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.

Power sets need not agree across transitive models

Example

If $M$ is a countable transitive model of ZF, then externally its internal power set of $ω$ is countable and misses an actual subset of $ω$ . This conditional example does not assert that ZF proves the existence of such an $M$ .

Facts & Assumptions

[F1]

Ordinals and omega in transitive models: In ambient ZF, ordinalhood is absolute between transitive membership domains containing the parameter. A transitive set model of ZF contains precisely the real finite ordinals as its natural numbers and has $ω^{M} = ω$ . Its ordinals form an initial segment of the actual ordinals.

[F2]

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.

[F3]

Ranks agree and hierarchy membership is absolute: If $M \subseteq N$ are transitive models of ZF and $x \in M$ , then ${rank}^{M} (x) = {rank}^{N} (x)$ . For every $α \in M \cap Ord$ , $(V_{α})^{M} = M \cap (V_{α})^{N}$ . Equality of the two internal power sets or stage sets is not asserted.

Verification

Given: A countable transitive $M ⊨ Z F$ and an external countability injection.

1.1

By F1, $ω^{M} = ω$ . Fix an external injection $h : M \to ω$ . Define $e (n)$ to be the unique $x \in M$ with $h (x) = n$ when there is one, and otherwise $\emptyset \in M$ . Thus $e : ω \to M$ is onto, without any further choice. Put $A_{n} = e (n)$ if $e (n) \subseteq ω$ , and $A_{n} = \emptyset$ otherwise.

F1given

2.1

Let $d = {n \in ω : n \notin A_{n}}$ . For every $k$ , $k \in d$ iff $k \notin A_{k}$ , so $d \neq A_{k}$ . If $d \in M$ , surjectivity gives $e (k) = d$ for some $k$ , whence $A_{k} = d$ because $d \subseteq ω$ , a contradiction. Therefore $d$ is a subset of the actual omega outside $M$ .

step 1.1algebra

3.1

Let $p$ be the internal power set of omega. Transitivity gives $p \subseteq M$ . Bounded subset agreement F2 and the internal power-set axiom give $p = {x \in M : x \subseteq ω}$ . Restricting $h$ to $p$ proves countability, and step 2.1 proves $d \notin p$ . Thus the hierarchy intersection identity in F3 does not imply equality of internal and external power sets.

F2F3step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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) — §3.5 distinction between bounded operation graphs and unbounded power-set quantification, local diagonal example (standard reference, not scraped)