On the non-existence of $\kappa$-mad families

Haim Horowitz; Saharon Shelah

arxiv: 1906.09538 · v1 · pith:AWDYUMBFnew · submitted 2019-06-23 · 🧮 math.LO

On the non-existence of kappa-mad families

Haim Horowitz , Saharon Shelah This is my paper

Pith reviewed 2026-05-25 18:03 UTC · model grok-4.3

classification 🧮 math.LO

keywords κ-mad familiessupercompact cardinalLaver indestructibilityforcingZFDC_κalmost disjoint families

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The pith

Starting from a Laver-indestructible supercompact cardinal, forcing produces a model of ZF + DC_κ with no κ-mad families.

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

The paper proves the consistency of there being no κ-mad families while dependent choice holds below κ. It begins in a ground model that contains a Laver-indestructible supercompact cardinal κ and applies a forcing construction to reach the desired model. A sympathetic reader cares because this shows that the existence of maximal almost disjoint families of size κ is not forced by ZF plus the limited choice axiom DC_κ alone.

Core claim

Starting from a model with a Laver-indestructible supercompact cardinal κ, we construct a model of ZF+DC_κ where there are no κ-mad families.

What carries the argument

The forcing construction over a Laver-indestructible supercompact cardinal that destroys all κ-mad families while preserving DC_κ.

If this is right

In the constructed model, no κ-mad family exists.
The non-existence of κ-mad families is consistent with ZF + DC_κ.
Maximality principles for almost disjoint families at level κ can fail under restricted choice.
Supercompact cardinals supply enough strength to control the existence of such families via forcing.

Where Pith is reading between the lines

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

The same style of argument might apply to other maximality statements at inaccessible cardinals.
Results like this separate combinatorial objects from choice axioms more sharply than previously known.
Weaker large-cardinal hypotheses might suffice if the forcing can be simplified.

Load-bearing premise

The ground model contains a Laver-indestructible supercompact cardinal κ.

What would settle it

A proof inside ZF + DC_κ that a κ-mad family must exist, or a verification that the forcing fails to remove every potential κ-mad family.

read the original abstract

Starting from a model with a Laver-indestructible supercompact cardinal $\kappa$, we construct a model of $ZF+DC_{\kappa}$ where there are no $\kappa$-mad families.

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 consistency of ZF + DC_κ with no κ-mad families from a Laver-indestructible supercompact, via a standard forcing construction.

read the letter

The core result is a model of ZF + DC_κ with no κ-mad families, built by forcing over a ground model that has a Laver-indestructible supercompact κ. This separates the non-existence of these maximal almost disjoint families from full choice at the cardinal κ while keeping dependent choice up to κ intact. The construction appears to be a direct extension of earlier mad family consistency work to the uncountable setting with a controlled amount of choice preserved. The abstract states the large cardinal starting point and the target theory clearly, and the approach follows the usual pattern of using indestructibility to keep the cardinal properties alive through the forcing that kills the mad families. That part looks routine and well-motivated for this kind of question. The main limitation is the strength of the assumption: a Laver-indestructible supercompact is a heavy hypothesis, even if it is the standard tool for these separations. The paper does not claim to weaken it, so the result stays conditional on that. Without seeing the full forcing details it is impossible to check the exact preservation of DC_κ or whether any side conditions on the mad families are handled cleanly, but nothing in the stated claim suggests an internal gap or circularity. This is aimed at set theorists who care about the boundary between choice principles and combinatorial objects like mad families at uncountable cardinals. It is the kind of targeted consistency result that merits a serious referee to verify the technical steps, even if the large cardinal cost is high.

Referee Report

0 major / 3 minor

Summary. The paper proves a consistency result: starting from a ground model with a Laver-indestructible supercompact cardinal κ, a forcing construction yields a model of ZF + DC_κ in which there are no κ-mad families.

Significance. If correct, the result establishes the consistency of the non-existence of κ-mad families together with DC_κ, conditional on a large-cardinal hypothesis. This is a standard-strength contribution to the study of mad families and choice principles at singular cardinals in ZF, extending known results about the failure of maximality principles under limited choice.

minor comments (3)

[Abstract and §1] The abstract and introduction should explicitly state the precise definition of a κ-mad family used in the paper (e.g., whether it requires maximality with respect to all subsets or only those of size <κ).
[§3] Notation for the forcing poset and the iteration should be introduced with a clear reference to the Laver preparation; a diagram or schematic of the iteration would improve readability.
[§4] The preservation argument for DC_κ under the forcing should include a brief reminder of the relevant theorem from the literature (e.g., the standard preservation of DC under <κ-closed forcing).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were provided in the report, so we have no points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a consistency result via an explicit forcing construction over a ground model containing a Laver-indestructible supercompact cardinal κ, yielding ZF + DC_κ with no κ-mad families. This relies on an external large-cardinal hypothesis and standard set-theoretic techniques rather than any self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The derivation is self-contained as a mathematical proof from stated assumptions, with no reduction of the result to its premises by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the existence of a Laver-indestructible supercompact cardinal as an external assumption; standard ZF axioms and forcing techniques are used but not invented here.

axioms (1)

domain assumption Existence of a Laver-indestructible supercompact cardinal κ
Explicitly stated as the starting hypothesis for the model construction.

pith-pipeline@v0.9.0 · 5536 in / 1072 out tokens · 24553 ms · 2026-05-25T18:03:44.076585+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

[HwSh:1090] Haim Horowitz and Saharon Shelah, Can you take T oernquist’s inac- cessible away? arXiv:1605.02419 [HwSh:1093] Haim Horowitz and Saharon Shelah, Transcendence bases, well-orderings of the reals and the axiom of choice, arXiv:1901.01508 [HwSh:1113] Haim Horowitz and Saharon Shelah, Madness and r egularity properties, arXiv:1704.08327 [HwSh:1145...

work page arXiv 1901

[1] [1]

[HwSh:1090] Haim Horowitz and Saharon Shelah, Can you take T oernquist’s inac- cessible away? arXiv:1605.02419 [HwSh:1093] Haim Horowitz and Saharon Shelah, Transcendence bases, well-orderings of the reals and the axiom of choice, arXiv:1901.01508 [HwSh:1113] Haim Horowitz and Saharon Shelah, Madness and r egularity properties, arXiv:1704.08327 [HwSh:1145...

work page arXiv 1901