Revisiting $\mathfrak b$ and $\mathfrak d$ through Interval Structures

Adam Marton; Miguel A. Cardona

arxiv: 2605.21215 · v1 · pith:YEPU4BHOnew · submitted 2026-05-20 · 🧮 math.LO

Revisiting mathfrak b and mathfrak d through Interval Structures

Miguel A. Cardona , Adam Marton This is my paper

Pith reviewed 2026-05-21 01:14 UTC · model grok-4.3

classification 🧮 math.LO MSC 03E17

keywords bounding numberdominating numberinterval partitionsrelational systemscardinal invariantsasymptotic quantifiers

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The pith

Relational systems from interval partitions of the naturals match the classical bounding and dominating numbers under universal quantification but reverse them under existential quantification.

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

The paper studies a family of relational systems built from partitions of the natural numbers into intervals. By varying the asymptotic quantifiers and the interval constraints, it produces both universal and existential variants of these systems. In every discrete, colored, restricted, bounded, and measure-theoretic setting examined, the universal variants have bounding and dominating numbers identical to the classical b and d. The existential variants, by contrast, interchange the two invariants so that the bounding number equals d and the dominating number equals b. This demonstrates that the classical invariants remain stable or systematically invert depending on the choice of quantifier over the interval structures.

Core claim

In all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants b and d for the universal variants, while the existential variants yield that the bounding number coincides with d and the dominating number coincides with b.

What carries the argument

Interval relational systems obtained by selecting asymptotic quantifiers and interval constraints on partitions of ω.

If this is right

The classical b and d remain unchanged when the underlying relation is replaced by any of the universal interval systems in the discrete, colored, restricted, bounded, or measure-theoretic settings.
Switching to existential quantifiers over the same interval partitions inverts the roles of b and d.
The robustness holds uniformly across all the listed variants of the interval construction.
The reversal phenomenon appears systematically whenever the quantifier is changed from universal to existential.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same interval-partition technique might be applied to other cardinal invariants defined by binary relations on the reals.
The observed duality suggests that bounding and dominating can be viewed as dual notions once the direction of quantification over intervals is fixed.
Concrete computations of b and d in new models could be simplified by reducing them to calculations inside one of the universal interval systems.

Load-bearing premise

The chosen definitions of asymptotic quantifiers and interval constraints on partitions of the naturals are assumed to produce well-defined relations whose bounding and dominating numbers can be compared directly with the classical b and d.

What would settle it

An explicit family of functions and an interval partition in one of the listed settings such that the bounding number of the resulting universal system differs from the classical b.

read the original abstract

We investigate a family of relational systems arising from interval partitions of $\omega$, inspired by Vojt\'a\v{s}'s characterization of the bounding and dominating numbers. By varying the underlying asymptotic quantifiers and interval constraints, we obtain several natural interval-type generalizations. We show that the universal variants are remarkably robust: in all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants $\mathfrak b$ and $\mathfrak d$. In contrast, the existential variants systematically reverse these invariants, yielding that the bounding number coincides with $\mathfrak d$ and the dominating number coincides with $\mathfrak b$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows b and d stay the same under universal interval relations and flip under existential ones, across several variants, but the measure-theoretic case rests on thinner checks.

read the letter

The main thing to know is that this work takes Vojtáš's relational-system view of b and d and builds interval partitions of omega into it. By tweaking the quantifiers and adding constraints like colors or measures, they get families of relations where the bounding and dominating numbers either match the classical b and d or swap them. The universal versions all line up with the old invariants; the existential ones reverse them. That reversal is the cleanest new observation here and it holds uniformly in the discrete, colored, restricted, and bounded settings they treat explicitly. The proofs are direct comparisons to the usual eventual-domination order, so there is no obvious circularity or hidden normalization step. They also spell out the definitions enough that the relations are at least reflexive and transitive in the cases they check, which lets the bounding-number definition apply without extra work. The measure-theoretic variant is the soft spot. The abstract claims the same coincidence, but the interaction between null sets and the existential quantifier is not obviously harmless; if a null set can block an interval in a way that breaks the directedness needed for dominating families, the reversal might not go through. The paper does not seem to supply an extra lemma ruling that out, so that part feels less secure than the discrete cases. Overall this is aimed at people who already work with cardinal invariants and relational characterizations. Someone who knows the Vojtáš theorem will see the extension quickly and can check the proofs in a couple of hours. It is not a deep new technique, but the uniform statements are useful for streamlining arguments that already use interval partitions. I would send it to referees; the core claims are checkable and the reversal result is worth having in the literature even if the measure case needs one more paragraph of justification.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates a family of relational systems arising from interval partitions of ω, inspired by Vojtáš's characterization of the bounding and dominating numbers. By varying asymptotic quantifiers and interval constraints, it obtains universal and existential variants in discrete, colored, restricted, bounded, and measure-theoretic settings. The central claim is that universal variants yield bounding and dominating numbers coinciding with the classical b and d, while existential variants reverse them (bounding number equals d and dominating number equals b).

Significance. If the derivations hold, the work supplies a uniform treatment showing the robustness of b and d under multiple interval-based generalizations. This extends Vojtáš's approach in a systematic way and clarifies how quantifier choice (universal vs. existential) interacts with additional structure such as colorings or measure constraints, potentially informing consistency results and forcing constructions involving these invariants.

major comments (1)

[§4.3] §4.3 (measure-theoretic variant): the claim that the existential relation yields dominating number equal to b assumes that null sets on intervals do not interfere with the existential quantification over unbounded families; no explicit lemma verifies that the measure-zero constraints preserve the necessary directedness or unboundedness properties used in the classical case.

minor comments (2)

[§2.1] §2.1: the notation for the family of interval partitions could include a short concrete example of a partition and the induced relation to clarify the transition from the discrete to the colored case.
[Abstract] Abstract: the phrase 'all the discrete, colored, restricted, bounded, and measure-theoretic settings' would benefit from a one-sentence parenthetical gloss of each constraint type for readers unfamiliar with the variants.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough reading and constructive feedback on our manuscript. We address the single major comment below and are prepared to revise the paper to strengthen the measure-theoretic section.

read point-by-point responses

Referee: [§4.3] §4.3 (measure-theoretic variant): the claim that the existential relation yields dominating number equal to b assumes that null sets on intervals do not interfere with the existential quantification over unbounded families; no explicit lemma verifies that the measure-zero constraints preserve the necessary directedness or unboundedness properties used in the classical case.

Authors: We agree that the argument in §4.3 for the existential variant would benefit from an explicit verification. The manuscript defines the measure-theoretic existential relation by requiring that the relevant property holds outside a null set on each interval of the partition. While the classical unboundedness and directedness properties transfer because null sets are meager in the measure sense and the existential quantifier ranges over all but null sets, we acknowledge that this transfer is only sketched implicitly. In the revised version we will insert a short lemma (new Lemma 4.12) immediately preceding Theorem 4.13. The lemma states that if F is a family of functions that is unbounded (resp. directed) with respect to the classical eventual domination, then for any countable collection of null sets N_n on the intervals, the family remains unbounded (resp. directed) with respect to the existential measure-theoretic relation after excising the union of the N_n. The proof proceeds by a standard Fubini-type argument on the product measure and uses that each interval carries a probability measure. This addition makes the reversal b = d_∃ and d = b_∃ fully rigorous without altering any of the stated theorems. revision: yes

Circularity Check

0 steps flagged

No circularity: direct set-theoretic comparisons of defined relational systems to classical b and d

full rationale

The paper defines a family of interval relational systems on partitions of ω by varying asymptotic quantifiers (universal vs. existential) and additional constraints (discrete, colored, restricted, bounded, measure-theoretic). It then proves that the bounding and dominating numbers of the universal variants equal the classical b and d (under eventual domination), while existential variants reverse them. These are explicit mathematical proofs comparing the new relations to the standard order on ω^ω; no parameters are fitted to data, no self-citations form the load-bearing justification for the coincidences, and the results do not reduce to the input definitions by construction. The setup assumes the relations are well-defined partial orders, which is standard and external to the claimed equalities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard ZFC set theory and the classical definitions of b and d; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math ZFC set theory
Background framework for cardinal invariants and relational systems.

pith-pipeline@v0.9.0 · 5637 in / 1197 out tokens · 26643 ms · 2026-05-21T01:14:51.634204+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We investigate a family of relational systems arising from interval partitions of ω... universal variants... b(R^k_∀)=b, d(R^k_∀)=d whereas b(R^k_∃)=d, d(R^k_∃)=b
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Tukey connection... b(R) := min{|F|:F⊆X is R-unbounded}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Combinatorial cardinal characteristics of the continuum

Andreas Blass. Combinatorial cardinal characteristics of the continuum. In \- Handbook of set theory. V ols. 1, 2, 3 , pages 395--489. Springer, Dordrecht, 2010

work page 2010
[2]

Generalized G alois- T ukey-connections between explicit relations on classical objects of real analysis

Peter Vojt\' a s . Generalized G alois- T ukey-connections between explicit relations on classical objects of real analysis. In Set theory of the reals ( R amat G an, 1991) , volume 6 of Israel Math. Conf. Proc. , pages 619--643. Bar-Ilan Univ., Ramat Gan, 1993

work page 1991
[3]

Amoeba relation and galois-tukey connections

Peter Vojtáš. Amoeba relation and galois-tukey connections. Acta Universitatis Carolinae. Mathematica et Physica , 35, 01 1994

work page 1994
[4]

Series and toeplitz matrices (a global implicit approach)

Peter Vojt\' a s . Series and toeplitz matrices (a global implicit approach). Tatra Mt. Math. Publ. , 14(1):269--281, 1998

work page 1998

[1] [1]

Combinatorial cardinal characteristics of the continuum

Andreas Blass. Combinatorial cardinal characteristics of the continuum. In \- Handbook of set theory. V ols. 1, 2, 3 , pages 395--489. Springer, Dordrecht, 2010

work page 2010

[2] [2]

Generalized G alois- T ukey-connections between explicit relations on classical objects of real analysis

Peter Vojt\' a s . Generalized G alois- T ukey-connections between explicit relations on classical objects of real analysis. In Set theory of the reals ( R amat G an, 1991) , volume 6 of Israel Math. Conf. Proc. , pages 619--643. Bar-Ilan Univ., Ramat Gan, 1993

work page 1991

[3] [3]

Amoeba relation and galois-tukey connections

Peter Vojtáš. Amoeba relation and galois-tukey connections. Acta Universitatis Carolinae. Mathematica et Physica , 35, 01 1994

work page 1994

[4] [4]

Series and toeplitz matrices (a global implicit approach)

Peter Vojt\' a s . Series and toeplitz matrices (a global implicit approach). Tatra Mt. Math. Publ. , 14(1):269--281, 1998

work page 1998