ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Bounded formulas and the direction of absoluteness

Example

Bounded graph agreement can be checked explicitly on $a = {0}$ , $b = {0, 1}$ and $f = {⟨ 0, 1 ⟩}$ . Here $f : a \to b$ is injective. Existence of such a graph is a separate existential assertion whose witness must be retained for upward transfer.

Facts & Assumptions

[F1]

Absolute basic set operations and relations: The graphs of empty set, subset, unordered pair, singleton, union, intersection (with $⋂ \emptyset = \emptyset$ ), difference, Kuratowski ordered pair, Cartesian product, relation domain/range, functionhood, evaluation and injection have $Δ_{0}$ definitions. Thus their values agree between transitive membership structures whenever the input and output sets are in the smaller domain. This is graph agreement, not an assertion that an arbitrary transitive domain is closed under these operations.

[F2]

Bounded formulas are absolute for transitive sets: If $M \subseteq N$ are nonempty transitive sets, every $Δ_{0}$ formula is absolute between them on parameter tuples from $M$ . No internal set-theory axioms are required. The analogous assertion for definable transitive classes is a formula-by-formula scheme.

[F3]

Sigma-one truth goes upward: For nonempty transitive $M \subseteq N$ , a literal existential block over a $Δ_{0}$ matrix transfers truth upward, and its universal dual transfers truth downward. For formulas classified only by ZF-provable equivalence, assume both structures satisfy ZF (or all axioms used in the equivalence proof).

Verification

Given: $0 = \emptyset$ , $1 = {0}$ , $a = {0}$ , $b = {0, 1}$ , $f = {⟨ 0, 1 ⟩}$ .

1.1

The formula for $x \subseteq y$ is $\forall u \in x (u \in y)$ ; for $z = {x, y}$ it is $x \in z \land y \in z \land \forall u \in z (u = x \lor u = y)$ . In the instance, the only element $0$ of $a$ belongs to $b$ , and the only elements of $b$ are $0, 1$ , so both tests hold for $a \subseteq b$ and $b = {0, 1}$ .

givenalgebra

2.1

Use the bounded $K (p, x, y)$ graph of F1. A bounded injection test is: every $p \in f$ has coordinates $x \in a, y \in b$ with $K (p, x, y)$ ; every $x \in a$ occurs in such a $p$ ; and for $p, q \in f$ and their coordinates in $a, b$ , equal first coordinates imply equal second coordinates and conversely. Here $⟨ 0, 1 ⟩ = {{0}, {0, 1}}$ is the sole pair, its first coordinate is $0$ , its value is $1 \in b$ , and comparing the only pair with itself verifies both uniqueness and injection.

F1step 1.1

3.1

In transitive domains containing $a, b, f$ , F2 preserves these bounded graph tests. If the smaller domain contains the displayed witness $f$ , F3 transfers the sentence $\exists f (f : a ↪ b)$ upward by retaining it. If only $a, b$ are present, graph absoluteness alone neither constructs $f$ in that domain nor supplies downward transfer of its existence.

F2F3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Freiburg, Course Notes for Set Theory and Independence Proofs (2024) — Propositions 3.5.6/3.5.8 pp51–52 (standard reference, not scraped)