construction The central feature of the construction is the separation of even- and odd-exponent positions in the positional representation of the index. Even-exponent positions contribute to the integer part of the represented number, while odd-exponent positions contribute to its fractional part. These two components form an interconnected, symmetric recursive structure. The construction establishes a reversible relationship between the index and the corresponding numerical representation. The method is formulated for every integer numeral base b ≥ 2. Its experimental extension investigates arbitrary positive real bases b > 1. Number Representation and Cantor's Diagonal Argument The research investigates the hypothesis that every existing, precisely describable number can be represented within the recursive enumeration system. It therefore re-examines the conventional distinction between rational, irrational, and transcendental numbers, as well as the mathematical interpretation of infinity and countability. A separate part of the research examines Cantor's diagonal argument. The author proposes that digit patterns produced through diagonal construction can also be located within the recursive indexing structure. These broader propositions are presented as research claims for mathematical examination, distinct from the established finite-level properties of the construction. Documented Research History 1–7 July 2020: Dated correspondence documenting early investigations of binary numeral representations, recursive patterns, ordering, enumeration, and relationships between numerical values and their indices. March 2025: Documented spreadsheet investigations involving π, √2, numerical relationships, and equation construction. 29 September 2026: The research manuscript The Recursive Structure of the Number Concept and Infinity develops the binary starting pattern, the separation of even- and odd-exponent place values, and the recursive use of quotient and remainder operations. October–November 2026: Further development and documentation of the construction, its extension across numeral bases, its symmetry, and its relationship to Cantor's diagonal argument. Supporting Research Materials The accompanying spreadsheets and interactive HTML documents present computational examples, recursive patterns, positional representations, and investigations across different numeral bases. The publication documents the author's independently developed mathematical construction, its research history, and the broader mathematical questions arising from it. Researcher and author: Szalontay Anna VeronikaIndependent researcher, Budapest, Hungary.