CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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No sigma-algebra is countably infinite

Statement

There is no countably infinite sigma-algebra.

Facts & Assumptions

Given: A putative countably infinite sigma-algebra $A$ on a set $X$ .

[L1]

Countably infinite means equinumerous with $N$ (Finite, countably infinite, countable, uncountable).

[L2]

A sigma-algebra with an injective sequence of members contains pairwise disjoint nonempty members indexed by $N$ (A sigma-algebra with a listed infinite subfamily contains a disjoint sequence of nonempty members).

[L3]

A sigma-algebra is closed under countable unions (Sigma-algebras).

[L4]

There is no surjection $N \to P (N)$ (Cantor's theorem: $A ≺ P (A)$ ), and injections both ways give a bijection (The Schröder-Bernstein theorem).

Proof

technique · contradiction

1.1assume-contraL1L2

Suppose, for contradiction, that $A$ is countably infinite. By [L1] its bijective listing and [L2] give pairwise disjoint nonempty sets $D_{n} \in A$ .

2.1step 1.1L1L3

As in the preceding theorem, $S \mapsto ⋃_{n \in S} D_{n}$ is an injection $P (N) \to A$ , using [L3] and disjointness. Composing with a bijection $A \to N$ gives an injection $P (N) \to N$ .

3.1step 2.1L4discharge-contradiction∎

The singleton map injects $N$ into $P (N)$ , so [L4] would give a bijection and hence a surjection $N \to P (N)$ , contradicting Cantor's theorem. Therefore no countably infinite sigma-algebra exists.

Depends on

Used by

FALSE: a countably infinite sigma-algebra exists False statement

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercise 2.8 (standard reference, not scraped)