Lean statement

Relative consistency of the Partition Principle with ordinal Choice and failure of Choice

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.PartitionConsistency.main_consistency

Lean statement file: PartitionConsistency.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 244 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main_consistency :    Consistent SetLanguage.ZF → Consistent SetLanguage.targetTheory := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

inductive type OAI.PartitionConsistency.Formula line 11

inductive Formula : ℕ → Type where  | bot {n} : Formula n  | mem {n} (x y : Fin n) : Formula n  | equal {n} (x y : Fin n) : Formula n  | imp {n} (p q : Formula n) : Formula n  | conj {n} (p q : Formula n) : Formula n  | disj {n} (p q : Formula n) : Formula n  | all {n} (p : Formula (n + 1)) : Formula n  | ex {n} (p : Formula (n + 1)) : Formula n

abbreviation OAI.PartitionConsistency.Formula.neg line 23

abbrev neg {n} (p : Formula n) : Formula n := p.imp .bot

abbreviation OAI.PartitionConsistency.Formula.iff line 24

abbrev iff {n} (p q : Formula n) : Formula n := (p.imp q).conj (q.imp p)

definition OAI.PartitionConsistency.Formula.liftRen line 27

def liftRen {n m} (r : Fin n → Fin m) : Fin (n + 1) → Fin (m + 1) :=  Fin.lastCases (Fin.last m) (fun i => (r i).castSucc)

definition OAI.PartitionConsistency.Formula.rename line 31

def rename {n m} (r : Fin n → Fin m) : Formula n → Formula m  | .bot => .bot  | .mem x y => .mem (r x) (r y)  | .equal x y => .equal (r x) (r y)  | .imp p q => .imp (rename r p) (rename r q)  | .conj p q => .conj (rename r p) (rename r q)  | .disj p q => .disj (rename r p) (rename r q)  | .all p => .all (rename (liftRen r) p)  | .ex p => .ex (rename (liftRen r) p)

abbreviation OAI.PartitionConsistency.Formula.raise line 41

abbrev raise {n} (p : Formula n) : Formula (n + 1) := rename Fin.castSucc p

definition OAI.PartitionConsistency.Formula.instantiate line 44

def instantiate {n} (p : Formula (n + 1)) (x : Fin n) : Formula n :=  rename (Fin.lastCases x id) p

definition OAI.PartitionConsistency.Formula.close line 48

def close : {n : ℕ} → Formula n → Formula 0  | 0, p => p  | _ + 1, p => close (.all p)

abbreviation OAI.PartitionConsistency.Sentence line 54
```
abbrev Sentence := Formula 0
```
abbreviation OAI.PartitionConsistency.Theory line 55
```
abbrev Theory := Set Sentence
```

inductive type OAI.PartitionConsistency.Derives line 61

inductive Derives : {n : ℕ} → Set (Formula n) → Formula n → Prop where

Show the remaining 38 lines

  | hyp {n} {Γ : Set (Formula n)} {p} : p ∈ Γ → Derives Γ p  | weaken {n} {Γ Δ : Set (Formula n)} {p} :      Γ ⊆ Δ → Derives Γ p → Derives Δ p  | impI {n} {Γ : Set (Formula n)} {p q} :      Derives (insert p Γ) q → Derives Γ (.imp p q)  | impE {n} {Γ : Set (Formula n)} {p q} :      Derives Γ (.imp p q) → Derives Γ p → Derives Γ q  | botE {n} {Γ : Set (Formula n)} {p} : Derives Γ .bot → Derives Γ p  | classical {n} {Γ : Set (Formula n)} {p} :      Derives (insert p.neg Γ) .bot → Derives Γ p  | conjI {n} {Γ : Set (Formula n)} {p q} :      Derives Γ p → Derives Γ q → Derives Γ (.conj p q)  | conjL {n} {Γ : Set (Formula n)} {p q} :      Derives Γ (.conj p q) → Derives Γ p  | conjR {n} {Γ : Set (Formula n)} {p q} :      Derives Γ (.conj p q) → Derives Γ q  | disjL {n} {Γ : Set (Formula n)} {p q} :      Derives Γ p → Derives Γ (.disj p q)  | disjR {n} {Γ : Set (Formula n)} {p q} :      Derives Γ q → Derives Γ (.disj p q)  | disjE {n} {Γ : Set (Formula n)} {p q r} :      Derives Γ (.disj p q) → Derives (insert p Γ) r →      Derives (insert q Γ) r → Derives Γ r  | allI {n} {Γ : Set (Formula n)} {p : Formula (n + 1)} :      Derives (Formula.raise '' Γ) p → Derives Γ (.all p)  | allE {n} {Γ : Set (Formula n)} {p : Formula (n + 1)} (t : Fin n) :      Derives Γ (.all p) → Derives Γ (p.instantiate t)  | exI {n} {Γ : Set (Formula n)} {p : Formula (n + 1)} (t : Fin n) :      Derives Γ (p.instantiate t) → Derives Γ (.ex p)  | exE {n} {Γ : Set (Formula n)} {p : Formula (n + 1)} {q : Formula n} :      Derives Γ (.ex p) → Derives (insert p (Formula.raise '' Γ)) q.raise →      Derives Γ q  | eqRefl {n} {Γ : Set (Formula n)} (t : Fin n) : Derives Γ (.equal t t)  | eqSubst {n} {Γ : Set (Formula n)} {s t : Fin n} (p : Formula (n + 1)) :      Derives Γ (.equal s t) → Derives Γ (p.instantiate s) →      Derives Γ (p.instantiate t)  | rename {n m} {Γ : Set (Formula n)} {p : Formula n} (r : Fin n → Fin m) :      Derives Γ p → Derives (Formula.rename r '' Γ) (Formula.rename r p)

definition OAI.PartitionConsistency.Theory.at line 102

def Theory.at (T : Theory) (n : ℕ) : Set (Formula n) :=  Formula.rename Fin.elim0 '' T

definition OAI.PartitionConsistency.Consistent line 107

def Consistent (T : Theory) : Prop := ∀ n, ¬ Derives (T.at n) .bot

definition OAI.PartitionConsistency.SetLanguage.subset line 118

def subset {n} (a b : Fin n) : Formula n :=  rename ![a, b] (Formula.all ((Formula.mem 2 0).imp (Formula.mem 2 1)) : Formula 2)

definition OAI.PartitionConsistency.SetLanguage.empty line 122

def empty {n} (a : Fin n) : Formula n :=  rename ![a] (Formula.all (Formula.mem 1 0).neg : Formula 1)

definition OAI.PartitionConsistency.SetLanguage.nonempty line 126

def nonempty {n} (a : Fin n) : Formula n :=  rename ![a] (Formula.ex (Formula.mem 1 0) : Formula 1)

definition OAI.PartitionConsistency.SetLanguage.unorderedPair line 130

def unorderedPair {n} (p x y : Fin n) : Formula n :=  rename ![p, x, y]    (Formula.all ((Formula.mem 3 0).iff ((Formula.equal 3 1).disj (Formula.equal 3 2))) : Formula 3)

definition OAI.PartitionConsistency.SetLanguage.successor line 135

def successor {n} (b a : Fin n) : Formula n :=  rename ![b, a]    (Formula.all ((Formula.mem 2 0).iff ((Formula.mem 2 1).disj (Formula.equal 2 1))) : Formula 2)

definition OAI.PartitionConsistency.SetLanguage.orderedPair line 140

def orderedPair {n} (p x y : Fin n) : Formula n :=  rename ![p, x, y]    (Formula.ex (Formula.ex ((unorderedPair 3 1 1).conj      ((unorderedPair 4 1 2).conj (unorderedPair 0 3 4)))) : Formula 3)

definition OAI.PartitionConsistency.SetLanguage.edge line 146

def edge {n} (f x y : Fin n) : Formula n :=  rename ![f, x, y]    (Formula.ex ((Formula.mem 3 0).conj (orderedPair 3 1 2)) : Formula 3)

definition OAI.PartitionConsistency.SetLanguage.functionOn line 153

def functionOn {n} (f d : Fin n) : Formula n :=  rename ![f, d]    ((Formula.all ((Formula.mem 2 0).imp        (Formula.ex (Formula.ex ((Formula.mem 3 1).conj (orderedPair 2 3 4)))))).conj      (Formula.all ((Formula.mem 2 1).imp        (Formula.ex ((edge 0 2 3).conj (Formula.all ((edge 0 2 4).imp (Formula.equal 4 3))))))) : Formula 2)

definition OAI.PartitionConsistency.SetLanguage.mapsTo line 161

def mapsTo {n} (f d r : Fin n) : Formula n :=  rename ![f, d, r]    ((functionOn 0 1).conj      (Formula.all (Formula.all ((Formula.mem 3 1).imp ((edge 0 3 4).imp (Formula.mem 4 2))))) : Formula 3)

definition OAI.PartitionConsistency.SetLanguage.surjection line 167

def surjection {n} (f d r : Fin n) : Formula n :=  rename ![f, d, r]    ((mapsTo 0 1 2).conj      (Formula.all ((Formula.mem 3 2).imp (Formula.ex ((Formula.mem 4 1).conj (edge 0 4 3))))) : Formula 3)

definition OAI.PartitionConsistency.SetLanguage.injection line 173

def injection {n} (f d r : Fin n) : Formula n :=  rename ![f, d, r]    ((mapsTo 0 1 2).conj      (Formula.all (Formula.all (.all        ((Formula.mem 3 1).imp ((Formula.mem 4 1).imp          (((edge 0 3 5).conj (edge 0 4 5)).imp (Formula.equal 3 4))))))) : Formula 3)

23 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: countable, model and transitive.

Theorem 1.2 of The Partition Principle does not imply Choice

Theorem

Let V be a countable transitive model of \ZFC. There are a set forcing \mathbb Q\in V, a V-generic filter G, and a transitive symmetric submodel W of V[G] such that

V\subseteq W\subseteq V[G],\qquad \Ord^W=\Ord^V,\qquad W\models\ZF+\PP+\ACwo+\neg\AC.

Every countable sequence of elements of V that belongs to W already belongs to V.

Section 1, Introduction Read it in the paper TeX source, line 29

Show the TeX

Let $V$ be a countable transitive model of $\ZFC$. There are a set forcing
$\mathbb Q\in V$, a $V$-generic filter $G$, and a transitive symmetric
submodel $W$ of $V[G]$ such that
\[
 V\subseteq W\subseteq V[G],\qquad
 \Ord^W=\Ord^V,\qquad
 W\models\ZF+\PP+\ACwo+\neg\AC.
\]
Every countable sequence of elements of $V$ that belongs to $W$ already
belongs to $V$.

Other possible paper matches, closest wording first

Theorem 3.3 of The Partition Principle does not imply Choice. Words of the Lean statement, its names split into words, that also appear in the theorem: internal, arbitrary, pure, formula, relation, axiom and five more.
Theorem 1.1 of The Partition Principle does not imply Choice. Words of the Lean statement, its names split into words, that also appear in the theorem: consistent, pp, zf, ac, neg and cwo. Found in the paper’s title, which its main theorem states: principle, partition and choice.
Proposition 3.7 of The Partition Principle does not imply Choice. Words of the Lean statement, its names split into words, that also appear in the theorem: indexed, nonempty, ordinal, family, function, cwo and two more.
Proposition 3.6 of The Partition Principle does not imply Choice. Words of the Lean statement, its names split into words, that also appear in the theorem: bound, surjection, nonempty, image and ordinal.
Theorem 7.5 of The Partition Principle does not imply Choice. Words of the Lean statement, its names split into words, that also appear in the theorem: ordinary, wellorderable, pp, family, function, ordinals and seven more.
Proposition 3.8 of The Partition Principle does not imply Choice. Words of the Lean statement, its names split into words, that also appear in the theorem: wellorderable, neg, ac and models.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 54 s).
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 244’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Stated: PartitionConsistency.main_consistency : Consistent ZF → Consistent (ZF ∪ {PP, ACwo, ¬AC}) (a hand-coded first-order Derives calculus, full ZF with arbitrary Separation/Replacement instances, PP = every internal surjection X ↠ Y yields an internal injection Y ↪ X, ACwo = choice for ordinal-indexed families), which is the summary's first two claims. The third claim, a transitive symmetric extension of any countable transitive model of ZFC with no new countable sequences, appears only as exists_model_partitionPrinciple_without_choice and needs an extra hypothesis (an internal strongly inaccessible K in the ground model, docs: 'containing an internal strongly inaccessible cardinal') and its conclusion lacks ordinal-indexed choice and the no-new-sequences property (docs: 'stronger transitive-model preservation assertions are outside').
Definitions the group asks readers to compare with the paper: Hand-written proof theory and set theory: Formula, Derives (natural deduction), Consistent T := ∀ n, ¬ Derives (T.at n) .bot, ZF := {p | IsZFAxiom p} with separation/replacement coded by variable-index arithmetic, PP, AC, ACwo. The theorem is an implication from Consistent ZF, so a coding error that made the encoded ZF inconsistent would make it vacuous; PartitionPrinciple uses its own SetFormula/Realize semantics on ZFSet. (axiom_files hit for LocalRecords.lean is a doc-comment false positive.)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: In the countable-transitive-model clause, the Lean adds an internal strongly inaccessible cardinal to the hypotheses and drops from the conclusion ACwo-in-N, "symmetric extension" and "no new countable sequences of ground-model elements". The relative-consistency clauses are fully stated.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the relative-consistency implication: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and failure of the Axiom of Choice.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

324 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 414 to its own declarations, 94 to Mathlib’s documentation. 47 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInamespace PartitionConsistency/-- Pure first-order membership syntax with equality. `n` variables are in scope.A quantifier binds the last variable in its body. There are no atoms/urelements,function symbols, constants, semantic predicates, or mathematical assumptionsin this syntax. -/inductive Formula : ℕ → Type where  | bot {n} : Formula n  | mem {n} (x y : Fin n) : Formula n  | equal {n} (x y : Fin n) : Formula n  | imp {n} (p q : Formula n) : Formula n  | conj {n} (p q : Formula n) : Formula n  | disj {n} (p q : Formula n) : Formula n  | all {n} (p : Formula (n + 1)) : Formula n  | ex {n} (p : Formula (n + 1)) : Formula nnamespace Formulaabbrev neg {n} (p : Formula n) : Formula n := p.imp .botabbrev iff {n} (p q : Formula n) : Formula n := (p.imp q).conj (q.imp p)/-- Under a binder, a renaming leaves the newly bound variable bound. -/def liftRen {n m} (r : Fin n → Fin m) : Fin (n + 1) → Fin (m + 1) :=  Fin.lastCases (Fin.last m) (fun i => (r i).castSucc)/-- Capture-avoiding renaming; in this relational language all terms are variables. -/def rename {n m} (r : Fin n → Fin m) : Formula n → Formula m  | .bot => .bot  | .mem x y => .mem (r x) (r y)  | .equal x y => .equal (r x) (r y)  | .imp p q => .imp (rename r p) (rename r q)  | .conj p q => .conj (rename r p) (rename r q)  | .disj p q => .disj (rename r p) (rename r q)  | .all p => .all (rename (liftRen r) p)  | .ex p => .ex (rename (liftRen r) p)abbrev raise {n} (p : Formula n) : Formula (n + 1) := rename Fin.castSucc p/-- Substitution of a variable term for the variable just quantified. -/def instantiate {n} (p : Formula (n + 1)) (x : Fin n) : Formula n :=  rename (Fin.lastCases x id) p/-- Universal closure of all the free variables. -/def close : {n : ℕ} → Formula n → Formula 0  | 0, p => p  | _ + 1, p => close (.all p)end Formulaabbrev Sentence := Formula 0abbrev Theory := Set Sentence/-- Classical first-order natural deduction with equality. Hypotheses may be aninfinite theory, but a derivation is a finite inductive proof. The `allI` and`exE` rules use a fresh last variable, and shift all assumptions to avoidcapture. We allow finitely many free variables in proofs of closed sentences. -/inductive Derives : {n : ℕ} → Set (Formula n) → Formula n → Prop where  | hyp {n} {Γ : Set (Formula n)} {p} : p ∈ Γ → Derives Γ p  | weaken {n} {Γ Δ : Set (Formula n)} {p} :      Γ ⊆ Δ → Derives Γ p → Derives Δ p  | impI {n} {Γ : Set (Formula n)} {p q} :      Derives (insert p Γ) q → Derives Γ (.imp p q)  | impE {n} {Γ : Set (Formula n)} {p q} :      Derives Γ (.imp p q) → Derives Γ p → Derives Γ q  | botE {n} {Γ : Set (Formula n)} {p} : Derives Γ .bot → Derives Γ p  | classical {n} {Γ : Set (Formula n)} {p} :      Derives (insert p.neg Γ) .bot → Derives Γ p  | conjI {n} {Γ : Set (Formula n)} {p q} :      Derives Γ p → Derives Γ q → Derives Γ (.conj p q)  | conjL {n} {Γ : Set (Formula n)} {p q} :      Derives Γ (.conj p q) → Derives Γ p  | conjR {n} {Γ : Set (Formula n)} {p q} :      Derives Γ (.conj p q) → Derives Γ q  | disjL {n} {Γ : Set (Formula n)} {p q} :      Derives Γ p → Derives Γ (.disj p q)  | disjR {n} {Γ : Set (Formula n)} {p q} :      Derives Γ q → Derives Γ (.disj p q)  | disjE {n} {Γ : Set (Formula n)} {p q r} :      Derives Γ (.disj p q) → Derives (insert p Γ) r →      Derives (insert q Γ) r → Derives Γ r  | allI {n} {Γ : Set (Formula n)} {p : Formula (n + 1)} :      Derives (Formula.raise '' Γ) p → Derives Γ (.all p)  | allE {n} {Γ : Set (Formula n)} {p : Formula (n + 1)} (t : Fin n) :      Derives Γ (.all p) → Derives Γ (p.instantiate t)  | exI {n} {Γ : Set (Formula n)} {p : Formula (n + 1)} (t : Fin n) :      Derives Γ (p.instantiate t) → Derives Γ (.ex p)  | exE {n} {Γ : Set (Formula n)} {p : Formula (n + 1)} {q : Formula n} :      Derives Γ (.ex p) → Derives (insert p (Formula.raise '' Γ)) q.raise →      Derives Γ q  | eqRefl {n} {Γ : Set (Formula n)} (t : Fin n) : Derives Γ (.equal t t)  | eqSubst {n} {Γ : Set (Formula n)} {s t : Fin n} (p : Formula (n + 1)) :      Derives Γ (.equal s t) → Derives Γ (p.instantiate s) →      Derives Γ (p.instantiate t)  | rename {n m} {Γ : Set (Formula n)} {p : Formula n} (r : Fin n → Fin m) :      Derives Γ p → Derives (Formula.rename r '' Γ) (Formula.rename r p)/-- Sentences are usable as open hypotheses with any finite supply of variables. -/def Theory.at (T : Theory) (n : ℕ) : Set (Formula n) :=  Formula.rename Fin.elim0 '' T/-- Ordinary syntactic consistency: there is no finite natural-deduction proofof falsity from the theory, with any finite supply of free variables. -/def Consistent (T : Theory) : Prop := ∀ n, ¬ Derives (T.at n) .botnamespace SetLanguageopen Formula/-! All abbreviations in this namespace expand to formulas of the pure languageof equality and membership. Functions are sets of Kuratowski pairs *inside thestructure*. In particular, no quantification over external Lean functions isused in PP, Choice, or ordinal-indexed Choice. -//-- `a ⊆ b`. -/def subset {n} (a b : Fin n) : Formula n :=  rename ![a, b] (Formula.all ((Formula.mem 2 0).imp (Formula.mem 2 1)) : Formula 2)/-- `a = ∅`. -/def empty {n} (a : Fin n) : Formula n :=  rename ![a] (Formula.all (Formula.mem 1 0).neg : Formula 1)/-- `a ≠ ∅`, stated as existence of a member. -/def nonempty {n} (a : Fin n) : Formula n :=  rename ![a] (Formula.ex (Formula.mem 1 0) : Formula 1)/-- `p = {x,y}`. -/def unorderedPair {n} (p x y : Fin n) : Formula n :=  rename ![p, x, y]    (Formula.all ((Formula.mem 3 0).iff ((Formula.equal 3 1).disj (Formula.equal 3 2))) : Formula 3)/-- `b = a ∪ {a}`. -/def successor {n} (b a : Fin n) : Formula n :=  rename ![b, a]    (Formula.all ((Formula.mem 2 0).iff ((Formula.mem 2 1).disj (Formula.equal 2 1))) : Formula 2)/-- Kuratowski ordered pair: `p = {{x},{x,y}}`. -/def orderedPair {n} (p x y : Fin n) : Formula n :=  rename ![p, x, y]    (Formula.ex (Formula.ex ((unorderedPair 3 1 1).conj      ((unorderedPair 4 1 2).conj (unorderedPair 0 3 4)))) : Formula 3)/-- The internal graph relation `⟨x,y⟩ ∈ f`. -/def edge {n} (f x y : Fin n) : Formula n :=  rename ![f, x, y]    (Formula.ex ((Formula.mem 3 0).conj (orderedPair 3 1 2)) : Formula 3)/-- `f` is a set graph of a function whose domain is exactly `d`.The first conjunct excludes non-pairs and graph entries outside the domain;the second conjunct asserts totality and single-valuedness. -/def functionOn {n} (f d : Fin n) : Formula n :=  rename ![f, d]    ((Formula.all ((Formula.mem 2 0).imp        (Formula.ex (Formula.ex ((Formula.mem 3 1).conj (orderedPair 2 3 4)))))).conj      (Formula.all ((Formula.mem 2 1).imp        (Formula.ex ((edge 0 2 3).conj (Formula.all ((edge 0 2 4).imp (Formula.equal 4 3))))))) : Formula 2)/-- Internal set function `f : d → r`. -/def mapsTo {n} (f d r : Fin n) : Formula n :=  rename ![f, d, r]    ((functionOn 0 1).conj      (Formula.all (Formula.all ((Formula.mem 3 1).imp ((edge 0 3 4).imp (Formula.mem 4 2))))) : Formula 3)/-- Internal set surjection `f : d ↠ r`. -/def surjection {n} (f d r : Fin n) : Formula n :=  rename ![f, d, r]    ((mapsTo 0 1 2).conj      (Formula.all ((Formula.mem 3 2).imp (Formula.ex ((Formula.mem 4 1).conj (edge 0 4 3))))) : Formula 3)/-- Internal set injection `f : d ↪ r`. -/def injection {n} (f d r : Fin n) : Formula n :=  rename ![f, d, r]    ((mapsTo 0 1 2).conj      (Formula.all (Formula.all (.all        ((Formula.mem 3 1).imp ((Formula.mem 4 1).imp          (((edge 0 3 5).conj (edge 0 4 5)).imp (Formula.equal 3 4))))))) : Formula 3)/-- Transitivity of a set (not transitivity of the external structure). -/def transitiveSet {n} (a : Fin n) : Formula n :=  rename ![a]    (Formula.all (Formula.all ((Formula.mem 1 0).imp ((Formula.mem 2 1).imp (Formula.mem 2 0)))) : Formula 1)/-- Membership is an internal strict linear order on `a`. -/def membershipLinear {n} (a : Fin n) : Formula n :=  rename ![a]    ((Formula.all ((Formula.mem 1 0).imp (Formula.mem 1 1).neg)).conj      ((Formula.all (Formula.all (Formula.all ((Formula.mem 1 0).imp ((Formula.mem 2 0).imp ((Formula.mem 3 0).imp        (((Formula.mem 1 2).conj (Formula.mem 2 3)).imp (Formula.mem 1 3)))))))).conj        (Formula.all (Formula.all ((Formula.mem 1 0).imp ((Formula.mem 2 0).imp          ((Formula.equal 1 2).disj ((Formula.mem 1 2).disj (Formula.mem 2 1)))))))) : Formula 1)/-- Every nonempty internal subset of `a` has a membership-minimal element.This does NOT assert external well-foundedness of the relation. -/def membershipWellFounded {n} (a : Fin n) : Formula n :=  rename ![a]    (Formula.all (((subset 1 0).conj (nonempty 1)).imp      (Formula.ex ((Formula.mem 2 1).conj (Formula.all ((Formula.mem 3 1).imp (Formula.mem 3 2).neg))))) : Formula 1)/-- A von Neumann ordinal: a transitive set internally wellordered by membership. -/def ordinal {n} (a : Fin n) : Formula n :=  (transitiveSet a).conj ((membershipLinear a).conj (membershipWellFounded a))/-- A choice function on the set `A` of nonempty sets. -/def choiceOn {n} (c A : Fin n) : Formula n :=  rename ![c, A]    ((functionOn 0 1).conj      (Formula.all ((Formula.mem 2 1).imp (Formula.ex ((edge 0 2 3).conj (Formula.mem 3 2))))) : Formula 2)/-- A graph `F` coding a family of nonempty sets indexed by ordinal `η`. -/def nonemptyFamily {n} (F η : Fin n) : Formula n :=  rename ![F, η]    ((functionOn 0 1).conj      (Formula.all (Formula.all ((Formula.mem 2 1).imp ((edge 0 2 3).imp (nonempty 3))))) : Formula 2)/-- `c(ξ) ∈ F(ξ)` for every `ξ ∈ η`, with `c` an internal function. -/def choiceForFamily {n} (c η F : Fin n) : Formula n :=  rename ![c, η, F]    ((functionOn 0 1).conj      (Formula.all (Formula.all ((Formula.mem 3 1).imp ((edge 2 3 4).imp        (Formula.ex ((edge 0 3 5).conj (Formula.mem 5 4))))))) : Formula 3)/-- The Partition Principle. The reverse injection is not required to be asection of the original surjection. Both `X` and `Y` range over all sets. -/def PP : Sentence :=  Formula.all (Formula.all ((Formula.ex (surjection 2 0 1)).imp (Formula.ex (injection 2 1 0))))/-- Ordinary Choice for every set of nonempty sets. -/def AC : Sentence :=  Formula.all ((Formula.all ((Formula.mem 1 0).imp (nonempty 1))).imp (Formula.ex (choiceOn 1 0)))/-- Choice for every ordinal-indexed family of nonempty sets, including ordinalswhich may fail to be well founded from outside a model. -/def ACwo : Sentence :=  Formula.all ((ordinal 0).imp    (Formula.all ((nonemptyFamily 1 0).imp (Formula.ex (choiceForFamily 2 0 1)))))/-! Full pure-set ZF, not a finite fragment and not ZFA. -/def extensionality : Sentence :=  Formula.all (Formula.all ((Formula.all ((Formula.mem 2 0).iff (Formula.mem 2 1))).imp (Formula.equal 0 1)))def emptySet : Sentence := Formula.ex (empty 0)def pairing : Sentence := Formula.all (Formula.all (Formula.ex (unorderedPair 2 0 1)))def union : Sentence :=  Formula.all (Formula.ex (Formula.all ((Formula.mem 2 1).iff (Formula.ex ((Formula.mem 3 0).conj (Formula.mem 2 3))))))def powerSet : Sentence := Formula.all (Formula.ex (Formula.all ((Formula.mem 2 1).iff (subset 2 0))))def infinity : Sentence :=  Formula.ex ((Formula.ex ((Formula.mem 1 0).conj (empty 1))).conj    (Formula.all ((Formula.mem 1 0).imp (Formula.ex ((Formula.mem 2 0).conj (successor 2 1))))))def foundation : Sentence :=  Formula.all ((nonempty 0).imp    (Formula.ex ((Formula.mem 1 0).conj (Formula.all ((Formula.mem 2 1).imp (Formula.mem 2 0).neg)))))/-- The instance of Separation for an entirely arbitrary formula `p(params,x)`.All parameters are universally closed. Neither the quantifier complexity of`p` nor the number of parameters is bounded. -/def separation {n} (p : Formula (n + 1)) : Sentence :=  let A : Fin (n + 3) := (Fin.last n).castSucc.castSucc  let B : Fin (n + 3) := (Fin.last (n + 1)).castSucc  let x : Fin (n + 3) := Fin.last (n + 2)  let r : Fin (n + 1) → Fin (n + 3) :=    Fin.lastCases x (fun i => i.castSucc.castSucc.castSucc)  close (Formula.all (Formula.ex (Formula.all ((Formula.mem x B).iff ((Formula.mem x A).conj (rename r p))))))/-- Substitution into a binary formula with `n` additional parameters. -/def relation {n m} (p : Formula (n + 2)) (params : Fin n → Fin m) (x y : Fin m) :    Formula m :=  rename (Fin.lastCases y (Fin.lastCases x params)) p/-- The instance of Replacement for an entirely arbitrary formula`p(params,x,y)`: if it is single-valued and total on `A`, its image is a set.As with Separation, the parameters are all universally closed. -/def replacement {n} (p : Formula (n + 2)) : Sentence :=  let unique : Formula (n + 3) :=    Formula.all ((relation p (fun i => i.castSucc.castSucc.castSucc.castSucc)      (Fin.last (n + 1)).castSucc.castSucc (Fin.last (n + 3))).imp      (Formula.equal (Fin.last (n + 3)) (Fin.last (n + 2)).castSucc))  let totalFunctional : Formula (n + 1) :=    Formula.all ((Formula.mem (Fin.last (n + 1)) (Fin.last n).castSucc).imp      (Formula.ex ((relation p (fun i => i.castSucc.castSucc.castSucc)        (Fin.last (n + 1)).castSucc (Fin.last (n + 2))).conj unique)))  let image : Formula (n + 1) :=    Formula.ex (Formula.all ((Formula.mem (Fin.last (n + 2)) (Fin.last (n + 1)).castSucc).iff      (Formula.ex ((Formula.mem (Fin.last (n + 3)) (Fin.last n).castSucc.castSucc.castSucc).conj        (relation p (fun i => i.castSucc.castSucc.castSucc.castSucc)          (Fin.last (n + 3)) (Fin.last (n + 2)).castSucc)))))  close (Formula.all (totalFunctional.imp image))/-- Exactly the axioms and all schema instances of pure first-order ZF. -/inductive IsZFAxiom : Sentence → Prop where  | extensionality : IsZFAxiom extensionality  | emptySet : IsZFAxiom emptySet  | pairing : IsZFAxiom pairing  | union : IsZFAxiom union  | powerSet : IsZFAxiom powerSet  | infinity : IsZFAxiom infinity  | foundation : IsZFAxiom foundation  | separation {n} (p : Formula (n + 1)) : IsZFAxiom (separation p)  | replacement {n} (p : Formula (n + 2)) : IsZFAxiom (replacement p)/-- Zermelo–Fraenkel set theory, without Choice. -/def ZF : Theory := {p | IsZFAxiom p}/-- ZF with the Partition Principle, ordinal-indexed Choice and failure of Choice. -/def targetTheory : Theory := ZF ∪ {PP, ACwo, AC.neg}end SetLanguage/-- Relative consistency of ZF with the Partition Principle, ordinal-indexedChoice and failure of Choice. -/theorem main_consistency :    Consistent SetLanguage.ZF → Consistent SetLanguage.targetTheory := by  sorryend PartitionConsistencyend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

The Partition Principle does not imply Choice

Theorem

If \ZF is consistent, then so is \ZF+\PP+\ACwo+\neg\AC.

Section 1, Introduction Read it in the paper TeX source, line 23

Show the TeX

If $\ZF$ is consistent, then so is $\ZF+\PP+\ACwo+\neg\AC$.

Theorem

Let V be a countable transitive model of \ZFC. There are a set forcing \mathbb Q\in V, a V-generic filter G, and a transitive symmetric submodel W of V[G] such that

V\subseteq W\subseteq V[G],\qquad \Ord^W=\Ord^V,\qquad W\models\ZF+\PP+\ACwo+\neg\AC.

Every countable sequence of elements of V that belongs to W already belongs to V.

Section 1, Introduction Read it in the paper TeX source, line 29

Show the TeX

Let $V$ be a countable transitive model of $\ZFC$. There are a set forcing
$\mathbb Q\in V$, a $V$-generic filter $G$, and a transitive symmetric
submodel $W$ of $V[G]$ such that
\[
 V\subseteq W\subseteq V[G],\qquad
 \Ord^W=\Ord^V,\qquad
 W\models\ZF+\PP+\ACwo+\neg\AC.
\]
Every countable sequence of elements of $V$ that belongs to $W$ already
belongs to $V$.

Configuration and file facts

File: lean/ComparatorChallenges/PartitionConsistency.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 244 link it.
Solution module: OAI.SetTheory.PartitionConsistency.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 47 local vocabulary declarations
Lines of Lean, without comments or blank lines: 200
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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