Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Logic and set theory

Uncountability of the Continuum

No sequence indexed by the natural numbers can list every real number: the full real line is uncountable.

Exact theorem

Exact Mathlib statement

theorem Cardinal.not_countable_real : ¬(Set.univ : Set ℝ).Countable

The theorem at a glance

Uncountability of the Continuum at a glance

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

The reals have continuum cardinality, strictly beyond countable infinity, so no countable list contains every real number. Explanatory diagram.
Detailed visual description

Infinite Boolean streams inject into the real line, while rational Cauchy-sequence representations provide the reverse cardinal control. These two bounds identify the real cardinal with the continuum; the strict inequality beyond countable infinity yields noncountability of all real numbers.

Statement structure

Statement and scope

Statement map for Uncountability of the ContinuumThe universal set of real numbers is not countable. Claim boundary: This page indexes Mathlib's declaration that the universal set of real numbers is not countable. The checked source reaches it through the nearby cardinal equality #ℝ = 𝔠 and the strict inequality ℵ₀ < 𝔠, but the selected endpoint itself is the noncountability statement; it does not separately assert the equality, identify one special failed enumeration, or say that every subset of ℝ is uncountable. The pinned upstream declaration is Cardinal.not_countable_real. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingThe universal set ofreal numbers is notcountable.Claim boundaryThis page indexesMathlib's declarationthat the universal setof real numbers is notcountable. The checkedsource reaches itthrough the nearbycardinal equality #ℝ = 𝔠and the strictinequality ℵ₀ < 𝔠, butthe selected endpointitself is thenoncountabilitystatement; it does notseparately assert theequality, identify onespecial failedenumeration, or say thatevery subset of ℝ isuncountable.Pinned declarationmathlib ·Cardinal.not_countable_realStatement map for Uncountability of the ContinuumThe universal set of real numbers is not countable. Claim boundary: This page indexes Mathlib's declaration that the universal set of real numbers is not countable. The checked source reaches it through the nearby cardinal equality #ℝ = 𝔠 and the strict inequality ℵ₀ < 𝔠, but the selected endpoint itself is the noncountability statement; it does not separately assert the equality, identify one special failed enumeration, or say that every subset of ℝ is uncountable. The pinned upstream declaration is Cardinal.not_countable_real. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingThe universal set ofreal numbers is notcountable.Claim boundaryThis page indexesMathlib's declarationthat the universal setof real numbers is notcountable. The checkedsource reaches itthrough the nearbycardinal equality #ℝ = 𝔠and the strictinequality ℵ₀ < 𝔠, butthe selected endpointitself is thenoncountabilitystatement; it does notseparately assert theequality, identify onespecial failedenumeration, or say thatevery subset of ℝ isuncountable.Pinned declarationmathlib ·Cardinal.not_countable_real

Read the exact Mathlib declaration

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

Distinct Boolean streams inject into distinct real values, while a separate reverse bound identifies the real cardinal with the continuum beyond countable infinity. Explanatory scientific diagram.
Detailed visual description

Four finite windows show distinct infinite-choice tapes entering from behind a clipped portal frame. Each tape connects directly to one solid curve, and the four curves terminate in preserved order at four separated points on the continuum band without crossing, merging, or sharing a destination. Above, continuous converging traces flow into a sequence mesh and return through a separate gold ribbon for the reverse cardinal bound. A conventional sparse solid ladder with two vertical rails and short horizontal rungs at right contrasts countable structure with the full continuum.

Why it matters

A mathematical landmark

The uncountability of the real numbers is one of the foundational discoveries of modern set theory: infinite sets can have genuinely different sizes. Mathlib's selected endpoint connects that idea to an exact universal-set noncountability statement, with the nearby source route identifying the real cardinal with the continuum.

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 page indexes Mathlib's declaration that the universal set of real numbers is not countable. The checked source reaches it through the nearby cardinal equality #ℝ = 𝔠 and the strict inequality ℵ₀ < 𝔠, but the selected endpoint itself is the noncountability statement; it does not separately assert the equality, identify one special failed enumeration, or say that every subset of ℝ is uncountable.

Source and local evidence

Where the theorem comes from

Existing declaration
Cardinal.not_countable_real in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.uncountability-continuum.v001
Source
Open the pinned upstream reference