False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: the union of an increasing sequence of monotone classes is a monotone class

Statement

The union of every increasing sequence of monotone classes on one ambient set is a monotone class.

Facts & Assumptions

Given: The ambient set $X : = N$ and $M_{n} : = P (n)$ , where $n = {0, \dots, n - 1}$ and in particular $M_{0} = {\emptyset}$ .

[L1]

A monotone class is closed under increasing countable unions and decreasing countable intersections (Monotone classes of sets).

Refutation

technique · counterexample

1.1

Every increasing or decreasing sequence in the finite family $M_{n}$ stabilizes, so its union or intersection belongs to $M_{n}$ . Thus each $M_{n}$ is a monotone class, and $M_{n} \subseteq M_{n + 1}$ .

L1algebra

2.1

The union $⋃_{n} M_{n}$ is the family of finite subsets of $N$ . The increasing sequence $0 \subseteq 1 \subseteq 2 \subseteq \dots$ lies in this union but has union $N$ , which is not finite. Hence the union is not a monotone class.

step 1.1L1∎

Depends on

Monotone classes of sets

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: 1 result over 1 level. 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

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercise 2.4 (standard reference, not scraped)