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.
A ZFA model is a ZF model
False statement
A ZFA model is a ZF model.
Facts & Assumptions
Given: The hypotheses, objects, and conventions in the Statement.
ZFA universes, atoms, pure sets, and the kernel distinguishes atoms from sets and identifies the pure kernel.
Jech–Sochor first embedding theorem gives a bounded atom-to-set simulation when the ZFA permutation model is presented inside an ambient ZFA+AC model.
Counterexample
In a ZFA model with distinct atoms , both have no members. Unrestricted ZF Extensionality would conclude , so the whole ZFA universe is not a ZF model.
The valid repairs are narrower: F1 says the pure kernel is a ZF model, and F2 transfers a prescribed power-iterate segment into a symmetric ZF model under its explicit ambient ZFA+AC premise. Neither identifies the entire atom universe with a ZF universe.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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 4 and 6 (standard reference, not scraped)