TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.

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The tail-flip hereditary-symmetric interpretation is a model of ZF

Statement

The tail-flip hereditary-symmetric interpretation $F_{t f}$ is a transitive inner model of ZF of the generic extension $V [G]$ .

Facts & Assumptions

Given: The tail-flip symmetric system over the transitive ground $V ⊨ Z F C + V = L$ and a $V$ -generic $G$ .

[F1]

The tail-flip hereditary-symmetric model defines $F_{t f}$ to be the hereditary-symmetric interpretation of that exact system and proves that each individual HS name is fixed by some $H_{m}$ . It expressly disclaims the former fixed-stage definability union.

[F2]

Hereditarily symmetric interpretations form a transitive ZF model proves that the hereditary-symmetric interpretation of any symmetric system over a transitive ZF ground is a transitive ZF model between the ground and the full generic extension.

Proof

technique · direct application of the general symmetric-model theorem

1.1

By F1, $(P, G, F)$ is a symmetric system and $F_{t f} = {H S}_{F}^{G}$ . The definition also supplies the exact finite-support fact later used by the tail argument; no finite-predicate presentation or equality of two model constructions is used in this step.

2.1

Apply F2 to this system. It gives $V \subseteq F_{t f} \subseteq V [G]$ , transitivity, and every ZF axiom and schema instance. Hence $F_{t f}$ is the claimed inner model of ZF.

F1F2step 1.1

3.1

The legacy identifier records why this symmetric model appears on the Feferman page: its infinite coordinate-tail automorphism is the transform used in Feferman's tail-complement theorem. Feferman's ZF-model theorem concerns his different ramified hierarchy. The present ZF conclusion comes solely from F2 and does not identify the two models or revive the false literal union from F1.

F1F2step 2.1∎

Depends on

Used by

The tail-flip symmetric model is finitely formalizable Lemma

Dependency tree · two levels

10 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

Solomon Feferman, Some applications of the notions of forcing and generic sets, ramified-model Theorem 4.9 and tail argument Theorem 4.12, printed pp. 341–344 (standard reference, not scraped)