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.
Semantic and formal inner-model theorem for L
Statement
For each fixed axiom of , ZF proves . If M is a transitive set model of ZF, its internally defined constructible class , viewed externally as a set with actual membership, satisfies and has exactly the ordinals of M. No existence of such M, or arithmetized consistency-transfer theorem, is asserted here.
Facts & Assumptions
Given: ZF. Collected the actual fixed-axiom derivations and compared guarded quantifier satisfaction with the external set L^M. Preserved the conditional set-model scope and excluded an unsupported Con-transfer claim.
Elementary ZF axioms inside L: Extensionality, Foundation, Empty Set, Pairing, Union, and Infinity hold in L.
Separation in the constructible universe: Every fixed Separation instance holds in L.
Internal Power Set in L: Internal Power Set holds in L.
Replacement in L: Every fixed Replacement instance holds in L.
Absoluteness, idempotence and minimality of L: Levels in a transitive ZF model agree below its height, and L satisfies V=L.
The constructible universe satisfies AC: AC has a ZF proof after relativization to L.
Relativization agrees with induced set satisfaction: Induced set satisfaction agrees with quantifier relativization for each fixed formula.
Soundness for arbitrary set signatures: Every ZF derivation is valid in each set model of ZF.
Proof
Fix one axiom sigma. F1 supplies the six basic ZF axioms, while F2, F3, and F4 supply Separation, Power Set, and Replacement; each schema instance uses only its fixed formula and finitely many ZF instances. F6 supplies AC and F5 supplies V=L. Thus for this sigma there is a ZF derivation of . This assertion is indexed externally by standard axioms and is not an internal truth assertion about all formulas.
Suppose now that M is a transitive set model of ZF. External Separation on M, using satisfaction of the fixed predicate defining constructibility, forms as a set. By F5 it is the union of the actual for ordinals alpha in M. Hence it is nonempty and transitive, all its ordinals belong to M, and every ordinal of M belongs to C (its successor stage is still indexed in M).
Evaluate each fixed derivation from step 1.1 in M. Its axioms hold there, and first-order inference preserves satisfaction; thus M satisfies the internally relativized sigma. Constructor comparison of that fixed formula, as in F7, identifies this with satisfaction in C: atoms are actual membership and each guarded quantifier ranges over exactly C. Therefore C satisfies each standard axiom of ZFC+V=L. Step 2.1 gives the same-ordinals conclusion. The result remains conditional on the supplied M, with no assertion that a model can be obtained from a consistency statement.