Grades 9–12FoundationsPlanned

Countability and Cantor's diagonal argument

Comparing infinite sets by bijections: ℕ, ℤ and ℚ are countable, every set is strictly smaller than its power set, and Cantor's diagonal argument shows that the real numbers are uncountable.

math.found.countability

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Builds on

Functions Proof: direct, contrapositive, contradiction and induction Real numbers and irrationality

Required prerequisites, all the way down: 14

You will learn to

Show that ℤ and ℚ are countable by constructing explicit enumerations or bijections.
Define the power set P(A) and prove Cantor's theorem that no function A → P(A) is surjective.
Reproduce Cantor's diagonal argument that [0, 1) is uncountable, handling numbers with two decimal expansions.

Definition

Sets A and B have the same cardinality if there is a bijection A → B.
A set is countable if it is finite or has the same cardinality as ℕ; otherwise it is uncountable.
The power set P(A) is the set of all subsets of A.

Theorem

ℤ, ℚ and ℕ × ℕ are countable.
A countable union of countable sets is countable (the general statement uses the axiom of countable choice).
Cantor's theorem: for every set A there is no surjection from A onto P(A).
ℝ is uncountable; in fact ℝ and P(ℕ) have the same cardinality.

Common misconceptions

Thinking a proper subset of an infinite set must be smaller: ℕ and the even numbers have the same cardinality.
Objecting that the diagonal number 'can just be added to the list'; the argument shows that every list misses some number.
Forgetting that 0.4999… = 0.5, which a careful diagonal argument must avoid (for example by never changing a digit to 0 or 9).

Leads to

Ordinals and cardinals Completeness, compactness and Löwenheim–Skolem Computability Measure theory and integration

Optional discoveries

Forcing and independence Is there a set strictly larger than ℕ but smaller than ℝ? This is the continuum hypothesis; Gödel (1940) and Cohen (1963) showed that, if ZFC is consistent, it can be neither proved nor refuted in ZFC.Wonder Trail →
Gödel's incompleteness theorems A diagonal idea reappears in Gödel's proof through a self-referential sentence; the resemblance is genuine in spirit, but Gödel's diagonal lemma is a different construction.
← Counting
← Sets
← Set-builder notation, quantifiers and infinite sets
← Rational numbers