False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-29

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: every injection of a set into itself is a bijection

Statement

FALSE. The statement

every injective function $f : A \to A$ from a set to itself is a bijection

for all sets $A$ .

The claim is plausible because it is true for finite $A$ : that is clause 4 of A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$ , and equality holds if and only if $B = A$ . What is easy to miss is that the proof of that clause uses finiteness twice, at the transport $\lvert f[A]\rvert = \lvert A\rvert$ and at the step "a subset of the same cardinality as the whole is the whole", and neither survives without it.

Facts & Assumptions

Given: The von Neumann naturals $\mathbb{N}$ with $0 = \varnothing$ and successor $\sigma$ (The natural numbers $\mathbb{N}$ (von Neumann)), and $A := \mathbb{N}$ , $f := \sigma$ .

[L1]

$(\mathbb{N},0,\sigma)$ satisfies the Peano axioms: $\sigma(n) \ne 0$ for every $n$ , and $\sigma$ is injective (The von Neumann naturals form a Peano system).

[L2]

Every nonzero natural is a successor (Every nonzero natural number is a successor), so the image of $\sigma$ is exactly $\mathbb{N}\setminus\{0\}$ .

[L3]

Injection, surjection, bijection (Injection, surjection, bijection): $f$ is surjective when its image is the whole codomain, and bijective when injective and surjective.

[L4]

For finite $A$ , every injection $A \to A$ is a bijection (A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$ , and equality holds if and only if $B = A$ , clause 4), the proof going through $\lvert f[A]\rvert = \lvert A\rvert$ and clause 3 of the same theorem (The cardinality $\lvert A\rvert$ of a finite set).

[L5]

$\mathbb{N}$ is not finite: $\mathbb{N} \not\approx n$ for every natural $n$ (The pigeonhole principle on $\mathbb{N}$ , claim 4, Finite, countably infinite, countable, uncountable, Equinumerous sets, $A \approx B$ and $A \preceq B$ ).

Refutation

technique · direct

1.1

The witness is the successor map $\sigma : \mathbb{N} \to \mathbb{N}$ . It is injective by [L1].

givenL1L3

2.1

It is not surjective: $0$ is not in its image, since $\sigma(n) \ne 0$ for every $n$ by [L1]. Equivalently, its image is $\mathbb{N}\setminus\{0\}$ by [L2], a proper subset of $\mathbb{N}$ .

step 1.1L1L2L3

3.1

So $\sigma$ is an injection of $\mathbb{N}$ into itself that is not a bijection, and the displayed statement is false.

step 1.1step 2.1L3

4.1

Finiteness is exactly the missing hypothesis. By [L4] the statement is true whenever $A$ is finite, and $\mathbb{N}$ is not finite by [L5]. In the proof of [L4] the hypothesis is spent at the transport of cardinality along the bijection $A \to f[A]$ , which presupposes $A$ finite, and then at the conclusion $f[A] = A$ from $\lvert f[A]\rvert = \lvert A\rvert$ , which is the clause of A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$ , and equality holds if and only if $B = A$ that fails here: $\sigma[\mathbb{N}]$ is a proper subset of $\mathbb{N}$ equinumerous with it.

step 3.1L4L5∎

Remarks

A set for which the statement fails is called Dedekind-infinite, and the refutation above exhibits $\mathbb{N}$ as one. Claim 5 of The pigeonhole principle on $\mathbb{N}$ says that no natural number is Dedekind-infinite, which is the finite half of the same picture.
The relation between the two notions of infinity — "not finite" and "Dedekind-infinite" — is a genuine question of set theory without choice, and it is treated on the countability page rather than here.
The surjective half fails too. The map $\mathbb{N} \to \mathbb{N}$ sending $0$ and $1$ to $0$ and $n \ge 2$ to $n-1$ is surjective and not injective, so neither half of clause 4 of A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$ , and equality holds if and only if $B = A$ survives the loss of finiteness.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Dedekind-infinite set (Wikipedia) (standard reference, not scraped)
Surjective function (Wikipedia) (standard reference, not scraped)
Pigeonhole principle (Wikipedia) (standard reference, not scraped)
Finite set (Wikipedia) (standard reference, not scraped)