Numberia

Numberia is a fictional mathematical framework used in puzzles, speculative mathematics, and certain works of fiction to study the properties of numbers and their relationships beyond conventional number theory. In Numberia, numbers are organized into structures called numbernets and governed by a compact set of axioms that emphasize additive and multiplicative symmetries, modular chords, and the interaction between growth rates and sparsity. The term derives from "number" and the -ia suffix common to disciplines, signaling a systematic field of study rather than a single theorem.

Foundational concepts include Numberia space, a formal set of integers equipped with an operation set that generalizes addition and multiplication; Numberia sequences, infinite lists defined by recurrence rules or morphisms; and Numberia primes, elements that cannot be factored within a given Numberia framework. Unlike standard number theory, Numberia often allows alternative axioms or rules to illustrate how different assumptions influence results, making it popular in thought experiments and puzzle design.

Historically, Numberia originated as a construct in mathematical fables and classroom demonstrations to explore axioms' consequences. In practice, it appears in recreational mathematics communities and in speculative fiction to pose questions about density, regularity, and computability.

Applications include puzzle design, algorithmic experiments, and didactic tools that encourage flexible thinking about numbers. Open questions in Numberia resemble those in standard theory but adapted to its axioms, such as the behavior of Numberia primes under various transformations or the characterization of Numberia sequences with prescribed constraints.