Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)
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.

Identity and successor in a normal ultrapower

Example

In ZFC, assume U is a normal measure on kappa and write Scott classes through their transitive collapse. For every alpha<kappa,

[cα]U=α,[id]U=κ,[ξ↦ξ+1]U=κ+1<jU(κ).

Facts & Assumptions

Given: ZFC. Computed constant, identity and successor classes and verified the strict bound from all coordinate successors remaining below kappa.

[F1]

Measurability, normal measures and elementary embeddings: Normality identifies the collapsed identity class with kappa.

[F2]

The critical point of a measurable ultrapower: Constant ordinal classes below kappa are fixed.

[F3]

Los schema for the universe ultrapower: Universe Los transfers each fixed first-order membership formula to the Scott ultrapower and hence through its collapse.

Verification

1.1

F2 gives [c_alpha]=j(alpha)=alpha for alpha<kappa; F1 gives [id]=kappa. At every coordinate xi, the ordinal xi+1 is the set xi union {xi}, namely the successor of xi. F3 transfers this fixed defining formula, so the collapsed class of xi maps to xi+1 is the successor of [id], exactly kappa+1.

F1F2F3
2.1

An infinite cardinal kappa is a limit ordinal: any infinite successor ordinal beta+1 is equinumerous with beta by shifting a countably infinite subset, so cannot be an initial ordinal. Therefore xi+1<kappa for every xi<kappa. The successor representative takes its values in kappa at all coordinates, so its class belongs to j(kappa) by coordinate membership. Step 1.1 identifies this member with kappa+1, proving the strict inequality.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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