Domain 01 · Mathematics Codex
Foundations & Logic
The bedrock of all mathematics — sets, logic, proof, and the limits of what mathematics can know.
Scope
Set theory (including ZFC axioms), mathematical logic, proof theory, model theory, type theory, category theory, computability theory, and incompleteness results.
Every other part of mathematics rests on foundations. Before you can prove anything rigorously, you need a precise definition of what a set is, what logical inference means, and what constitutes a valid proof. This domain contains the machinery that makes certainty possible — and the results (particularly Gödel's incompleteness theorems) that define its limits.
Entries in this domain
Build status: This domain is in active build. Live entries are marked below. All other entries are planned and pending — see build status for the full picture.
-
Set
Live
A collection of distinct objects — the foundational concept from which all modern mathematics is built.
-
ZFC Axioms
Pending
The nine axioms of Zermelo-Fraenkel set theory with Choice — the standard foundation of modern mathematics.
-
Proof
Pending
A finite sequence of logical steps that establishes a mathematical statement with certainty.
-
Gödel's Incompleteness Theorems
Pending
Any sufficiently powerful formal system either contains true statements it cannot prove, or is inconsistent.
-
Function
Pending
A rule that assigns to each element in one set exactly one element in another set.