Generalized Reaping Numbers in Set Theory

• Generalized reaping numbers are set theoretic invariants that extend classical splitting and reaping notions to higher-cardinal, Boolean algebra, and density-sensitive frameworks.
• They are characterized using methods like partition trees, canonical functions, and local-to-global diagonalization to reveal intricate combinatorial and cofinality phenomena.
• The study connects these invariants with ultrafilter bases and open problems on cofinality constraints, bridging combinatorial set theory with Boolean algebra and measure 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 (Shelah, 2014, Lambie-Hanson, 2022, Brendle et al., 2024, Valderrama, 26 Jun 2025).

1. Classical prototype and the unreaped-family viewpoint

In the classical setting, a set $x\in[\omega]^\omega$0 splits $x\in[\omega]^\omega$1 if both $x\in[\omega]^\omega$2 and $x\in[\omega]^\omega$3 are infinite. A family $x\in[\omega]^\omega$4 is splitting if every infinite $x\in[\omega]^\omega$5 is split by some $x\in[\omega]^\omega$6, and the splitting number $x\in[\omega]^\omega$7 is the least size of such a family. Dually, a set $x\in[\omega]^\omega$8 reaps a family $x\in[\omega]^\omega$9 if for every $y\in[\omega]^\omega$0, either $y\in[\omega]^\omega$1 or $y\in[\omega]^\omega$2. The reaping number $y\in[\omega]^\omega$3 is the least $y\in[\omega]^\omega$4 such that no single $y\in[\omega]^\omega$5 reaps $y\in[\omega]^\omega$6; equivalently, $y\in[\omega]^\omega$7 is the least size of an unreaped family, or the minimal size of a family that cannot be simultaneously split by one set (Shelah, 2014, Brendle et al., 2024).

This admits an equivalent Boolean-algebraic formulation in $y\in[\omega]^\omega$8. If $y\in[\omega]^\omega$9 denotes the equivalence class of $|y\cap x|=|y\setminus x|=\aleph_0$0 modulo finite, then the classical invariants may be written as

$|y\cap x|=|y\setminus x|=\aleph_0$1

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 (Brendle et al., 2024).

The classical invariant is also linked to eventual domination. On $|y\cap x|=|y\setminus x|=\aleph_0$2, one defines

$|y\cap x|=|y\setminus x|=\aleph_0$3

and the dominating number $|y\cap x|=|y\setminus x|=\aleph_0$4 is the least size of a dominating family in $|y\cap x|=|y\setminus x|=\aleph_0$5. Shelah’s theorem that $|y\cap x|=|y\setminus x|=\aleph_0$6 implies $|y\cap x|=|y\setminus x|=\aleph_0$7 shows that even the classical reaping number is subject to nontrivial structural restrictions on its cofinality (Shelah, 2014).

2. Higher-cardinal reaping numbers $|y\cap x|=|y\setminus x|=\aleph_0$8

For an infinite cardinal $|y\cap x|=|y\setminus x|=\aleph_0$9, the generalized reaping number is defined by replacing $\mathfrak{r}_\kappa$0 with $\mathfrak{r}_\kappa$1, $\mathfrak{r}_\kappa$2 with $\mathfrak{r}_\kappa$3, and “finite” with “of size $\mathfrak{r}_\kappa$4”. Thus for $\mathfrak{r}_\kappa$5, $\mathfrak{r}_\kappa$6 splits $\mathfrak{r}_\kappa$7 iff

$\mathfrak{r}_\kappa$8

A family $\mathfrak{r}_\kappa$9 is unreaped if there is no single $[\kappa]^\kappa$0 that splits every element of $[\kappa]^\kappa$1, and

$[\kappa]^\kappa$2

A standard diagonal argument yields $[\kappa]^\kappa$3 for every infinite $[\kappa]^\kappa$4 (Lambie-Hanson, 2022).

At singular $[\kappa]^\kappa$5 of uncountable cofinality $[\kappa]^\kappa$6, Lambie-Hanson places $[\kappa]^\kappa$7 into a Galvin-Hajnal framework. One fixes an increasing continuous sequence $[\kappa]^\kappa$8 converging to $[\kappa]^\kappa$9, and interprets $\mathfrak{r}(B)$0 as the cofinality of the quasiorder $\mathfrak{r}(B)$1 with $\mathfrak{r}(B)$2 and

$\mathfrak{r}(B)$3

with corresponding local structures $\mathfrak{r}(B)$4 at each $\mathfrak{r}(B)$5. The projection maps are

$\mathfrak{r}(B)$6

In this formalism, $\mathfrak{r}(B)$7 and $\mathfrak{r}(B)$8 (Lambie-Hanson, 2022).

The main theorem specialized to reaping numbers states that if $\mathfrak{r}(B)$9 is an ordinal such that a canonical function $B$0 exists, $B$1 is stationary, and

$B$2

then

$B$3

and if $B$4, then

$B$5

This is a Galvin-Hajnal-type and hence Silver-type restriction on $B$6: stationary many local upper bounds at the $B$7-levels force a global upper bound at $B$8 (Lambie-Hanson, 2022).

The proof uses canonical functions, Jech’s variation on Galvin-Hajnal, and density parameters such as the lower density $B$9. In the reaping case, the base step constructs global witnesses from local non-splitting data by combining dense families $\mathfrak{r}_X$0 with repeated applications of Fodor’s lemma. This places generalized reaping numbers firmly within the PCF-flavored structure theory of singular cardinals (Lambie-Hanson, 2022).

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

Shelah proved the classical theorem

$\mathfrak{r}_X$1

Equivalently, if $\mathfrak{r}_X$2, then $\mathfrak{r}_X$3. The result is purely combinatorial and works in ZFC (Shelah, 2014).

The proof begins by assuming toward contradiction that $\mathfrak{r}_X$4 and $\mathfrak{r}_X$5, writing

$\mathfrak{r}_X$6

for a strictly increasing sequence $\mathfrak{r}_X$7 with each $\mathfrak{r}_X$8. One fixes a witnessing reaping family $\mathfrak{r}_X$9, and for each $\omega$0 lets $\omega$1. The core construction is a binary tree of partitions

$\omega$2

such that for each level $\omega$3, $\omega$4 is a partition of $\omega$5 into infinite sets, each level refines the previous one, and each node can be split so as to handle all members of the subfamily $\omega$6. The point is that any family of size $\omega$7 is not reaping, so it can be split (Shelah, 2014).

From the partition tree one defines sets $\omega$8 indexed by $\omega$9 and $x\in[\omega]^\omega$00, together with functions $x\in[\omega]^\omega$01 satisfying

$x\in[\omega]^\omega$02

A further property is that if $x\in[\omega]^\omega$03, then $x\in[\omega]^\omega$04 is finite. Since each family $x\in[\omega]^\omega$05 has cardinality $x\in[\omega]^\omega$06, it is not dominating; therefore one can choose escaping functions $x\in[\omega]^\omega$07. After removing earlier intersections, one obtains infinite sets $x\in[\omega]^\omega$08, and then

$x\in[\omega]^\omega$09

These are disjoint infinite subsets of $x\in[\omega]^\omega$10 that split every $x\in[\omega]^\omega$11, contradicting that $x\in[\omega]^\omega$12 witnesses $x\in[\omega]^\omega$13 (Shelah, 2014).

For generalized reaping numbers, the significance of the argument is methodological. The same paper explicitly proposes the higher-cardinal definitions

$x\in[\omega]^\omega$14

and

$x\in[\omega]^\omega$15

where

$x\in[\omega]^\omega$16

It then notes that one would like to ask whether an analogue of Shelah’s theorem holds: $x\in[\omega]^\omega$17 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 $x\in[\omega]^\omega$18-versus-$x\in[\omega]^\omega$19 combinatorics (Shelah, 2014).

4. Boolean-algebraic reaping numbers and reduced powers

A different generalization replaces subsets of $x\in[\omega]^\omega$20 by elements of an arbitrary Boolean algebra $x\in[\omega]^\omega$21. Writing $x\in[\omega]^\omega$22, the reaping relation is

$x\in[\omega]^\omega$23

Thus $x\in[\omega]^\omega$24 is decided by $x\in[\omega]^\omega$25: it is either contained in $x\in[\omega]^\omega$26 or disjoint from $x\in[\omega]^\omega$27. The reaping number of $x\in[\omega]^\omega$28 is

$x\in[\omega]^\omega$29

which the authors identify with the weak density of $x\in[\omega]^\omega$30. If $x\in[\omega]^\omega$31 is atomless, the corresponding splitting number is

$x\in[\omega]^\omega$32

For $x\in[\omega]^\omega$33, these recover the classical $x\in[\omega]^\omega$34 and $x\in[\omega]^\omega$35 (Brendle et al., 2024).

The same work introduces an almost-refinement order on maximal antichains. If $x\in[\omega]^\omega$36 and $x\in[\omega]^\omega$37 is c.c.c., then

$x\in[\omega]^\omega$38

where $x\in[\omega]^\omega$39. The relational system $x\in[\omega]^\omega$40 turns out to mediate between antichain combinatorics and reaping phenomena. In particular,

$x\in[\omega]^\omega$41

and hence

$x\in[\omega]^\omega$42

If $x\in[\omega]^\omega$43 is non-atomic and $x\in[\omega]^\omega$44-finite c.c., then

$x\in[\omega]^\omega$45

so the almost-refinement structure realizes the classical bounding/dominating complexity of $x\in[\omega]^\omega$46 (Brendle et al., 2024).

For the Cohen algebra

$x\in[\omega]^\omega$47

Monk’s classical fact is that

$x\in[\omega]^\omega$48

The nontrivial behavior appears after passing to reduced powers. The paper proves

$x\in[\omega]^\omega$49

and consequently

$x\in[\omega]^\omega$50

For reduced powers $x\in[\omega]^\omega$51, one has

$x\in[\omega]^\omega$52

and for c.c.c. $x\in[\omega]^\omega$53,

$x\in[\omega]^\omega$54

In the Cohen case this yields

$x\in[\omega]^\omega$55

and, as a precise computation,

$x\in[\omega]^\omega$56

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 (Brendle et al., 2024).

The Boolean-algebraic framework also ties reaping to ultrafilter bases. For any infinite $x\in[\omega]^\omega$57,

$x\in[\omega]^\omega$58

Moreover, for any Boolean algebra $x\in[\omega]^\omega$59,

$x\in[\omega]^\omega$60

and if $x\in[\omega]^\omega$61 is complete, atomless, and c.c.c., then

$x\in[\omega]^\omega$62

Applying this to $x\in[\omega]^\omega$63 gives

$x\in[\omega]^\omega$64

while a suitable parametrized diamond principle implies

$x\in[\omega]^\omega$65

These conclusions exhibit reaping numbers as a bridge between antichain combinatorics, ideals such as $x\in[\omega]^\omega$66, and ultrafilter numbers (Brendle et al., 2024).

5. Density-sensitive reaping numbers on $x\in[\omega]^\omega$67

A third development keeps the base set $x\in[\omega]^\omega$68 but changes the notion of splitting. For $x\in[\omega]^\omega$69,

$x\in[\omega]^\omega$70

If these coincide, their common value is $x\in[\omega]^\omega$71; otherwise one writes $x\in[\omega]^\omega$72. Likewise, for $x\in[\omega]^\omega$73 with $x\in[\omega]^\omega$74 infinite,

$x\in[\omega]^\omega$75

and if these agree, the common value is $x\in[\omega]^\omega$76; otherwise $x\in[\omega]^\omega$77. Densities therefore take values in

$x\in[\omega]^\omega$78

For nonempty $x\in[\omega]^\omega$79, the generalized reaping number $x\in[\omega]^\omega$80 is the least size of a family $x\in[\omega]^\omega$81 such that for every $x\in[\omega]^\omega$82 and every infinite-coinfinite $x\in[\omega]^\omega$83, there exists $x\in[\omega]^\omega$84 with

$x\in[\omega]^\omega$85

The dual invariant $x\in[\omega]^\omega$86 is the least size of a family $x\in[\omega]^\omega$87 such that for every $x\in[\omega]^\omega$88 there are $x\in[\omega]^\omega$89 and $x\in[\omega]^\omega$90 with $x\in[\omega]^\omega$91 (Valderrama, 26 Jun 2025).

These invariants are connected to permutation-based density-distribution numbers $x\in[\omega]^\omega$92 and $x\in[\omega]^\omega$93. A key lemma yields the inequalities

$x\in[\omega]^\omega$94

The sharpest identifications occur when $x\in[\omega]^\omega$95 contains an extreme density. For all $x\in[\omega]^\omega$96 such that $x\in[\omega]^\omega$97 or $x\in[\omega]^\omega$98,

$x\in[\omega]^\omega$99

In particular,

$y\in[\omega]^\omega$00

Thus density-sensitive reaping at the endpoints $y\in[\omega]^\omega$01 and $y\in[\omega]^\omega$02 collapses to the category cardinals (Valderrama, 26 Jun 2025).

For interior densities $y\in[\omega]^\omega$03, the behavior is subtler. The paper recalls that for every $y\in[\omega]^\omega$04, $y\in[\omega]^\omega$05, and proves the corresponding permutation identity

$y\in[\omega]^\omega$06

It also establishes the ZFC bounds

$y\in[\omega]^\omega$07

for nonempty $y\in[\omega]^\omega$08, and, when $y\in[\omega]^\omega$09,

$y\in[\omega]^\omega$10

Hence for $y\in[\omega]^\omega$11,

$y\in[\omega]^\omega$12

Another upper bound is

$y\in[\omega]^\omega$13

This can be strict: in the Hechler model, $y\in[\omega]^\omega$14 and likewise $y\in[\omega]^\omega$15, while $y\in[\omega]^\omega$16, and $y\in[\omega]^\omega$17 there (Valderrama, 26 Jun 2025).

The oscillation case links generalized reaping to classical block-splitting invariants. Writing $y\in[\omega]^\omega$18 and $y\in[\omega]^\omega$19, the paper proves

$y\in[\omega]^\omega$20

This suggests that oscillatory density reaping sits between composite invariants built from the classical splitting, reaping, bounding, and dominating numbers (Valderrama, 26 Jun 2025).

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
$y\in[\omega]^\omega$21	$y\in[\omega]^\omega$22	Galvin-Hajnal-type upper bounds at singular $y\in[\omega]^\omega$23
Boolean algebra $y\in[\omega]^\omega$24	$y\in[\omega]^\omega$25	$y\in[\omega]^\omega$26
Relative density on $y\in[\omega]^\omega$27	$y\in[\omega]^\omega$28	If $y\in[\omega]^\omega$29 or $y\in[\omega]^\omega$30, then $y\in[\omega]^\omega$31

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 $y\in[\omega]^\omega$32 into a global upper bound on $y\in[\omega]^\omega$33 (Lambie-Hanson, 2022). 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 (Brendle et al., 2024). A third is the replacement of inclusion-based splitting by analytic predicates such as relative asymptotic density, producing invariants whose values are governed by $y\in[\omega]^\omega$34, $y\in[\omega]^\omega$35, and strong measure zero (Valderrama, 26 Jun 2025). Shelah’s cofinality theorem supplies a complementary theme: comparison with a dominating-type invariant can force reaping numbers to have restricted cofinality (Shelah, 2014).

Several open directions are explicit in the cited work. Shelah’s note asks, in effect, whether a higher-cardinal analogue

$y\in[\omega]^\omega$36

can be established for regular uncountable $y\in[\omega]^\omega$37, but does not prove it (Shelah, 2014). Lambie-Hanson identifies the ultrafilter number $y\in[\omega]^\omega$38 as a prominent singular-cardinal characteristic not covered by the Galvin-Hajnal framework and asks whether a version of the main theorem holds for $y\in[\omega]^\omega$39 (Lambie-Hanson, 2022). The Boolean-algebraic work explicitly asks whether

$y\in[\omega]^\omega$40

for the random algebra $y\in[\omega]^\omega$41 (Brendle et al., 2024). In the density-sensitive setting, the paper leaves open whether

$y\in[\omega]^\omega$42

and asks whether $y\in[\omega]^\omega$43 (Valderrama, 26 Jun 2025).

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 (Shelah, 2014, Lambie-Hanson, 2022, Brendle et al., 2024, Valderrama, 26 Jun 2025).