Axoms

Axoms are a term used in discussions of foundational theory to denote a family of basic propositions within an imagined formal system called Axomic Logic. Unlike standard axioms, Axoms are designed to be modular and context-sensitive, enabling different collections to coexist and interact within a single framework. The aim is to study how changing starting points affects theorems and models.

Origin and concept: The term appears in speculative fiction and thought experiments since the early 21st century, where authors use Axoms to illustrate how foundational commitments shape theory. In these works, researchers select a set of Axoms to derive a coherent theory of a domain such as geometry, computation, or physics, and then explore the consequences of relaxing some Axoms.

Structure and examples: An Axom consists of a primitive condition and an inference rule that determines how new statements are derived. Axoms are categorized by context, such as spatial or temporal Axoms, and may be combined to form hybrid systems. Example Axoms address minimal existence, local consistency, or compatibility with a chosen model.

Applications and reception: In academic and fictional contexts, Axoms serve as a pedagogical tool to illustrate model dependence and the impact of assumptions. They are used to compare alternative logical universes and to simulate how different starting points yield different conclusions. Critics argue that Axoms risk mixing literary device with formal rigor if treated as mere postulates rather than precise rules.

See also: Axiom, Postulate, Model theory, Logic, Foundations of mathematics.