Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31‡ rests on unproved material (inherited)
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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Assuming choice, refuted: every ccc space is separable

Statement

Every ccc space is separable.

Facts & Assumptions

Given: The Axiom of Choice and an index set II with ∣I∣>2ℵ0|I|>2^{\aleph_0}.

[L1]

Under choice every Cantor cube 2I2^I satisfies ccc (Under choice, every Cantor cube 2I2^I satisfies ccc).

[L2]

Under choice, ∣I∣>2ℵ0|I|>2^{\aleph_0} implies that 2I2^I is not separable (Under choice, if ∣I∣>2ℵ0|I|>2^{\aleph_0}, then the Cantor cube 2I2^I is not separable).

Refutation

technique · direct
1.1

Let X=2IX=2^I with its product topology.

given
1.2

The space XX satisfies the hypothesis of the proposed implication because it is ccc by [L1].

L1
2.1

The same space fails the proposed conclusion because it is not separable by [L2].

step 1.1L2
3.1

Thus XX is a ccc nonseparable space, which refutes the statement.

step 1.2step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 14 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