Independence of the outer-measure free-set question

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ch_counterexample_reconstruction: Reconstructs the construction, written out in both 2026 notes and attributed by them to Hechler, of a family of countable bounded sets under CH with no infinite independent set.

evidence/: Review records for the Problem 501 reconstruction pages: independent focused reviews and a distinct grade, with no executable evidence.

glazer_lemma_2_1_reconstruction: Reconstructs the Tonelli counting argument showing that, when column sections are uniformly bounded, the points of an infinite-measure set whose forbidden rows leave infinite measure form a measurable set of positive measure.

glazer_lemma_2_2_reconstruction: Reconstructs the one-step update of the recursion: removing a selected point's row, column and fiber from an infinite-measure pool leaves a measurable pool of infinite measure.

glazer_lemma_4_1_reconstruction: Reconstructs, with the standard random-forcing facts stated as imports, the reading of a name for a point of a standard Borel space by a Borel function of countably many generic coordinates; also fixes the measure-algebra conventions used by the later forcing pages.

glazer_lemma_4_3_reconstruction: Reconstructs the elementary-submodel and Fodor argument showing that, under CH, every family of omega_2 countable sets has a Delta-subsystem of size omega_2.

glazer_lemma_4_5_reconstruction: Reconstructs the fresh-coordinate Fubini argument showing that the generic points on uncountably many disjoint petals are forced to form a set of outer measure one, with the factorization lemma imported and the joint-measurability point labeled.

glazer_proposition_4_4_reconstruction: Reconstructs the homogenization of omega_2 names under CH: a size-omega_2 subfamily is read by one Borel map from a common countable root and pairwise disjoint petals of one isomorphism type.

glazer_theorem_1_1_reconstruction: Reconstructs the assembly of the main theorem from the forcing interface and the ZFC core, and Corollary 1.2, the relative consistency of both answers to the first question of Problem 501 from the consistency of ZFC.

glazer_theorem_3_2_reconstruction: Reconstructs the forcing-free core: a family admitting a profile certificate has an infinite independent set, by running the selection recursion on a Borel graph built from open envelopes, without ever treating the relation x in A_y as measurable.

glazer_theorem_5_1_reconstruction: Reconstructs the assembly of a profile certificate in the omega_2 random-real extension of a CH ground: random points in each unit interval, names for open envelopes, homogeneous reading, fresh-profile fullness, and a Borel truncation of the read codes.

lee_lemma_2_1_reconstruction: Reconstructs the selection lemma: against a total extension of Lebesgue measure, every set of infinite measure contains a point whose set of forbidding indices leaves infinite measure, by a counting contradiction through the section inequality.

lee_lemma_3_1_reconstruction: Reconstructs the one-sided Fubini inequality for an arbitrary subset of the plane: the upper integral of the total-measure vertical sections is at most the integral of the outer measures of the horizontal sections.

lee_theorem_1_1_reconstruction: Reconstructs the recursion that builds an infinite independent set from the selection lemma under the Full Measure Extension Axiom, and Corollary 1.2, the independence of the first question of Problem 501 relative to a measurable cardinal.

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This folder holds author-recorded reconstructions of the proofs behind the independence of the first question of Problem 501: whether every family $(A_y)_{y\in\mathbb R}$ of bounded sets of Lebesgue outer measure below one has an infinite independent set, an infinite $X\subseteq\mathbb R$ with $x\notin A_y$ for all distinct $x,y\in X$. Write $P$ for the positive assertion. Two self-published 2026 notes prove the positive direction consistent, and both write out the counterexample under CH for the negative direction. Each page names its source pages and result labels, writes out the essential deductions, cites its external inputs as imported theorems, and labels what it leaves out. None is an independent review; nothing here changes the problem's status or assigns a tier.

The random-real argument (Glazer (2026)), which makes $P$ independent of ZFC relative to $\mathrm{Con}(\mathrm{ZFC})$, is reconstructed in the paper's own two halves. The forcing-free core is Lemma 2.1, Lemma 2.2 and Theorem 3.2 (profile certificates give free sets). The forcing module is Lemma 4.1 (countable Borel reading, with the measure-algebra conventions), Lemma 4.3 ($\Delta$-systems under CH), Proposition 4.4 (homogeneous reading), Lemma 4.5 (fresh profiles are outer full, with Lemma 4.2 imported) and Theorem 5.1 (the forcing interface). The assembly and Corollary 1.2 are on the Theorem 1.1 page.

The measure-extension argument (Lee (2026)), which gives the same conclusion from a countably additive extension of Lebesgue measure to all subsets of $\mathbb R$, hence independence relative to a measurable cardinal, is reconstructed as Lemma 3.1 (the section inequality), Lemma 2.1 (selection) and the [[research/erdos_501/lee_theorem_1_1_reconstruction|Theorem 1.1 page]] with Corollary 1.2.

The CH counterexample page reconstructs the construction both notes write out for the negative direction.

Where things stand

Reconstructed. Every labeled result whose proof is in one of the two held PDFs: from Glazer's note, Lemmas 2.1, 2.2, 4.1, 4.3, 4.5, Proposition 4.4, Theorems 3.2, 5.1 and 1.1 and Corollary 1.2; from Lee's note, Lemmas 3.1 and 2.1, Theorem 1.1 and Corollary 1.2; and the CH construction of both. Glazer's Lemma 4.2 (factorization of the measure algebra over disjoint coordinate sets) is imported as the source imports it, from Laczkovich and Miller, and the standard random-forcing facts, Fodor's theorem, the count of Borel maps, the constructible universe and Fremlin's equiconsistency are cited as imports where used.

Labeled points. The joint measurability behind the Fubini display in Glazer's Lemma 4.5 is not argued in the source; the page closes it with a repository-supplied remark (shrink to a closed set, or use that coanalytic sets are universally measurable). Glazer's Section 3 fixes "the standard coding" of open sets without exhibiting one, and Theorem 5.1 fixes a measure-preserving map with null fibers without exhibiting one; both pages supply an instance as a compilation fill. Lee's Lemma 3.1 asserts a positive weight of integral at most one without displaying one; the page displays one. None of these is an author-issued correction, and no gap in either argument was found at the level of the sources.

Reviewed. Each reconstruction page was independently reviewed, as it stood at 2026-09-28T05:03:27Z, by a focused review filed under evidence/verify/, and the thirteen reports were graded by a distinct grade. The graded verdicts, fidelity then argument, as the grade records them: the CH counterexample page, faithful with corrections (C1), sound; Glazer Lemma 2.1, faithful, sound; Lemma 2.2, faithful, sound; Lemma 4.1, faithful with corrections (C2), sound with C2 naming the absoluteness fact the general case rests on; Lemma 4.3, faithful with corrections (C3, in the Boundary paragraph only), sound; Lemma 4.5, faithful with corrections (C4), argument as filed defective at the two appeals to (R2) and sound as corrected by C4; Proposition 4.4, faithful with corrections (C5), sound, the corrected sentence not being load-bearing; Theorem 1.1, faithful, sound; Theorem 3.2, faithful with corrections (C6), sound; Theorem 5.1, faithful, sound; Lee Lemma 2.1, faithful, sound; Lemma 3.1, faithful with corrections (C7), sound; Theorem 1.1, faithful, sound. Every report passed; none was graded void. The corrections C1--C7 were applied, so the current text of the seven corrected pages differs from the reviewed text at the places the grade names. No tier is assigned and the problem's status is unchanged. The reviews do not compare either note's Lean development with the paper, and the Hechler attribution of the CH construction remains as recorded on the problem page. After the review, line wrapping was normalized on the reconstruction pages; no formula or sentence changed.

Mechanism. Both proofs run the same greedy recursion: keep a pool of infinite measure, pick a point whose set of forbidding indices leaves the pool of infinite measure, then delete its own set, the points it forbids and its fiber; the selection step is a Tonelli or Fubini count against sections of measure uniformly below one. The notes differ only in how the nonmeasurable relation $x\in A_y$ is made countable: Lee measures it with a total extension of Lebesgue measure and a one-sided section inequality, while Glazer replaces each $A_y$ by a Borel-coded open envelope valid on an outer-measure-one set of random profiles, produced by $\Delta$-system homogenization of countably supported names.