Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14
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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.

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Every ccc forcing is countably closed

False statement

Every ccc forcing is countably closed (σ-closed).

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Closure and chain conditions of Cohen forcing proves that Add⁡(ω,1) is ccc.

Counterexample

1.1

Let P=Add⁡(ω,1). It is countable, hence ccc, as also recorded in F1. Define pn={((0,k),0):k<n}. Then pn+1≤pn, but a common lower bound would contain ⋃npn, an infinite function, and so would not be a condition. Therefore P is not countably closed.

F1
2.1

The explicit descending chain already refutes the claim without appealing to a B-page example. Under F1's strict <κ convention, its assertion that this forcing is ω-closed concerns only finite descending sequences and is not the advertised countable closure.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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