Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: 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 two local hypotheses at omega

Example

In ZF, CH(ω) together with CH(P(ω)) implies P(ω)≈ω1. These are two separate local hypotheses.

Facts & Assumptions

[F1]

Specker’s two-local-GCH theorem: The two local hypotheses at a set containing omega give its power set equinumerous with its Hartogs number.

[F2]

ω1 is uncountable, every ordinal below it is at most countable, it is a cardinal and a limit ordinal, and its existence is a theorem of ZF: The Hartogs number of omega is omega_1, the least uncountable ordinal.

Verification

Given: The objects and hypotheses in the statement.

1.1

Apply Specker with X=ω and the identity injection of omega. The two premises concern respectively sizes between omega and its power set, and sizes between that power set and its own power set. The conclusion is P(ω)≈h(ω).

F1
2.1

By the definition and characterization of the first uncountable ordinal, h(ω)=ω1. Substitution gives the asserted bijection, and hence well-orders the power set.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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