Logic and foundations · existing Mathlib declaration

Well-Ordering Theorem

For any type α, regardless of its size or prior structure, there is some way to compare every pair of elements linearly such that the resulting strict order is well founded.

well-orders
linear orders
well-founded relations
cardinals
pullback orders
choice principles

The theorem at a glance

Well-Ordering Theorem — existence without canonicity

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

An ivory and forest-green theorem poster shows the same arbitrary collection before and after placement on one open-ended total-order path, states the exact existential LinearOrder and WellFoundedLT conclusion, and excludes canonicity, computability, and relation extension. — Every type admits some linear order whose strict relation is well founded.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Exactly twelve distinct tokens appear in a three-by-four set and reappear once each in a strict vertical order, with one-to-one gold connector lines. — The same sample elements are organized into a strict order without adding, losing, or merging any element.

Complementary intuition

A mathematical landmark

The well-ordering theorem is a central choice principle: it makes transfinite methods available on an arbitrary type after choosing a suitable order. Mathlib's declaration captures the exact existence of a linear order with well-founded strict comparison.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

theorem exists_wellOrder (α : Type u) : ∃ (_ : LinearOrder α), WellFoundedLT α

Selected evidence declaration text

theorem exists_wellOrder (α : Type u) : ∃ (_ : LinearOrder α), WellFoundedLT α

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's theorem that every type α admits some LinearOrder whose strict relation is well founded. It is an existential result only: the selected declaration does not provide a computable or canonical order, preserve or extend a relation already on α, or identify the chosen order with a particular ordinal.

Proof Atlas did not originate the well-ordering theorem or Mathlib's declaration.
The theorem asserts existence; it does not choose a canonical, unique, or computationally recoverable order.
The selected declaration does not say that its order extends or preserves any pre-existing relation on α.
The generated explanation and visuals are not proof evidence.

Provenance

Origin and evidence stay separate

Existing declaration: exists_wellOrder in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.well-ordering-theorem.v001
Source handling: Verified upstream reference; no mirrored source package