Alphabeta MathHow 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.
Pincus transfer interfaces and preservation limits
Remark
Pincus transfer enlarges the interface to specified finite conjunctions of injectively boundable statements and, in a standard formulation, the Boolean Prime Ideal theorem. Those syntactic hypotheses must be verified for each intended consumer; this page does not use that stronger theorem.
The limitation in Jech–Sochor transfer for certified atom-blind boundable sentences is real. The atom-sensitive ZFA assertion “there are two distinct objects with no members” cannot hold in a transitive ZF model, where Extensionality makes the empty set unique. Thus neither Jech–Sochor nor Pincus transfer preserves arbitrary ZFA truth, and later pages may not infer conjunction or BPI preservation merely from this orientation remark.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
3 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
- Jech, The Axiom of Choice, Chapters 6 and 9 (standard reference, not scraped)