Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Grothendieck universe closure convention

Definition

A Grothendieck universe, in the closure convention used here, is a nonempty transitive set U such that: if x,y∈U then {x,y}∈U; if x∈U then P(x)∈U; and if I∈U and f is any set function on I with f(i)∈U for every i∈I, then ⋃i∈If(i)∈U. The indexing function need not itself belong to U.

The additional condition ω∈U is imposed only when explicitly stated. This is a definition by closure conditions, not an assertion that a universe containing any prescribed set exists. Transitivity means that every member of a member of U is itself a member of U.

Conventions and prerequisites: Minimality and closure laws of TC.

Depends on

Used by

Dependency tree · two levels

2 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