Existing in Mathlib · Logic and foundations

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

Exact theorem

Exact Mathlib statement

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

The theorem at a glance

Well-Ordering Theorem — existence without canonicity

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

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.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains 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.

Why it matters

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.

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

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

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.

Source and local evidence

Where the theorem comes from

Existing declaration: exists_wellOrder in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.well-ordering-theorem.v001
Source: Open the pinned upstream reference