Cantor's Theorem and Uncountability

Why This Matters

Not all infinite sets are the same size. The natural numbers, the integers, and the rationals are all countable: you can list them. The real numbers are not. This distinction is not a curiosity. It is the reason that complexity measures like VC dimension exist.

The set of all computer programs is countable (they are finite strings over a finite alphabet). The set of all functions from $\mathbb{R}$ to $\{0, 1\}$ is uncountable. So most functions cannot be computed by any program. In ML terms: the hypothesis class of all possible classifiers is uncountably infinite, which is why you must restrict to a learnable subclass and measure its complexity.

Core Definitions

Definition

Countable Set

A set $A$ is countable if there exists an injection $f: A \to \mathbb{N}$. Equivalently, $A$ is countable if it is finite or if there exists a bijection $f: A \to \mathbb{N}$ (countably infinite).

Definition

Uncountable Set

A set $A$ is uncountable if it is infinite and there is no bijection between $A$ and $\mathbb{N}$. No listing $a_1, a_2, a_3, \ldots$ can exhaust $A$.

Definition

Cardinality

Two sets $A$ and $B$ have the same cardinality, written $|A| = |B|$, if there exists a bijection $f: A \to B$. We write $|A| \leq |B|$ if there exists an injection from $A$ to $B$. The Cantor-Bernstein theorem says: if $|A| \leq |B|$ and $|B| \leq |A|$, then $|A| = |B|$.

Definition

Power Set$\mathcal{P}(A)$

The power set of $A$, written $\mathcal{P}(A)$, is the set of all subsets of $A$:

$\mathcal{P}(A) = \{B : B \subseteq A\}$

For finite $A$ with $|A| = n$, we have $|\mathcal{P}(A)| = 2^n$.

Main Theorems

Theorem

Uncountability of the Reals

Statement

The set $\mathbb{R}$ of real numbers is uncountable. More precisely, the interval $[0, 1]$ is uncountable.

Intuition

No matter how you try to list the reals in $[0, 1]$ as a sequence $r_1, r_2, r_3, \ldots$, there is always a real number your list misses. The diagonal argument constructs this missing number explicitly.

Proof Sketch

Assume for contradiction that $[0, 1]$ is countable. List all elements as $r_1, r_2, r_3, \ldots$, writing each in decimal as $r_i = 0.d_{i1}d_{i2}d_{i3}\ldots$. Construct a new number $x = 0.x_1 x_2 x_3 \ldots$ where $x_i \neq d_{ii}$ (and $x_i \notin \{0, 9\}$ to avoid issues with non-unique representations). Then $x \in [0, 1]$ but $x \neq r_i$ for every $i$, since $x$ and $r_i$ differ in the $i$-th decimal digit. This contradicts the assumption that the list contains all of $[0, 1]$.

Why It Matters

This is the first proof that infinity is not a single concept. There are strictly more real numbers than natural numbers. The argument pattern (diagonalization) reappears throughout mathematics and computer science: the halting problem, Godel's incompleteness theorems, and the existence of non-computable functions all use variants of it.

Failure Mode

The proof does not work for the rationals. Every rational has a terminating or repeating decimal, and the number constructed by the diagonal argument is generically irrational. The rationals are countable despite being dense in $\mathbb{R}$.

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Theorem

Cantor's Theorem

Statement

For any set $A$, there is no surjection from $A$ onto $\mathcal{P}(A)$. Therefore $|A| < |\mathcal{P}(A)|$: the power set is strictly larger.

Intuition

You cannot pair up elements of $A$ with subsets of $A$ so that every subset gets paired. There are always leftover subsets. The proof constructs one such leftover subset explicitly using diagonalization.

Proof Sketch

Let $f: A \to \mathcal{P}(A)$ be any function. Define $D = \{a \in A : a \notin f(a)\}$. Claim: $D$ is not in the range of $f$. If $D = f(b)$ for some $b$, then $b \in D \iff b \notin f(b) = D$, a contradiction. So $f$ is not surjective. Since the injection $a \mapsto \{a\}$ shows $|A| \leq |\mathcal{P}(A)|$, we conclude $|A| < |\mathcal{P}(A)|$.

Why It Matters

Cantor's theorem guarantees an infinite hierarchy of infinite cardinalities: $|\mathbb{N}| < |\mathcal{P}(\mathbb{N})| < |\mathcal{P}(\mathcal{P}(\mathbb{N}))| < \cdots$. There is no largest infinity. This is also the template for Russell's paradox (consider the set of all sets that do not contain themselves), which motivated axiomatic set theory.

Failure Mode

The theorem requires the axioms of separation (to form the set $D$) and power set (for $\mathcal{P}(A)$ to exist). In weaker set theories or constructive settings, the precise formulation matters.

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Connection to Computability

The set of all Turing machines (equivalently, all programs) is countable: each program is a finite string, and the set of finite strings over any finite alphabet is countable. But the set of all functions $f: \mathbb{N} \to \{0, 1\}$ has cardinality $2^{|\mathbb{N}|} = |\mathcal{P}(\mathbb{N})|$, which by Cantor's theorem is uncountable.

Therefore: most functions from $\mathbb{N}$ to $\{0, 1\}$ are not computable. No program computes them. This is not a conjecture; it follows directly from a counting argument.

Connection to ML

The hypothesis class of all binary classifiers on $\mathbb{R}^d$ (all functions $h: \mathbb{R}^d \to \{0, 1\}$) has cardinality $2^{|\mathbb{R}^d|}$, which is uncountably infinite. The ERM generalization bound for finite hypothesis classes gives a gap of $\sqrt{\log|\mathcal{H}| / n}$, which is undefined when $|\mathcal{H}|$ is an infinite cardinal (a cardinal does not have a real-valued logarithm), so the finite-class bound simply does not apply.

This is why complexity measures exist. VC dimension, Rademacher complexity, and covering numbers are all ways to assign a finite "effective size" to an infinite hypothesis class, making generalization bounds non-vacuous.

Common Confusions

Watch Out

The rationals are dense but countable

Density and countability are independent properties. The rationals are dense in $\mathbb{R}$ (between any two reals there is a rational), yet $\mathbb{Q}$ is countable. Density is a topological property; countability is a set-theoretic one.

Watch Out

Cantor's theorem does not say which cardinality the reals have

Cantor proved $|\mathbb{R}| > |\mathbb{N}|$, but did not determine whether $|\mathbb{R}| = \aleph_1$ (the next cardinal after $\aleph_0 = |\mathbb{N}|$). The claim $|\mathbb{R}| = \aleph_1$ is the Continuum Hypothesis, which is independent of ZFC. It can be neither proved nor disproved from the standard axioms.

Exercises

ExerciseCore

Problem

Prove that the set of all finite binary strings $\{0, 1\}^*$ is countable.

Hint

Reveal Solution

ExerciseCore

Problem

Using Cantor's theorem, prove that there exists a function $f: \mathbb{N} \to \{0, 1\}$ that is not computable by any Turing machine.

Hint

Reveal Solution

ExerciseAdvanced

Problem

Let $\mathcal{H}$ be the set of all binary classifiers on $[0,1]$. Show that $|\mathcal{H}|$ is uncountable and explain why the finite-class ERM bound does not apply.

Hint

Reveal Solution

References

Canonical:

• Halmos, Naive Set Theory (1960), Sections 1-4 and 25
• Enderton, Elements of Set Theory (1977), Chapter 6

Connection to CS:

• Sipser, Introduction to the Theory of Computation (2013), Section 4.2

Connection to ML:

• Shalev-Shwartz & Ben-David, Understanding Machine Learning (2014), Chapter 2

• Munkres, Topology (2000), Chapter 1 (set theory review)

Next Topics

• Zermelo-Fraenkel set theory: the axiomatic framework that makes Cantor's arguments rigorous

Continue on ComputationPath

• Godel's incompleteness theorems: another limit result using diagonalization