TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How 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.

Jech–Sochor transfer for certified atom-blind boundable sentences

Statement

Let $M$ be a transitive model of $Z F A + A C$ with atom set $A$ and pure kernel $K$ , and let $V = {H S}_{F}^{M}$ be the permutation submodel defined by an $M$ -internal normal group/filter system. Fix an iterate height $α$ and a $K$ -generic outer universe for the pure forcing of Jech–Sochor first embedding theorem at that height. Let $W$ and $A^{*}$ be its symmetric ZF model and atom-image set.

A transfer certificate for a sentence $T$ consists of a fixed formula $θ$ in the many-sorted incidence structure naming finitely many levels $P^{β} (A)$ for $β \leq α$ and, if needed, finitely many pure-kernel parameters shown to be fixed by the embedding. Each quantifier is restricted to one named carried sort; every atomic membership/equality test, including one involving a pure parameter, is certified to be preserved by the corresponding isomorphism. The base sort is treated as opaque; the formula never tests whether an atom image has members outside the carried structure. The typed formula itself, with any carried parameters, has the same truth value on corresponding source and target tuples. If a global sentence $T$ is sought, a certificate may prove that $T$ in $V$ is equivalent to $θ$ on the source sorts and that $T$ in $W$ is equivalent to the same $θ$ on the corresponding $A^{*}$ sorts. Then $V ⊨ T$ implies $W ⊨ T$ (and the converse holds in these fixed models). A sentence merely called boundable by Boundable sentences over an atom set has no such transfer conclusion without this additional certificate. In particular, the assertion that two distinct objects have no members is atom-sensitive and is excluded; no whole-universe elementary embedding is claimed.

Facts & Assumptions

Given: The ambient ZFA+AC presentation, its specified permutation system, the $K$ -generic outer universe, and the finite typed transfer certificate in the Statement.

[F1]

Boundable sentences over an atom set defines general relative-rank boundability. That condition alone does not certify preservation under an atom-to-set embedding; the typed certificate in the Statement is an additional hypothesis.

[F2]

Jech–Sochor first embedding theorem gives, in the stated generic outer universe, a membership isomorphism through any prescribed iterate, respecting its lower levels.

Proof

1.1

Apply F2 at height $α$ to obtain the bijections on every sort named by $θ$ . Pure-kernel parameters are fixed by the recursive translation of pure sets, with their mixed atomic incidences checked as part of the certificate. The carried bijections preserve equality and membership between the named objects. They assert nothing about members of an image $a^{*}$ lying outside those sorts. General boundability in F1 supplies no missing preservation claim; the certificate explicitly limits the formula to this carried incidence structure.

F1F2

2.1

Induct on the finite typed formula $θ$ . Atomic equality and membership are preserved by step 1.1, and Boolean connectives follow immediately. For a quantifier over a named sort, F2 is onto that sort, so every possible target witness has exactly one source preimage and the induction hypothesis applies in both directions. Thus $θ$ has the same truth value in the source and target incidence structures. No quantifier ranges over the uncarried members of a base-sort image.

F2step 1.1

3.1

Step 2.1 already gives the parameterized typed-formula assertion. When the two extra equivalences to a global $T$ are supplied, compose them with step 2.1 to get $V ⊨ T$ if and only if $W ⊨ T$ . A one-way target consequence of a transported typed assertion may also be inferred without a global source equivalence. The atom-sensitive two-empty-objects formula fails the certificate: two atoms can be memberless in ZFA, while distinct memberless sets violate Extensionality in ZF, so no target equivalence to the same typed incidence formula can be proved. The only Choice hypothesis is the ambient ZFA+AC premise of F2; $W$ need not satisfy AC.

F1F2step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources

Jech, The Axiom of Choice, Chapter 6 Problem 1, p. 95 (standard reference, not scraped)