Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

The aleph equation carries no well-orderability assertion

Statement

False claim: “The equation 2ℵα=ℵα+1 says only that there is no intermediate size; it makes no assertion that P(ℵα) can be well-ordered.” Here the equation is interpreted literally as equinumerosity with the indicated initial ordinal.

Refutation

Given: The objects and hypotheses in the statement.

1.1

Under the stated interpretation, the equation asserts a bijection b:P(ℵα)→ℵα+1. Define U<V iff b(U)<b(V) in the ordinal order.

given
2.1

This is a linear order, and any nonempty family of subsets has a least member: take the inverse image of its image’s least ordinal. Thus the equation explicitly asserts well-orderability, refuting the claim. The equation is expressible in ZF; its being expressible does not remove this mathematical content.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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