REVIEW 3 major objections 3 minor 1 references
More nonamalgamable forcing extensions
T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Nonamalgamable forcing patterns extend to any poset projecting onto a wide poset.
desk verdict Solid extension of the HHK+19 blockchain results to wide projections and filter-based Mathias forcing; Theorem 19 has a repairable cone-projection gap and the Mathias coding arguments need more detail. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central mechanism is the concept of a projection between forcing notions, used to form 'tagged conditions' (q,p) where q is a condition in the target forcing and p is a condition in the projected forcing below π(q). By maintaining a descending sequence of tagged conditions that is infinitely often strong in the sense of the ≤* relation, the authors arrange that the filter generated by the working parts q projects exactly onto the filter generated by the tags p, allowing independent control of the original generics and the projected generics. The projected generics carry the coding information about the catastrophic real. For the Mathias case, the coding uses the oscillation of the stems: after each common point of the two generic reals, the ordering of the next elements encodes one bit of the catastrophic real.
What would settle it
Run the recursive construction of Theorem 23 in a concrete countable model M with a specified catastrophic real and with F the cofinite filter (so M_F is Cohen forcing). If at any stage a point enters the intersection of the stems for some i∈B outside the designated coding steps — say because a coordinate is first mentioned — then the decoded bits of z will be corrupted. A reader could check whether such an accidental intersection arises in the first few steps of the recursion.
Extended reading notes
Core claim
The central claim is that, given a family of subsets of an index set defined by finite obstacles, and a family of posets each of which adds a generic for a common wide poset of sufficient size, one can find generic filters for the original posets that realize the three-part pattern: generic existence for every allowed subfamily, nonamalgamability for every forbidden subfamily, and exact intersection of extensions of allowed subfamilies. The same three-part pattern is established for filter-based Mathias forcing, using a different coding scheme based on oscillation points in the stems. The construction simultaneously builds the generics and codes a ground-model-catastrophic real into the projected generics (or the stems of the Mathias reals), so that any model containing all the generic objects for a forbidden index set would contain the catastrophic real and therefore could not lie in the generic multiverse.
Load-bearing premise
The construction relies on the claim that no extraneous coding signals appear in the generic objects — for the Cohen/tag constructions, that noncoding extensions never create a new row of 1s across an obstacle, and for the Mathias construction, that the only points common to the generic reals are the coding points.
Editorial extensions
If this is right
- If Theorem 19 holds, nonamalgamability is not a special feature of wide posets themselves; any forcing that adds a generic for a wide poset of appropriate size exhibits the same patterns.
- Filter-based Mathias forcing, which is σ-centered and therefore ccc, realizes the full nonamalgamability pattern, adding a new class of examples outside the wide-poset framework.
- The construction allows mixing Cohen-type forcings with filter-based Mathias forcing, giving nonamalgamable pairs such as one Cohen real and one Mathias real.
- The intersection property (3) provides a strong form of exactness: the common elements of any two allowed extensions are exactly the elements of the extension for the intersection of the index sets.
Reading between the lines
- The same method might apply to any family of posets that all project onto a common wide poset without requiring equal cardinality, by using a more flexible coding scheme.
- The robustness of the coding could be tested by attempting to replace the fragile 'no accidental row of 1s' or 'only coding points in the intersection' conditions with an error-correcting code, which might make nonamalgamability easier to achieve in other settings.
- The open question about random real forcing (Question 20) could be attacked by trying to find a projection from random forcing to a wide forcing, or by proving that no such projection exists, following the approach of this paper.
- The exact intersection property suggests a connection to the structure of grounds in the generic multiverse; one could investigate whether such patterns correspond to a lattice of models generated by the allowed subfamilies.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends the nonamalgamability results of HHK+19 in two directions. Theorem 17 proves the finite-obstacle pattern for families of posets that add Cohen reals, using projection tags and a coding scheme into rows of 1s. Theorem 19 relaxes the hypothesis from wide posets to posets that project onto wide posets. Section 4 introduces coding via Todorčević oscillation for filter-based Mathias forcing, proving nonamalgamability for a pair of generics (Proposition 22), for finite-obstacle families with a fixed filter (Theorem 23), and for families of filters linearly ordered by inclusion (Theorem 24). The overarching method is the by-now standard one of coding a catastrophic real z into the projected generics while preserving genericity and the intersection property.
Significance. If the technical gaps are repaired, the paper makes a solid contribution: it broadens the class of forcings known to exhibit arbitrary finite-obstacle nonamalgamability patterns, and it provides a genuinely different coding mechanism for Mathias-type forcings, including mixed Cohen/Mathias examples. The paper is careful in setting up the projection terminology and the tagged-condition framework, and it correctly identifies the cone-homogeneity issue for Cohen forcing as a special feature. The authors also honestly discuss limitations such as the open random-real question. However, the manuscript currently contains several load-bearing claims that are not adequately proved, so the strength of the contribution cannot be fully assessed until those are fixed.
major comments (3)
- [§3.2, Theorem 19, first paragraph of the proof] The proof begins 'By Proposition 9 we have, for each i∈I, a projection π_i:Q_i→B_i'. Proposition 9 only yields a projection into a cone B_i↾p_i for some p_i∈B(P_i). Since wide posets are not in general cone-homogeneous, the step from a cone to the whole Boolean completion is unjustified; the argument in Corollary 10 relies on the special homogeneity of Cohen forcing. This gap is load-bearing because the entire tagged-condition construction takes tags in B_I. The gap is repairable: because P_i is wide of size κ, any cone below a condition is also wide of size κ (the maximal antichain of size κ from the width property lies inside the cone), and Q_i still adds a generic for P_i↾p_i. Replacing each P_i by such a cone, and each B_i by B_i↾p_i, preserves the hypotheses and lets the proof go through. This replacement must be stated explicitly.
- [§3.1, Claim 18] The assertion 'Consequently, if A∈A then p′≤p is a noncoding extension' is not justified. If B∈B is an obstacle not contained in A, then p may already have a row of 1s on B\A, and p′ may add 1s on B∩A at that same height; then p′ has a row of 1s across B that p lacked, even though p′\p has no 1s on I\A. The bullet points in the claim only control where new 1s are added, not whether pre-existing partial rows are completed. This is load-bearing because the decoding argument in Theorem 17 relies on the claim that the only rows of 1s across an obstacle are the ones added in the obstacle case. The construction needs an additional invariant, or a different choice of p′, to ensure that noncoding steps never complete a partial row across an obstacle.
- [§4, Proposition 22 and Theorem 23] The proof of Proposition 22 states that 'our construction ensured that the only points in the intersection of the two generic reals are the coding points' without giving the necessary induction. One must verify at each step that the new coding point lies above both current stems and that the subsequent one-point extensions are made from upper parts that exclude all points of the other stem; otherwise extraneous common elements could appear and corrupt the decoding of z. The same invariant is needed in Theorem 23 for intersections over finite obstacles; the sentence 'Inspecting the construction...' is not a full proof. Since nonamalgamability in this section is obtained precisely by decoding z from these intersections, this gap affects the central claim. Theorem 24, whose proof transfers verbatim, inherits the same concern.
minor comments (3)
- [§2.2, proof of Proposition 9] There is a typo in the cone-density verification: 'π(¯q≤p)' should read 'π(¯q)≤p'.
- [§3.2, proof of Theorem 19, near the end of Case 2.2.2] The phrase 'build >p_{n+1}≤c_n' should read 'build p_{n+1}≤c_n'.
- [§3.1, proof of Theorem 17] In the decoding paragraph, the sentence 'the only time we could have added such an index ℓ was in the obstacle case' depends on the noncoding-extension property; once Claim 18 is fixed, this sentence should be made into an explicit reference to the invariant.
Circularity Check
No circularity: the paper proves its results from assumptions and independently established lemmas, with no fitted inputs or self-referential derivations.
full rationale
The paper's derivation chain is self-contained relative to its hypotheses. Theorem 17 is proved directly via tagged-condition bookkeeping; Theorem 19 invokes Proposition 9 only to obtain a coordinatewise projection and then constructs the generics and coding points explicitly, rather than importing the target nonamalgamability pattern. Lemma 6 is taken from HHK+19, but it is a technical black-box lemma (no J-extension decides rho in chi) and is not the main theorem being extended; it is independently established in that prior paper, so citing it is legitimate support rather than circularity. The Mathias constructions (Propositions 22, Theorems 23-24) use a coding scheme based on Todorcevic oscillation and verify decoding from the constructed generics; there is no fitted parameter later renamed a prediction. The proof of Theorem 19 does contain a gap flagged by a close reading: 'By Proposition 9 we have, for each i, a projection pi_i: Q_i -> B_i' is stronger than Proposition 9's cone-valued conclusion, and cone-homogeneity of arbitrary wide B(P_i) is not established. Similarly, Proposition 22's assertion that the only intersection points are coding points is asserted rather than fully verified in the text. These are correctness/verification concerns about the proof as written, not instances where a claimed prediction reduces by definition to an input, a fitted quantity, or a self-citation chain. Accordingly, no circular step meeting the required evidentiary standard is present.
Assumptions & free parameters
assumptions (6)
- standard math ZFC set theory, with forcing as usual
- domain assumption M is a countable transitive model of ZFC
- standard math There exists a catastrophic real z for M
- standard math Lemma 6 (HHK+19, Lemma 2.9)
- standard math Proposition 9 (projection from a generic-adding poset to a cone of the Boolean completion)
- domain assumption For Theorem 24: the filters F_i are linearly ordered under inclusion
Cite this review
Pith. "Pith review of More nonamalgamable forcing extensions." pith.science (2026). https://pith.science/paper/WW6EEPHP
@misc{pith2026250522372,
author = {Pith},
title = {Pith review of: More nonamalgamable forcing extensions},
year = {2026},
howpublished = {\url{https://pith.science/paper/WW6EEPHP}},
note = {Machine review of arXiv:2505.22372}
}
read the original abstract
We extend the results of arXiv:1808.01509 on nonamalgamable forcing extensions to families of posets with wide projections. We also use a different coding method to obtain nonamalgamable extensions by filter-based Mathias forcing.
Reference graph
Works this paper leans on
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[1]
[Abr10] Uri Abraham,Proper forcing, Handbook of set theory. Vols. 1, 2, 3, Springer, Dor- drecht, 2010, pp. 333–394. MR 2768684 [Cum10] James Cummings,Iterated forcing and elementary embeddings, Handbook of set the- ory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 775–883. MR 2768691 [FH09] Gunter Fuchs and Joel David Hamkins,Degrees of rigidity for Sou...
work page 2009
Reviewed August 7, 2026 · model on record in the stance chip above.
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