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Two $\mathfrak{b}$ or not two $\mathfrak{b}$?

T0 review · 1 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For every ideal I, the Rothberger number b_R(I) is an upper bound for the alternative ideal bounding number b(I), and the two numbers coincide for several natural classes of ideals.

desk verdict A genuinely new inequality between two ideal versions of the bounding number, with clean proofs and one easily fixable gap in a technical lemma. read the letter →

arxiv 2507.03734 v1 pith:XNYAOT5C submitted 2025-07-04 math.LO math.CO

Rafa{l} Filipów , Adam Kwela This is my paper · ORCID
classification math.LOmath.CO MSC 03E1703E0503E3503E15
keywords boundingnumberRothbergergapidealonomegacardinalcharacteristicsP-idealBairepropertyCONVCTBL
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper compares two ideal-based generalizations of the classical bounding number b. One, written b(I), measures the smallest unbounded family inside a restricted poset of functions that take each value only on an ideal-small set; the other, the Rothberger number b_R(I), measures the smallest size of an I-Rothberger gap, a pair of orthogonal families with no separating set. The paper's central result is that b(I) ≤ b_R(I) for every ideal I. It then computes both numbers for several classes: they both equal ω1 for every ideal between the convergence ideal CONV and the countable-set ideal CTBL, and they both equal b for every P-ideal with the Baire property. It also proves b_R(NULL) ≥ p, which consistently rules out a NULL-Rothberger gap of size add(NULL), answering a question from the literature.

What carries the argument

The central device is a pair of matching characterizations. Theorem 3.1 characterizes b(I) as the least size of a family of sequences of pairwise disjoint I-small sets that 'catches' every partition of ω into I-small sets. Theorem 4.1 characterizes b_R(I) the same way except that disjointness is required simultaneously across all sequences in the family. Because the simultaneous condition is stronger, any family witnessing b_R(I) witnesses b(I), yielding b(I)≤b_R(I). The later sections use Fubini products and direct sums to transfer values between ideals, and a density construction with perfect nowhere dense sets to build the ω1-sized gaps for the CONV–CTBL interval.

What would settle it

Try to find an ideal I with b_R(I)<b(I); the proof of Theorem 3.2 would then fail, so a concrete check is whether a Rothberger gap of size κ forces an unbounded family of size κ in (D_I,≥_I). For the CONV–CTBL equality, check the simultaneous-choice step in Lemma 6.2: if it cannot be justified, a model with b_R(CONV)>ω1 would refute Theorem 6.3.

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Extended reading notes

Core claim

The discovery is that the two seemingly different ideal versions of the bounding number are ordered: the poset-based number never exceeds the gap-based number. The proof works by showing that if λ < b(I), then no I-Rothberger gap of size λ can exist: given any candidate gap, the characterization of b(I) produces a separating set. The same comparison tool also yields exact evaluations: b(I)=b_R(I)=ω1 for all ideals with CONV⊆I⊆CTBL, b(I)=b_R(I)=b for P-ideals with the Baire property, and b_R(NULL)≥p, with consistency results showing b(I)<b_R(I) and d<b(I)=b_R(I) are both possible.

Load-bearing premise

The proof of the ω1 result for ideals between CONV and CTBL depends on being able to choose, for every n and every countable ordinal α, a sequence inside the nth perfect set with the required limit point, and the paper does not spell out why all these choices can be made simultaneously.

Editorial extensions

If this is right

  • For every ideal I, the Rothberger number b_R(I) is an upper bound on b(I); in particular any ideal with a small Rothberger gap has a small unbounded family in the restricted function poset.
  • For every ideal I with CONV⊆I⊆CTBL, both numbers are exactly ω1, so b(CONV)=b_R(CONV)=ω1 and b(CTBL)=b_R(CTBL)=ω1; this can be strictly below b.
  • For every P-ideal with the Baire property, b_R(I)=b(I)=b, extending the known equality for analytic P-ideals.
  • b_R(NULL)≥p, and it is consistent that no NULL-(ω,add(NULL))-gap exists, settling the open question.
  • It is consistent that b=b(I)<b_R(I)≤c, that b<b(I)<b_R(I)≤c, and that d<b(I)=b_R(I)≤c, so the two numbers can differ by a controlled amount.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The inequality b(I)≤b_R(I) suggests a general pattern: the gap-based version of an ideal cardinal is the stronger one, so any lower bound proved for b_R(I) automatically applies to b(I); one could test whether the same ordering holds for other pairs of ideal invariants defined through gaps versus posets.
  • The construction behind Lemma 5.2 shows b_R(I⊗{∅})=b(I), so any ideal with a large b(I) can be converted into an ideal with equal Rothberger number; iterating this product might produce longer chains of distinct ideal bounding numbers.
  • The CONV–CTBL interval result raises the question whether all 'small' definable ideals have both numbers equal to ω1; a natural target would be the ideal of nowhere dense sets, where b_R is known to be add(M).
  • The paper leaves open whether a Borel ideal can have b(I)<b_R(I)≤c; if such an ideal exists, it would show the inequality is strict inside the definable realm, and the product/direct-sum toolbox here is a plausible route to build it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 6 minor

Summary. The paper compares two ideal generalizations of the classical bounding number b: the invariant b(I) introduced earlier by the authors, defined as the least size of an unbounded family in (D_I, ≥_I), and the Rothberger number b_R(I), defined as the least κ for which there exists an I-(ω,κ)-gap. The main result is Theorem 3.2, which proves b(I) ≤ b_R(I) for every ideal I, using a direct argument based on the authors' earlier characterization of b(I). The paper also proves a structural characterization of b_R(I) (Theorem 4.1) in the style of the b(I) characterization, derives consistency results showing that b(I) and b_R(I) can differ in various ways (Theorems 5.3 and 5.6), computes b(I)=b_R(I)=ω1 for every ideal between CONV and CTBL (Theorem 6.3), proves b_R(I)=b(I)=b for every P-ideal with the Baire property (Theorem 6.6), and answers a question of Kankaanpää by showing b_R(NULL) ≥ p (Theorem 7.1). The exposition is clear and the central arguments are generally rigorous, with the main proofs being self-contained modulo cited theorems from the literature.

Significance. The inequality b(I) ≤ b_R(I) is a clean and useful unifying result: it relates two previously separate ideal versions of the bounding number and yields immediate computations for several concrete ideals. The characterization in Theorem 4.1 is a valuable technical tool, and the consistency results provide a fairly complete picture of the possible relationships between b(I), b_R(I), b, d, and c. The paper also settles an open question of Kankaanpää about NULL-Rothberger gaps. The proofs are mostly detailed and check out; in particular, the proof of Theorem 3.2 is clever and correct, and the use of previously published characterizations is appropriate. The main weakness is an under-specified construction in Lemma 6.2, which supports Theorem 6.3; however, the gap is standard and readily repairable, and it does not affect the central inequality or the other main results.

major comments (1)
  1. [Lemma 6.2] The construction of the sets A_n and the sequences x_{n,α} is incomplete as written. First, 'A_n is a perfect nowhere dense set' cannot be interpreted literally for a subset of Q, since perfect subsets of [0,1] are uncountable; the authors presumably mean that the closure of A_n is perfect and nowhere dense, and this should be stated. Second, the proof requires that for each α < ω1 the set {y_{n,α} : n ∈ ω} be dense in [0,1] in order to conclude that the closure of B_α \ C is [0,1]; but picking y_{n,α} ∈ U_n for a fixed enumeration of a base only ensures that each y_{n,α} lies in the corresponding U_n, which does not imply density. This can be fixed by enumerating the base with infinitely many repetitions and choosing the A_n and the limits so that for each basic open set U and each α there are infinitely many n with y_{n,α} ∈ U, but the manuscript does not say this. Third, the line 'B_α \ C = [0,1]' is false as an equality of sets (B_α is countable), and should be replaced by the intended statement that the closure of B_α \ C is [0,1], or equivalently that B_α \ C ∉ CTBL. These points need to be addressed, since Lemma 6.2 underpins Theorem 6.3, one of the advertised main results.
minor comments (6)
  1. [Corollary 3.4] The introductory sentence says 'As a collorary of Theorem 4.1', but the proof actually uses Theorem 3.2 and cited results on b_R(I); the reference to Theorem 4.1 appears inaccurate.
  2. [Section 6 title] The title contains the typo 'Rathberger' and should read 'Rothberger'.
  3. [Section 2] The phrase 'On can also show' should be 'One can also show'.
  4. [Proposition 2.2(3)] The text reads 'I is a Borel ideals and and there is X'; this should be 'I is a Borel ideal and there is X'.
  5. [Section 4 preamble] The displayed formula before Theorem 4.1 has a typographical error in the indexing: '{Eαn :n<ωα<κ}' should be '{E^α_n : n < ω, α < κ}'. A similar notational issue appears in the statement of Theorem 4.1 where the index sets are written in a compressed way.
  6. [Lemma 6.2] The notation A_n is used ambiguously for both the subset of Q and its closure; the authors should consistently distinguish between a set and its closure, for example by writing cl(A_n) when referring to the closure.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central inequality b(I) ≤ bR(I) is proved from the published characterization of b(I) and a direct gap-separation argument, not from the target statement itself.

full rationale

The derivation chain is self-contained modulo published theorems. Theorem 3.2 ('b(I) ≤ bR(I) for every ideal I') is proved directly from Theorem 3.1, the characterization b(I)=min{|E|: ...} quoted from [9, Theorem 3.10]. The proof takes a putative (ω,λ)-gap, forms E = {{A∩B : A∈A} : B∈B}, applies the characterization in contrapositive to obtain a partition {C_n}∈P_I with ⋃_n(C_n∩⋃_{i≤n}E^B_{A_i})∈I for every B, and then verifies that C=⋃_n(A_n∩⋃_{i<n}C_i) I-separates (A,B). The inclusion B\C ⊆ ⋃_n(C_n∩⋃_{i≤n}E^B_{A_i}) checks out by the partition property. This is not a restatement of the target: Theorem 3.1 concerns b(I), while Theorem 3.2 adds the Rothberger-gap side. The later characterization Theorem 4.1 could also yield the inequality, but it is not needed. The paper's other self-citations—[9, Theorems 4.2, 4.5, 5.1, 5.13] and [15, Theorem 4.2, Corollary 3.7(b)]—carry published proofs and are used for lower bounds and consistency results (e.g., Theorem 7.1 uses [15, Corollary 3.7(b)] p ≤ b(NULL) plus Theorem 3.2), so they are independent support rather than circular imports. The only soft spot is Lemma 6.2: the simultaneous choice of sequences x_{n,α} in the countable dense sets A_n with pairwise distinct limits y_{n,α}∈U_n, and the density of {y_{n,α}:n} used to conclude B_α\C=[0,1], is terse; the density follows because y_{n,α}∈U_n for a fixed base, and the choice is standard (e.g., take A_n=Q∩D_n for disjoint perfect nowhere dense D_n and choose ω1 distinct limits in D_n). This is a gap in exposition, not circularity. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work to force a choice. Open questions 2.3 and 5.1 are genuine open questions, not hidden assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard set theory plus several cited theorems, mostly from the authors' own prior work and from Canjar and Talagrand. No parameters are fitted and no new entities are postulated; new ideals are built from existing ones via standard constructions.

assumptions (5)
  • standard math ZFC (standard set theory), including the consistency of the relevant forcing models
    The paper works in ZFC and proves consistency results in Cohen forcing models, citing [4,5,6,15].
  • domain assumption Theorem 3.1 from [9] characterizing b(I)
    Theorem 3.2 uses this characterization as a black box.
  • domain assumption Talagrand's characterization of ideals with the Baire property (Theorem 21 in [20])
    Lemma 6.4 relies on it to find intervals witnessing I-positivity.
  • domain assumption Canjar's results on values of b(J) for maximal ideals in Cohen models [4,5,6]
    Theorems 5.3 and 5.6 depend on these consistency facts.
  • domain assumption Kwela's result p ≤ b(NULL) [15, Corollary 3.7(b)]
    Theorem 7.1 relies on it.

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Pith. "Pith review of Two $\mathfrak{b}$ or not two $\mathfrak{b}$?." pith.science (2026). https://pith.science/paper/XNYAOT5C

@misc{pith2026250703734,
  author       = {Pith},
  title        = {Pith review of: Two $\mathfrakb$ or not two $\mathfrakb$?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XNYAOT5C}},
  note         = {Machine review of arXiv:2507.03734}
}
abstract

The paper is devoted to comparison of two generalizations of the bounding number $\mathfrak{b}$.

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