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.
Relative consistency of ZF with failure of Choice
Statement
Externally, implies . The implication uses separately fixed finite-fragment model proofs; no PA-verified uniform symmetric-model proof transformer or transitive model of full ZF is asserted.
Facts & Assumptions
Given: One hypothetical finite contradiction proof from .
Formal consistency of ZFC plus GCH relative to ZF gives , and hence consistency of ZFC.
Fixed finite-fragment verification for the basic Cohen symmetric model gives a ZFC proof of a set model for every externally fixed finite target fragment.
The Axiom of Choice is the sentence negated in the target.
Proof
The hypothetical contradiction proof uses a finite list of axioms of ZF and the negation of F3. Fix this list externally. F2 provides a ZFC proof that a set model of exists. Soundness for the particular finite contradiction proof gives a ZFC proof that no such model exists. Thus target inconsistency implies inconsistency of ZFC.
F1 makes ZFC consistent whenever ZF is consistent. Contraposition in step 1.1 therefore proves the stated external relative-consistency implication. Source Choice is available in the ambient ZFC construction; the symmetric target refutes it. Each target proof is handled separately, without a claim that PA verifies a uniform map on proof codes.
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
- Jech, The Axiom of Choice, Theorem 5.16 (standard reference, not scraped)