---
title: Generalized Reaping Numbers in Set Theory
url: https://www.emergentmind.com/topics/generalized-reaping-numbers
type: topic
---

# Generalized Reaping Numbers in Set Theory

Generalized reaping numbers are extensions of the classical reaping number
\[
\mathfrak{r}=\min\Bigl\{|\mathcal{R}|:\mathcal{R}\subseteq[\omega]^\omega\text{ is unreaped}\Bigr\},
\]
where a set \(x\in[\omega]^\omega\) splits \(y\in[\omega]^\omega\) iff \(|y\cap x|=|y\setminus x|=\aleph_0\). In current usage, the phrase covers at least three distinct but related developments: higher-cardinal invariants \(\mathfrak{r}_\kappa\) on \([\kappa]^\kappa\), Boolean-algebraic invariants \(\mathfrak{r}(B)\) defined from a reaping relation on an arbitrary Boolean algebra \(B\), and density-sensitive invariants \(\mathfrak{r}_X\) defined via prescribed relative asymptotic densities on \(\omega\). In each setting, the invariant measures the least size of a family that cannot be simultaneously split, reaped, or decided by a single witness [1401.4649][2202.06655][2410.18595][2506.21059].

## 1. Classical prototype and the unreaped-family viewpoint

In the classical setting, a set \(S\subseteq\omega\) splits \(X\subseteq\omega\) if both \(X\cap S\) and \(X\setminus S\) are infinite. A family \(\mathcal{S}\subseteq[\omega]^\omega\) is splitting if every infinite \(X\subseteq\omega\) is split by some \(S\in\mathcal{S}\), and the splitting number \(\mathfrak{s}\) is the least size of such a family. Dually, a set \(R\subseteq\omega\) reaps a family \(\mathcal{F}\subseteq[\omega]^\omega\) if for every \(X\in\mathcal{F}\), either \(X\subseteq^*R\) or \(X\subseteq^*\omega\setminus R\). The reaping number \(\mathfrak{r}\) is the least \(|\mathcal{F}|\) such that no single \(R\subseteq\omega\) reaps \(\mathcal{F}\); equivalently, \(\mathfrak{r}\) is the least size of an unreaped family, or the minimal size of a family that cannot be simultaneously split by one set [1401.4649][2410.18595].

This admits an equivalent Boolean-algebraic formulation in \(\mathcal{P}(\omega)/\mathrm{fin}\). If \([X]_{\mathrm{fin}}\) denotes the equivalence class of \(X\subseteq\omega\) modulo finite, then the classical invariants may be written as
\[
\mathfrak{r}=\mathfrak{r}(\mathcal{P}(\omega)/\mathrm{fin}),\qquad
\mathfrak{s}=\mathfrak{s}(\mathcal{P}(\omega)/\mathrm{fin}).
\]
This reformulation is conceptually important because it isolates the relational content of reaping and makes possible the later passage to arbitrary Boolean algebras and to reduced powers [2410.18595].

The classical invariant is also linked to eventual domination. On \(\omega^\omega\), one defines
\[
f\le^* g \Longleftrightarrow (\exists n_0)(\forall n\ge n_0)\ f(n)\le g(n),
\]
and the dominating number \(\mathfrak{d}\) is the least size of a dominating family in \((\omega^\omega,\le^*)\). Shelah’s theorem that \(\mathfrak{d}>\mathfrak{r}\) implies \(\operatorname{cf}(\mathfrak{r})>\omega\) shows that even the classical reaping number is subject to nontrivial structural restrictions on its cofinality [1401.4649].

## 2. Higher-cardinal reaping numbers \(\mathfrak{r}_\kappa\)

For an infinite cardinal \(\lambda\), the generalized reaping number is defined by replacing \(\omega\) with \(\lambda\), \([\omega]^\omega\) with \([\lambda]^\lambda\), and “finite” with “of size \(<\lambda\)”. Thus for \(x,y\in[\lambda]^\lambda\), \(x\) splits \(y\) iff
\[
|y\cap x|=|y\setminus x|=\lambda.
\]
A family \(\mathcal{Y}\subseteq[\lambda]^\lambda\) is unreaped if there is no single \(x\in[\lambda]^\lambda\) that splits every element of \(\mathcal{Y}\), and
\[
\mathfrak{r}_\lambda=
\min\Bigl\{|\mathcal{Y}|:\mathcal{Y}\subseteq[\lambda]^\lambda\text{ is unreaped}\Bigr\}.
\]
A standard diagonal argument yields \(\mathfrak{r}_\lambda>\lambda\) for every infinite \(\lambda\) [2202.06655].

At singular \(\kappa\) of uncountable cofinality \(\theta=\operatorname{cf}(\kappa)>\omega\), Lambie-Hanson places \(\mathfrak{r}_\kappa\) into a Galvin-Hajnal framework. One fixes an increasing continuous sequence \(\langle\kappa_i\mid i<\theta\rangle\) converging to \(\kappa\), and interprets \(\mathfrak{r}_\kappa\) as the cofinality of the quasiorder \((X,\preceq)\) with \(X=[\kappa]^\kappa\) and
\[
x\preceq y \iff x\text{ does not split }y,
\]
with corresponding local structures \((X_i,\preceq_i)\) at each \(\kappa_i\). The projection maps are
\[
\pi_i(x)=
\begin{cases}
x\cap\kappa_i & \text{if }|x\cap\kappa_i|=\kappa_i,\\
\kappa_i & \text{otherwise}.
\end{cases}
\]
In this formalism, \(\mathfrak{r}_\kappa=\operatorname{cf}(X,\preceq)\) and \(\mathfrak{r}_{\kappa_i}=\operatorname{cf}(X_i,\preceq_i)\) [2202.06655].

The main theorem specialized to reaping numbers states that if \(\beta\) is an ordinal such that a canonical function \(\varphi^\theta_\beta\) exists, \(S\subseteq\theta\) is stationary, and
\[
\mathfrak{r}_{\kappa_i}\le \kappa_i^{+\varphi^\theta_\beta(i)}
\quad\text{for all }i\in S,
\]
then
\[
\mathfrak{r}_\kappa \le \kappa^{+\beta}+\underline{d(\theta,\kappa)}+d(\mathrm{NS}_\theta\restriction S),
\]
and if \(\beta<\omega\), then
\[
\mathfrak{r}_\kappa \le \kappa^{+\beta}+\underline{d(\theta,\kappa)}.
\]
This is a Galvin-Hajnal-type and hence Silver-type restriction on \(\mathfrak{r}_\kappa\): stationary many local upper bounds at the \(\kappa_i\)-levels force a global upper bound at \(\kappa\) [2202.06655].

The proof uses canonical functions, Jech’s variation on Galvin-Hajnal, and density parameters such as the lower density \(\underline{d(\theta,\kappa)}\). In the reaping case, the base step constructs global witnesses from local non-splitting data by combining dense families \(W_j\subseteq[T\times\kappa_j]^\theta\) with repeated applications of Fodor’s lemma. This places generalized reaping numbers firmly within the PCF-flavored structure theory of singular cardinals [2202.06655].

## 3. Cofinality constraints and Shelah’s partition-tree template

Shelah proved the classical theorem
\[
\mathfrak{d}>\mathfrak{r}\ \Longrightarrow\ \operatorname{cf}(\mathfrak{r})>\omega.
\]
Equivalently, if \(\operatorname{cf}(\mathfrak{r})=\omega\), then \(\mathfrak{d}\le \mathfrak{r}\). The result is purely combinatorial and works in ZFC [1401.4649].

The proof begins by assuming toward contradiction that \(\mathfrak{d}>\mathfrak{r}\) and \(\operatorname{cf}(\mathfrak{r})=\omega\), writing
\[
\mathfrak{r}=\sup_{n<\omega}\lambda_n
\]
for a strictly increasing sequence \(\langle\lambda_n:n<\omega\rangle\) with each \(\lambda_n<\mathfrak{r}\). One fixes a witnessing reaping family \(\mathcal{A}=\{A_\alpha:\alpha<\mathfrak{r}\}\subseteq[\omega]^\omega\), and for each \(n\) lets \(\mathcal{A}_n=\{A_\alpha:\alpha<\lambda_n\}\). The core construction is a binary tree of partitions
\[
\langle A^*_\eta:\eta\in 2^{<\omega}\rangle
\]
such that for each level \(n\), \(\{A^*_\eta:\eta\in 2^n\}\) is a partition of \(\omega\) into infinite sets, each level refines the previous one, and each node can be split so as to handle all members of the subfamily \(\mathcal{A}_n\). The point is that any family of size \(<\mathfrak{r}\) is not reaping, so it can be split [1401.4649].

From the partition tree one defines sets \(B_{n,f}\) indexed by \(n\in\omega\) and \(f\in\omega^\omega\), together with functions \(g_{n,\alpha}\in\omega^\omega\) satisfying
\[
f\not\le^* g_{n,\alpha}\ \Longrightarrow\ |A_\alpha\cap B_{n,f}|=\aleph_0
\qquad (\alpha<\lambda_n).
\]
A further property is that if \(n_1\neq n_2\), then \(B_{n_1,f_1}\cap B_{n_2,f_2}\) is finite. Since each family \(\{g_{n_i,\alpha}:\alpha<\mathfrak{r}\}\) has cardinality \(<\mathfrak{d}\), it is not dominating; therefore one can choose escaping functions \(f_i\). After removing earlier intersections, one obtains infinite sets \(B_i\), and then
\[
B^0=\bigcup_{i\text{ even}}B_i,\qquad
B^1=\bigcup_{i\text{ odd}}B_i.
\]
These are disjoint infinite subsets of \(\omega\) that split every \(A_\alpha\), contradicting that \(\mathcal{A}\) witnesses \(\mathfrak{r}\) [1401.4649].

For generalized reaping numbers, the significance of the argument is methodological. The same paper explicitly proposes the higher-cardinal definitions
\[
\mathfrak{r}_\kappa=
\min\Bigl\{|\mathcal{A}|:\mathcal{A}\subseteq[\kappa]^\kappa,\ \neg\exists X\subseteq\kappa\ \forall A\in\mathcal{A}\bigl(|A\cap X|=|A\setminus X|=\kappa\bigr)\Bigr\}
\]
and
\[
\mathfrak{d}_\kappa=
\min\bigl\{|\mathcal{F}|:\mathcal{F}\subseteq\kappa^\kappa \text{ is cofinal in }(\kappa^\kappa,\le^*_\kappa)\bigr\},
\]
where
\[
f\le^*_\kappa g \Longleftrightarrow |\{\alpha<\kappa:f(\alpha)>g(\alpha)\}|<\kappa.
\]
It then notes that one would like to ask whether an analogue of Shelah’s theorem holds:
\[
\mathfrak{d}_\kappa>\mathfrak{r}_\kappa \ \Rightarrow\ \mathrm{cf}(\mathfrak{r}_\kappa)>\kappa.
\]
The paper does not claim or prove this statement. A plausible implication is that the partition-tree plus domination-diagonalization method is a template for higher-cardinal analogues, but any such extension requires checking that the finite-versus-infinite combinatorics can be replaced by \(<\kappa\)-versus-\(\kappa\) combinatorics [1401.4649].

## 4. Boolean-algebraic reaping numbers and reduced powers

A different generalization replaces subsets of \(\omega\) by elements of an arbitrary Boolean algebra \(B\). Writing \(B^+=B\setminus\{0\}\), the reaping relation is
\[
R(B)=(B,\ R,\ B^+),
\qquad
b\,R\,r \Longleftrightarrow \text{either }r\le b\text{ or }r\wedge b=0.
\]
Thus \(r\) is decided by \(b\): it is either contained in \(b\) or disjoint from \(b\). The reaping number of \(B\) is
\[
\mathfrak{r}(B)=\mathfrak{d}(R(B))
=\min\bigl\{|Y|:Y\subseteq B^+\text{ and }(\forall b\in B)(\exists r\in Y)\, b\,R\,r\bigr\},
\]
which the authors identify with the weak density of \(B\). If \(B\) is atomless, the corresponding splitting number is
\[
\mathfrak{s}(B)=\mathfrak{b}(R(B)).
\]
For \(B=\mathcal{P}(\omega)/\mathrm{fin}\), these recover the classical \(\mathfrak{r}\) and \(\mathfrak{s}\) [2410.18595].

The same work introduces an almost-refinement order on maximal antichains. If \(A,B\in\mathrm{Part}(B)\) and \(B\) is c.c.c., then
\[
A\le^* B \Longleftrightarrow \exists F\subseteq A\text{ finite}\ \ A_F\le B,
\]
where \(A_F=(A\setminus F)\cup\{\bigvee F\}\). The relational system \(\mathrm{Part}^*(B)\) turns out to mediate between antichain combinatorics and reaping phenomena. In particular,
\[
(B^+,\ge,B^+)^\omega \le_T \mathrm{Part}^*(B)\le_T \mathrm{Part}(B),
\]
and hence
\[
\mathfrak{d}(\mathrm{Part}^*(B))=\mathfrak{d}(\mathrm{Part}(B)).
\]
If \(B\) is non-atomic and \(\sigma\)-finite c.c., then
\[
(\omega^\omega,\le^*,\omega^\omega)\le_T \mathrm{Part}^*(B),
\]
so the almost-refinement structure realizes the classical bounding/dominating complexity of \(\omega^\omega\) [2410.18595].

For the Cohen algebra
\[
\mathbb{C}_\omega=\mathcal{B}(2^\omega)/\mathcal{M},
\]
Monk’s classical fact is that
\[
\mathfrak{r}(\mathbb{C}_\omega)=\mathfrak{s}(\mathbb{C}_\omega)=\aleph_0.
\]
The nontrivial behavior appears after passing to reduced powers. The paper proves
\[
(\mathrm{nwd}(2^\omega),\subseteq,\mathrm{nwd}(2^\omega))
\equiv_T \mathrm{Part}^*(\mathbb{C}_\omega)\equiv_T \mathrm{Part}(\mathbb{C}_\omega),
\]
and consequently
\[
\mathrm{cof}(\mathcal{M})=\mathfrak{d}(\mathrm{Part}^*(\mathbb{C}_\omega)).
\]
For reduced powers \({}^\omega B/\mathrm{Fin}\), one has
\[
R(\mathcal{P}(\omega)/\mathrm{fin})\le_T R({}^\omega B/\mathrm{Fin}),
\]
and for c.c.c. \(B\),
\[
R({}^\omega B/\mathrm{Fin})\le_T \mathrm{Part}^*(B)\ ;\ R(\mathcal{P}(\omega)/\mathrm{fin}).
\]
In the Cohen case this yields
\[
D(2^{<\omega})\le_T R({}^\omega\mathbb{C}_\omega/\mathrm{Fin})
\le_T D(2^{<\omega})\ ;\ R(\mathcal{P}(\omega)/\mathrm{fin}),
\]
and, as a precise computation,
\[
\mathfrak{r}({}^\omega\mathbb{C}_\omega/\mathrm{Fin})=\mathfrak{r}+\mathrm{cof}(\mathcal{M}),
\qquad
\mathfrak{s}({}^\omega\mathbb{C}_\omega/\mathrm{Fin})=
\min\{\mathfrak{s},\mathrm{add}(\mathcal{M})\}.
\]
Thus the generalized reaping number of the reduced power is exactly the sum of the classical reaping number and the cofinality of the meagre ideal [2410.18595].

The Boolean-algebraic framework also ties reaping to ultrafilter bases. For any infinite \(B\),
\[
\mathfrak{r}(B)\le \mathfrak{u}(B).
\]
Moreover, for any Boolean algebra \(B\),
\[
\mathfrak{u}\le \mathfrak{u}({}^\omega B/\mathrm{Fin}),
\]
and if \(B\) is complete, atomless, and c.c.c., then
\[
\mathfrak{u}({}^\omega B/\mathrm{Fin})=\mathfrak{u}(B).
\]
Applying this to \(\mathbb{C}_\omega\) gives
\[
\mathrm{cof}(\mathcal{M})\le \mathfrak{u}(\mathbb{C}_\omega),
\]
while a suitable parametrized diamond principle implies
\[
\mathfrak{u}(\mathbb{C}_\omega)=\aleph_1.
\]
These conclusions exhibit reaping numbers as a bridge between antichain combinatorics, ideals such as \(\mathcal{M}\), and ultrafilter numbers [2410.18595].

## 5. Density-sensitive reaping numbers on \(\omega\)

A third development keeps the base set \(\omega\) but changes the notion of splitting. For \(A\subseteq\omega\),
\[
\underline{d}(A)=\liminf_{n\to\infty}\frac{|A\cap n|}{n},
\qquad
\overline{d}(A)=\limsup_{n\to\infty}\frac{|A\cap n|}{n}.
\]
If these coincide, their common value is \(d(A)\in[0,1]\); otherwise one writes \(d(A)=\textsf{osc}\). Likewise, for \(A,B\subseteq\omega\) with \(B\) infinite,
\[
\underline{d}_B(A)=\liminf_{n\to\infty}\frac{|A\cap B\cap n|}{|B\cap n|},
\qquad
\overline{d}_B(A)=\limsup_{n\to\infty}\frac{|A\cap B\cap n|}{|B\cap n|},
\]
and if these agree, the common value is \(d_B(A)\); otherwise \(d_B(A)=\textsf{osc}\). Densities therefore take values in
\[
\textsf{all}=[0,1]\cup\{\textsf{osc}\}.
\]
For nonempty \(X\subseteq\textsf{all}\), the generalized reaping number \(\mathfrak{r}_X\) is the least size of a family \(\mathcal{R}\subseteq[\omega]^\omega\) such that for every \(r\in X\) and every infinite-coinfinite \(S\subseteq\omega\), there exists \(R\in\mathcal{R}\) with
\[
d_R(S)\neq r.
\]
The dual invariant \(\mathfrak{s}_X\) is the least size of a family \(\mathcal{S}\subseteq[\omega]^\omega\) such that for every \(R\in[\omega]^\omega\) there are \(S\in\mathcal{S}\) and \(r\in X\) with \(d_R(S)=r\) [2506.21059].

These invariants are connected to permutation-based density-distribution numbers \(\mathfrak{dd}_{X,Y}\) and \(\mathfrak{dd}^{\textsf{rel}}_{X,Y}\). A key lemma yields the inequalities
\[
\mathfrak{dd}_{X,\textsf{all}}\le \mathfrak{r}_X,
\qquad
\mathfrak{s}_X\le \mathfrak{dd}_{X,\textsf{all}}^\perp.
\]
The sharpest identifications occur when \(X\) contains an extreme density. For all \(X\subseteq[0,1]\) such that \(0\in X\) or \(1\in X\),
\[
\mathfrak{s}_X=\operatorname{cov}(\mathcal{M}),
\qquad
\mathfrak{r}_X=\operatorname{non}(\mathcal{M}).
\]
In particular,
\[
\mathfrak{s}_0=\operatorname{cov}(\mathcal{M}),
\qquad
\mathfrak{r}_0=\operatorname{non}(\mathcal{M}).
\]
Thus density-sensitive reaping at the endpoints \(0\) and \(1\) collapses to the category cardinals [2506.21059].

For interior densities \(X\subseteq(0,1)\), the behavior is subtler. The paper recalls that for every \(\rho\in(0,1)\), \(\mathfrak{r}_\rho=\mathfrak{r}_{1/2}\), and proves the corresponding permutation identity
\[
\mathfrak{dd}_{\{1/2\},\textsf{all}}=\mathfrak{dd}_{\{r\},\textsf{all}}
\qquad (r\in(0,1)).
\]
It also establishes the ZFC bounds
\[
\operatorname{cov}(\mathcal{N})\le \mathfrak{r}_X,
\qquad
\mathfrak{r}_X\le \operatorname{non}(\mathcal{M})
\]
for nonempty \(X\subseteq[0,1]\), and, when \(X\subseteq(0,1)\),
\[
\operatorname{non}(\mathcal{SN})\le \mathfrak{s}_X,
\qquad
\mathfrak{r}_X\le \operatorname{cov}(\mathcal{SN}).
\]
Hence for \(X=(0,1)\),
\[
\operatorname{cov}(\mathcal{N})\le \mathfrak{r}_{(0,1)}\le \operatorname{cov}(\mathcal{SN}).
\]
Another upper bound is
\[
\mathfrak{r}_X\le \mathfrak{dd}_{X,[0,1]}.
\]
This can be strict: in the Hechler model, \(\mathfrak{r}_{1/2}=\aleph_1\) and likewise \(\mathfrak{r}_{(0,1)}=\aleph_1\), while \(\mathfrak{dd}_{\rho,[0,1]}\ge \mathfrak{b}\), and \(\mathfrak{b}>\aleph_1\) there [2506.21059].

The oscillation case links generalized reaping to classical block-splitting invariants. Writing \(\mathfrak{fr}=\min\{\mathfrak{d},\mathfrak{r}\}\) and \(\mathfrak{fs}=\max\{\mathfrak{b},\mathfrak{s}\}\), the paper proves
\[
\mathfrak{fr}\le \mathfrak{dd}_{\{\textsf{osc}\},\textsf{all}},
\qquad
\mathfrak{dd}_{\{\textsf{osc}\},\textsf{all}}^\perp\le \mathfrak{fs}.
\]
This suggests that oscillatory density reaping sits between composite invariants built from the classical splitting, reaping, bounding, and dominating numbers [2506.21059].

## 6. Comparative structure, consequences, and open directions

The current literature supports three principal meanings of “generalized reaping number,” each with a different ambient category and a different notion of what it means to fail simultaneous splitting.

| Setting | Invariant | Representative structural result |
|---|---|---|
| \([\kappa]^\kappa\) | \(\mathfrak{r}_\kappa\) | Galvin-Hajnal-type upper bounds at singular \(\kappa\) |
| Boolean algebra \(B\) | \(\mathfrak{r}(B)=\mathfrak{d}(R(B))\) | \(\mathfrak{r}({}^\omega\mathbb{C}_\omega/\mathrm{Fin})=\mathfrak{r}+\mathrm{cof}(\mathcal{M})\) |
| Relative density on \(\omega\) | \(\mathfrak{r}_X\) | If \(0\in X\) or \(1\in X\), then \(\mathfrak{r}_X=\operatorname{non}(\mathcal{M})\) |

Across these frameworks, several recurring mechanisms appear. One is the passage from local to global control: Lambie-Hanson’s theorem converts stationary many local bounds on \(\mathfrak{r}_{\kappa_i}\) into a global upper bound on \(\mathfrak{r}_\kappa\) [2202.06655]. Another is the extraction of reaping from auxiliary structures: almost refinement of maximal antichains controls reaping in reduced powers of Boolean algebras, especially for the Cohen algebra [2410.18595]. A third is the replacement of inclusion-based splitting by analytic predicates such as relative asymptotic density, producing invariants whose values are governed by \(\mathcal{M}\), \(\mathcal{N}\), and strong measure zero [2506.21059]. Shelah’s cofinality theorem supplies a complementary theme: comparison with a dominating-type invariant can force reaping numbers to have restricted cofinality [1401.4649].

Several open directions are explicit in the cited work. Shelah’s note asks, in effect, whether a higher-cardinal analogue
\[
\mathfrak{d}_\kappa>\mathfrak{r}_\kappa \Rightarrow \operatorname{cf}(\mathfrak{r}_\kappa)>\kappa
\]
can be established for regular uncountable \(\kappa\), but does not prove it [1401.4649]. Lambie-Hanson identifies the ultrafilter number \(\mathfrak{u}_\kappa\) as a prominent singular-cardinal characteristic not covered by the Galvin-Hajnal framework and asks whether a version of the main theorem holds for \(\mathfrak{u}_\kappa\) [2202.06655]. The Boolean-algebraic work explicitly asks whether
\[
\mathrm{cof}(\mathcal{N}) \le \mathfrak{d}(\mathrm{Part}^*(\mathbb{B}_\omega))
\]
for the random algebra \(\mathbb{B}_\omega\) [2410.18595]. In the density-sensitive setting, the paper leaves open whether
\[
\mathfrak{r}_{\mathbb{Q}\cap(0,1)}=\mathfrak{r}_{(0,1)},
\qquad
\mathfrak{dd}_{\mathbb{Q}\cap(0,1),\textsf{all}}=\mathfrak{dd}_{(0,1),\textsf{all}},
\]
and asks whether \(\mathfrak{r}_{\textsf{osc}}\ge \operatorname{cov}(\mathcal{N})\) [2506.21059].

Taken together, these results show that generalized reaping numbers are not a single invariant but a family of closely related cardinal characteristics organized around a common schema: choose a notion of splitting or decision, define unreaped families relative to that notion, and study the minimal size of a witness to unsplittability. The higher-cardinal, Boolean-algebraic, and density-sensitive versions differ substantially in method, but each reveals structural restrictions that are invisible in the bare classical definition alone [1401.4649][2202.06655][2410.18595][2506.21059].

Source: https://www.emergentmind.com/topics/generalized-reaping-numbers