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Subject and independence

Role: independent reviewer in a fresh context, commissioned for refutation and given only the assignment. The reviewer took no part in writing the page, the sibling reconstruction pages or the library card, and opened no other review.

Subject: path wiki/research/erdos_501/lee_theorem_1_1_reconstruction.md as it stood at 2026-09-28T05:03:27Z, read whole as of that time.

Artifact: the six-page second-version PDF beside the library card Lee (2026), file lee_2026_relative_independence_erdos_problem_501.pdf, date line "June 1, 2026"; its physical page numbers coincide with the printed ones. Physical pages 1--3 were read clause by clause in the text layer and on page images rendered at 110 dpi (pages 1, 2 and 3), every displayed formula being checked on the images: the statements of Theorem 1.1 and Corollary 1.2 and the remark between them (p. 1); the comparison (1), the statement of Lemma 2.1, displays (2) and (3), the definition of $G$, the recursion and the induction (p. 2); the independence argument (p. 3). Pages 4--6 were read in the text layer only, to confirm that the corollary is proved nowhere else in the note and that Appendix A (pp. 5--6) is the counterexample under CH.

Allowed material read: the Statement section of the Lemma 2.1 page and, because the page under review cites it, that page's Definitions (the comparison $\nu\le m^*$); the Statement section of the CH counterexample page; the Statement section of the Glazer Theorem 1.1 page, for the cross-link check only; the provenance paragraph of the Lee library card; the statement of Problem 501; and the wiki sections named in the assignment (whole-claim report, audit checklist, source fidelity, page mechanics).

Exposures, none of which changed a finding: (a) the library card was displayed whole, so its Bears-on, Read-status, Overview, Lean-files and Relation sections were seen, and they carry status-adjacent sentences about supersession and site adoption; (b) the Lemma 2.1 page was displayed whole, including its Proof and Boundary; (c) the CH counterexample page and the Glazer Theorem 1.1 page were displayed with their Source and Standing paragraphs; (d) the problem page has no Statement heading, and the region before its assessment section, displayed to reach the statement, contains the page's Status paragraph, its Source paragraph, its References and its Formalization paragraph, while its frontmatter description also names the status; (e) the wider "canonical failure modes" section of the verification wiki was displayed together with the audit checklist; (f) a directory listing showed the file names of four other reviews under evidence/verify/, none of which was opened. Nothing among the private working files, no evidence program and no web search was consulted.

Restatement

Convention. The source writes $\subset$ for inclusion allowing equality; $m$ and $m^*$ are Lebesgue measure and Lebesgue outer measure on $\mathbb R$; a measure extending Lebesgue measure is a countably additive $\nu\colon\mathcal P(\mathbb R)\to[0,\infty]$ with $\nu(S)=m(S)$ for every Lebesgue measurable $S$; FMEA asserts that such a $\nu$ exists. A set $X$ is independent for a family $(A_y)_{y\in\mathbb R}$ when no member of $X$ lies in the set indexed by a different member of $X$: $x\notin A_y$ for all $x,y\in X$ with $x\ne y$, which covers both orders of every pair.

Theorem 1.1. Work in ZFC and assume FMEA. Let $(A_y)_{y\in\mathbb R}$ be any family of subsets of $\mathbb R$ indexed by all reals, with no boundedness and no measurability assumed, such that $m^*(A_y)<1$ for every $y\in\mathbb R$; the bound $1$ is the same for every $y$ and strict. Then there is an infinite $X\subseteq\mathbb R$ such that $x\notin A_y$ whenever $x,y\in X$ and $x\ne y$. Consequence, stated by the source as a remark: since the bounded families form a subclass, ZFC + FMEA proves $P$, the positive answer to the first question of Problem 501, which asks the same for families of bounded sets with $m^*(A_y)<1$.

Corollary 1.2. If the theory ZFC + FMEA is consistent, then $P$ is independent of ZFC: ZFC proves neither $P$ nor $\neg P$. In particular, if ZFC together with the existence of a measurable cardinal is consistent, then $P$ is independent of ZFC.

Checklist

• Quantifiers and scope. Pass. "For every $y\in\mathbb R$" is carried from the source's "with Lebesgue outer measure $<1$" into the page's hypothesis; the recursion uses it at every stage through Lemma 2.1 and at $f(n)$ through $\nu(A_{f(n)})<1$. The conclusion is proved for all distinct pairs in both orders ($f(j)\notin A_{f(i)}$ and $f(i)\notin A_{f(j)}$ for $i<j$). The base case $n=0$ is stated with $C_{f\restriction0}=\mathbb R$. Nothing is "almost all".
• Circularity. Pass. $G$ is total on $\mathbb R^{<\omega}$ because of the default value $r_0$, so $f$ exists by the recursion theorem before the induction begins; the induction hypothesis $\nu(C_{f\restriction n})=\infty$ is not the conclusion; and Lemma 2.1 is proved on its own page from the section inequality without Theorem 1.1.
• Model and convention changes. Pass. $\subseteq$ renders the source's $\subset$; the CH page's $\lambda^*$ and this page's $m^*$ are both Lebesgue outer measure; $\nu$ has the same type and extension property as in the source; no relaxed or averaged object replaces the family.
• Finite and statistical overreach. Inapplicable: no finite case, sample or heuristic is used anywhere.
• Uniformity. Inapplicable: no constant, error term or exchange of limits appears. The one uniform bound, $m^*(A_y)<1$, is a hypothesis; the recursion uses it only at the strength "finite", and Lemma 2.1 uses it at the strength "$<1$".
• Extremal conclusions. Inapplicable: the theorem asserts the existence of an infinite independent set and claims no extremum or sharpness.
• Consequences and composition. Pass. "So the theorem implies $P$ under FMEA": bounded families are a subclass, checked. "So $\mathrm{Con}(\mathrm{ZFC}+P)$": from $\mathrm{ZFC}+\mathrm{FMEA}\vdash P$, checked. "Hence $\mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})$": the imported Gödel step, named as imported. "So $\mathrm{Con}(\mathrm{ZFC}+\neg P)$": uses the CH page's Statement at exactly the strength "CH implies $\neg P$". Lemma 2.1 is consumed at exactly its statement with all three hypotheses present ($\nu$ a total extension of $m$, $m^*(A_y)<1$ for all $y$, $\nu(C)=\infty$). The composition inherits Lemma 2.1 and the CH counterexample at author-recorded standing and the equiconsistency from a source that is not held; the page's standing sentence claims no more.
• Computation. Inapplicable: the page contains no computation.
• Reproduction. Inapplicable: the page states no rerun command and no coverage claim.
• Source and verdict fidelity. Pass with one suggested correction. Both statements match the source in every hypothesis, quantifier and conclusion; every locator was checked (Theorem 1.1, Corollary 1.2 and the boundedness remark on p. 1; Section 2 on pp. 2--3; displays (1), (2) and (3) on p. 2; the reference to Fremlin's notes with "1D(e), 2E" on p. 1); the standing sentence claims author-recorded standing only. The corollary's derivation is supplied by the page and not labeled as such (F1), and the Boundary sentence about Glazer's argument reads more strongly than that page's Statement (F2).

Weakest steps

1. The measure of the next pool. Let $S=C_{f\restriction n}\setminus B_{f(n)}$ and $T=A_{f(n)}\cup\{f(n)\}$. Applying $\mathbb R\setminus(U\cup V)=(\mathbb R\setminus U)\setminus V$ to the union defining $C_{f\restriction(n+1)}$, with $U$ the union over $i<n$ and $V=A_{f(n)}\cup B_{f(n)}\cup\{f(n)\}$, and then once more inside,

$$C_{f\restriction(n+1)} =C_{f\restriction n}\setminus\bigl(A_{f(n)}\cup B_{f(n)}\cup\{f(n)\}\bigr) =S\setminus T.$$

Since $S\subseteq(S\setminus T)\cup T$ and $\nu$ is defined on every subset of $\mathbb R$ and finitely subadditive, $\infty=\nu(S)\le\nu(S\setminus T)+\nu(T)$. Further $\nu(T)\le\nu(A_{f(n)})+\nu(\{f(n)\})\le m^*(A_{f(n)})+0<1$, using the comparison and $\nu(\{p\})=m(\{p\})=0$. Hence $\nu(S\setminus T)=\infty$. The comparison itself: if $S'\subseteq\bigcup_iJ_i$ with open intervals $J_i$, countable subadditivity and the extension property give $\nu(S')\le\sum_i\nu(J_i)=\sum_im(J_i)$, and the infimum over such covers is $m^*(S')$. Composition: this is the induction hypothesis at $n+1$, which is exactly what Lemma 2.1 needs at the next stage; without it the pool $Q_{f\restriction(n+1)}$ could be empty and $f(n+1)$ would fall to the default $r_0$, which need not lie in any pool.

2. Nonemptiness of the pool and the location of $f(n)$. Given $\nu(C_{f\restriction n})=\infty$, Lemma 2.1 with $C=C_{f\restriction n}$ (its hypotheses: $\nu$ a total extension of $m$; $m^*(A_y)<1$ for all $y$; $\nu(C)=\infty$) yields $a\in C_{f\restriction n}$ with $\nu(C_{f\restriction n}\setminus B_a)=\infty$, that is $a\in Q_{f\restriction n}$ by (3). A nonempty subset of $\mathbb R$ has a $\preceq$-least element, so $G(f\restriction n)$ is that element; it lies in $Q_{f\restriction n}\subseteq C_{f\restriction n}$ and satisfies $\nu(C_{f\restriction n}\setminus B_{f(n)})=\infty$. Composition: this supplies both inputs of step 1, and $f(n)\in C_{f\restriction n}$ is also the fact used as $f(j)\in C_{f\restriction j}$ when the independent set is checked. For $i<j$, $C_{f\restriction j}\subseteq C_{f\restriction(i+1)}$ because the union removed grows with the sequence, so $f(j)$ avoids $\{f(i)\}$, $A_{f(i)}$ and $B_{f(i)}$; and $f(j)\notin B_{f(i)}$ unfolds, by $B_{f(i)}=\{y:f(i)\in A_y\}$, to $f(i)\notin A_{f(j)}$. Injectivity of $f$ makes $X$ infinite.

3. The corollary. Positive half: Theorem 1.1 is proved in ZFC + FMEA and its conclusion restricted to bounded families is $P$, so every model of ZFC + FMEA is a model of ZFC + $P$, and ZFC does not prove $\neg P$, since otherwise ZFC + FMEA would prove both $P$ and $\neg P$. Negative half: a model of ZFC + FMEA is a model of ZFC; its constructible universe is a model of ZFC + CH (Gödel, imported); ZFC + CH proves $\neg P$ by the CH counterexample; so ZFC + $\neg P$ has a model and ZFC does not prove $P$. Second sentence: the imported direction, from the consistency of a measurable cardinal to the consistency of ZFC + FMEA, feeds the first sentence. Composition: the two halves are independent of each other and each consumes exactly one page-level input.

Strongest attack

The attack aimed at the corollary, where reconstructions of independence results most often slip. First, FMEA refutes CH: under CH no total extension of Lebesgue measure exists (a classical theorem), so the negative half cannot be read off inside a model of FMEA, and a reconstruction that argued "the model of FMEA also gives $\neg P$" would be wrong. The page does not do this: it passes from $\mathrm{Con}(\mathrm{ZFC}+\mathrm{FMEA})$ to $\mathrm{Con}(\mathrm{ZFC})$ and only then to $\mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})$ through the constructible universe, which is the correct route, and it names the Gödel step as imported. Second, the "in particular" clause could use the wrong direction of the equiconsistency; the page uses the direction from the consistency of a measurable cardinal to $\mathrm{Con}(\mathrm{ZFC}+\mathrm{FMEA})$, which is the direction the clause needs. Third, "independent of ZFC" needs both non-provabilities, and the page derives both. The attack failed; its residue is the labeling defect F1, since the source proves the corollary only by one sentence and the page's derivation is supplied.

A second attack tried to make the recursion circular or the default value $r_0$ load-bearing: the induction might presuppose that the pools stay nonempty. It failed because $G$ is total, $f$ exists by the recursion theorem before any measure is computed, and the induction proves $\nu(C_{f\restriction n})=\infty$ from the previous stage alone, after which Lemma 2.1, not the recursion, supplies nonemptiness. A third attack looked for a dropped order of the independence pairs; both orders are derived, as shown in weakest step 2.

Premises

• Lemma 2.1, consumed from the Lemma 2.1 page. Interface: for $\nu$ a countably additive measure on $\mathcal P(\mathbb R)$ extending Lebesgue measure and $(A_y)_{y\in\mathbb R}$ with $m^*(A_y)<1$ for every $y$, every $C\subseteq\mathbb R$ with $\nu(C)=\infty$ contains an $x$ with $\nu(C\setminus B_x)=\infty$. Standing: author-recorded reconstruction. Reading depth: its Statement and Definitions; the source's statement on p. 2 read on the page image; the source's proof (pp. 4--5) not verified here.
• The comparison $\nu(S)\le m^*(S)$ for every $S\subseteq\mathbb R$: the source's display (1), p. 2; re-derived in weakest step 1.
• CH implies $\neg P$, consumed from the Statement of the CH counterexample page. Standing: author-recorded reconstruction; the source's Appendix A (pp. 5--6) read in the text layer only; not re-verified here.
• Gödel's theorem that the constructible universe of any model of ZFC satisfies ZFC + CH, hence $\mathrm{Con}(\mathrm{ZFC})$ implies $\mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})$: standard, not held, named as imported by the page.
• The equiconsistency of FMEA with a measurable cardinal, cited by the source to Fremlin's notes on real-valued-measurable cardinals, 1D(e) and 2E: not held in the library, so the cited version could not be compared; only the direction from the consistency of a measurable cardinal to $\mathrm{Con}(\mathrm{ZFC}+\mathrm{FMEA})$ is used; named as imported by the page.
• Background ZFC: a well-ordering of $\mathbb R$ (the axiom of choice) and the recursion theorem on $\omega$.
• Explicit assumptions beyond these: none. No batch acceptance order.

Findings

F1. Severity: suggested. Location: "## Proof of Corollary 1.2", "Theorem 1.1 gives $\mathrm{ZFC}+\mathrm{FMEA}\vdash P$, so ...". Defect: the source proves the corollary only by the sentence "Combining Theorem 1.1 with Hechler's theorem gives the following consequence" (p. 1, the paragraph before Corollary 1.2) and never mentions the constructible universe or the step from $\mathrm{Con}(\mathrm{ZFC})$ to $\mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})$. The page presents a full derivation under a proof heading without stating that the derivation is supplied here; the Standing paragraph lists Gödel's theorem as imported for the corollary but does not say that the source does not invoke it. Witness: p. 1, the sentence after "see [3, 1D(e), 2E]". Replacement: open the section with "The source justifies the corollary in one sentence, by combining Theorem 1.1 with Hechler's theorem (p. 1); the derivation below is supplied here."

F2. Severity: note. Location: Boundary, "The measure-extension hypothesis is what Glazer's argument ... removes by forcing." Defect: the sentence can be read as saying that Glazer's argument proves the theorem's conclusion outright. The Statement on the Glazer Theorem 1.1 page proves it inside $M[G]$, the $\omega_2$-random-real extension of a model of CH, so FMEA is replaced by a forcing-extension hypothesis, and what is removed is the large-cardinal assumption of Corollary 1.2. Witness: that page's Statement. Replacement: "Glazer's argument ... replaces it by working inside the $\omega_2$-random-real extension of a model of CH, which brings the consistency assumption of Corollary 1.2 down from a measurable cardinal to $\mathrm{Con}(\mathrm{ZFC})$."

F3. Severity: note. Location: "By the comparison $\nu\le m^*$ recorded on the Lemma 2.1 page". Defect: at this step the source cites its display (1) on p. 2; the page's locator points only to the sibling page, where the comparison sits in the Definitions section rather than in the Statement. Witness: p. 2, "By (1), $\nu(A_{f(n)})\le m^*(A_{f(n)})<1$". Replacement: "By the comparison $\nu\le m^*$ (the source's (1), p. 2, recorded in the Definitions of the Lemma 2.1 page)".

F4. Severity: note. Location: Standing, "the equiconsistency of FMEA with a measurable cardinal". Defect: the notes cited are not held in the library, and only the direction from a measurable cardinal to FMEA is used; the paragraph says neither. Witness: p. 1, "FMEA is equiconsistent with the existence of a measurable cardinal; see [3, 1D(e), 2E]"; no library folder holds the notes. Replacement: "... 1D(e) and 2E, not held; only the direction from the consistency of a measurable cardinal to $\mathrm{Con}(\mathrm{ZFC}+\mathrm{FMEA})$ is used; ...".

F5. Severity: note. Location: Statement, "Boundedness is not assumed, so the theorem implies $P$ under FMEA." Defect: the sentence sits inside the bold Theorem 1.1 paragraph, while in the source it is the remark following the theorem, not part of it. Witness: p. 1, "Note that Theorem 1.1 does not assume boundedness. Hence Theorem 1.1 implies P under FMEA." Replacement: start a new paragraph, "Remark (source, p. 1). Boundedness is not assumed, so the theorem implies $P$ under FMEA."

Verdict

Source fidelity: faithful. The statements of Theorem 1.1 and Corollary 1.2, the definitions, the displays (2) and (3), the recursion and the induction match the artifact at the stated pages and labels, and no hypothesis, quantifier, constant or boundary case is changed; no correction is required.

The argument as reconstructed: sound. Every deduction was re-derived; the inputs are consumed at exactly their stated strength.

Limitations: Lemma 2.1 and the CH counterexample were consumed at their page Statements and not re-verified; the equiconsistency cited to Fremlin's notes and Gödel's theorem are imported and their sources are not held; pages 4--6 of the artifact were read in the text layer only; the Lean files accompanying the source were not read or built.

This focused review assigns no tier and changes no status.