Alphabeta Math
RemarkRemark: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09
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.

Choice ledger and arbitrary-language boundary

Choice boundary

The proofs of Completeness for explicitly countable set languages and Compactness for explicitly countable languages are in classical ZF for a language supplied with an injection into ω. Their decisions are definable tests of consistency, not computable tests; minimum syntax codes avoid any countable choice of enumerations or representatives.

The general size bounds in Downward Löwenheim–Skolem with parameters and Upward Löwenheim–Skolem, including elementary extensions explicitly assume AC. Arbitrary witness hulls use a well-order to choose witnesses, and the stated general-language Henkin theorem retains its choice hypothesis. This records the assumptions actually used, without claiming their optimality.

Arbitrary-language compactness over ZF and its Boolean-prime-ideal comparison are assigned to the later Boolean-algebras/Stone-duality page. No version of that comparison is asserted or used as a prerequisite here. In particular no result from the recorded Set Theory Beyond Choice catalogue supplies the present proofs.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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