Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17
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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: every monotone class is an algebra

Statement

Every monotone class on X is an algebra of subsets of X.

Facts & Assumptions

Given: The set X:={1,2} and the family M:={∅,{1},X}.

[L1]

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

[L2]

An algebra is closed under complements relative to its ambient set (Algebras of subsets).

Refutation

technique · counterexample
1.1

Every increasing or decreasing sequence in the finite chain ∅⊆{1}⊆X stabilizes, so its union or intersection belongs to M. Thus M is a monotone class by [L1].

L1algebra
2.1

The complement X∖{1}={2} is absent from M, so [L2] shows that M is not an algebra.

step 1.1L2∎

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: 2 results 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