# Cardinal characteristic of the continuum

A **cardinal characteristic of the continuum** (also called a cardinal invariant) is a cardinal number, usually defined as the minimal size of a set of reals with a given property, that describes the combinatorial, topological, or analytic structure of the real line and related spaces such as the Baire space \( \omega^{\omega} \) and the power set \( \mathcal{P}(\omega) \).<sup>[1](https://eprints.whiterose.ac.uk/id/eprint/118463/1/cichon-large%28revised%29.pdf)</sup><sup> • </sup><sup>[2](https://arxiv.org/html/math/9407203)</sup> These cardinals typically take values between \( \aleph_1 \), the first uncountable cardinal, and \( \mathfrak{c} = 2^{\aleph_0} \), the cardinality of the continuum.<sup>[1](https://eprints.whiterose.ac.uk/id/eprint/118463/1/cichon-large%28revised%29.pdf)</sup><sup> • </sup><sup>[2](https://arxiv.org/html/math/9407203)</sup> The theory consists of two kinds of results: ZFC-provable equations and inequalities between pairs of characteristics, and independence results showing that other relations cannot be proved in ZFC.<sup>[2](https://arxiv.org/html/math/9407203)</sup> Because every characteristic equals \( \aleph_1 \) if the [continuum hypothesis](https://www.edgepedia.org/continuum-hypothesis) (CH) holds, the theory is of interest mainly when CH fails.<sup>[2](https://arxiv.org/html/math/9407203)</sup>

| Key fact | Detail |
|---|---|
| Definition | Minimal size of a set of reals with a given property, describing the combinatorial structure of the real line; values usually lie between \( \aleph_1 \) and \( \mathfrak{c} \)<sup>[1](https://eprints.whiterose.ac.uk/id/eprint/118463/1/cichon-large%28revised%29.pdf)</sup> |
| Bounding and dominating numbers | \( \mathfrak{b} \) is the smallest size of an unbounded family in \( (\omega^{\omega}, \leq^{*}) \); \( \mathfrak{d} \) the smallest size of a dominating family<sup>[2](https://arxiv.org/html/math/9407203)</sup><sup> • </sup><sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup> |
| Cichoń's diagram | Twelve characteristics associated with the null ideal \( \mathcal{N} \), meager ideal \( \mathcal{M} \), countable sets, and σ-compact subsets of the irrationals, with provable inequalities between them<sup>[4](https://www.dmg.tuwien.ac.at/kellner/publications/CMWLC.pdf)</sup> |
| Two ZFC equalities | \( \operatorname{add}(\mathcal{M}) = \min(\mathfrak{b}, \operatorname{cov}(\mathcal{M})) \) and \( \operatorname{cof}(\mathcal{M}) = \max(\mathfrak{d}, \operatorname{non}(\mathcal{M})) \)<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup> |
| Completeness | Any assignment of \( \aleph_1 \) and \( \aleph_2 \) to the diagram's entries consistent with its inequalities is consistent with ZFC<sup>[5](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2042-01.pdf)</sup><sup> • </sup><sup>[4](https://www.dmg.tuwien.ac.at/kellner/publications/CMWLC.pdf)</sup> |
| Cichoń's maximum | Consistent with ZFC, first with large cardinals (GKS19) and later without them, that all non-dependent entries are pairwise different<sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup><sup> • </sup><sup>[4](https://www.dmg.tuwien.ac.at/kellner/publications/CMWLC.pdf)</sup> |
| Widest separation | A model with 15 classical characteristics pairwise different has been constructed by forcing with preserved splitting families<sup>[7](https://link.springer.com/article/10.1007/s11856-021-2237-7)</sup> |

## Formal definitions and principal examples

The prototype characteristics concern the partial order \( \leq^{*} \) of eventual domination on \( \omega^{\omega} \), where \( f \leq^{*} g \) means \( g(n) \geq f(n) \) for all sufficiently large \( n \). A family \( D \subseteq \omega^{\omega} \) is dominating if every \( g \in \omega^{\omega} \) satisfies \( g \leq^{*} f \) for some \( f \in D \), and the **dominating number** \( \mathfrak{d} \) is the smallest size of such a family; the **bounding number** \( \mathfrak{b} \) is the smallest size of a family not bounded by any single function.<sup>[2](https://arxiv.org/html/math/9407203)</sup> Equivalently, \( \mathfrak{b} = \operatorname{add}(\mathcal{K}) = \operatorname{non}(\mathcal{K}) \) and \( \mathfrak{d} = \operatorname{cov}(\mathcal{K}) = \operatorname{cof}(\mathcal{K}) \), where \( \mathcal{K} \) is the σ-ideal generated by compact subsets of the irrationals.<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup>

For a σ-ideal \( \mathcal{I} \) on the reals, four characteristics are defined: \( \operatorname{add}(\mathcal{I}) \), the smallest number of \( \mathcal{I} \)-sets whose union is not in \( \mathcal{I} \); \( \operatorname{cov}(\mathcal{I}) \), the smallest number of \( \mathcal{I} \)-sets needed to cover the reals; \( \operatorname{non}(\mathcal{I}) \), the smallest cardinality of a set not in \( \mathcal{I} \); and \( \operatorname{cof}(\mathcal{I}) \), the smallest size of a cofinal family in \( \mathcal{I} \).<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup> The two most important ideals are \( \mathcal{N} \), the null ([Lebesgue measure](https://www.edgepedia.org/lebesgue-measure) zero) sets, and \( \mathcal{M} \), the meager (first category) sets.<sup>[1](https://eprints.whiterose.ac.uk/id/eprint/118463/1/cichon-large%28revised%29.pdf)</sup> Thus \( \operatorname{cov}(\mathcal{N}) \) is the smallest number of null sets covering the reals, with \( \aleph_0 < \operatorname{cov}(\mathcal{N}) \leq 2^{\aleph_0} \), and \( \operatorname{cov}(\mathcal{N}) = 2^{\aleph_0} \) under CH.<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup> For the ideal \( \mathsf{ctbl} \) of countable sets the values are fixed in ZFC: \( \operatorname{add}(\mathsf{ctbl}) = \operatorname{non}(\mathsf{ctbl}) = \aleph_1 \) and \( \operatorname{cov}(\mathsf{ctbl}) = \operatorname{cof}(\mathsf{ctbl}) = 2^{\aleph_0} \).<sup>[4](https://www.dmg.tuwien.ac.at/kellner/publications/CMWLC.pdf)</sup>

Other widely studied numbers include the **splitting number** \( \mathfrak{s} \), the smallest size of a family of subsets of \( \omega \) that splits every infinite set into two infinite pieces, and the **reaping number** \( \mathfrak{r} \), the smallest size of an unsplit family.<sup>[2](https://arxiv.org/html/math/9407203)</sup> The **pseudointersection number** \( \mathfrak{p} \) is the minimum size of a family of infinite subsets of \( \omega \) with the strong finite intersection property but no infinite pseudointersection, and the **tower number** \( \mathfrak{t} \) is the minimum size of a tower; \( \mathfrak{p} \leq \mathfrak{t} \) is easy to see.<sup>[8](https://shelah.logic.at/files/95517/E74.pdf)</sup> For the four numbers \( \mathfrak{b}, \mathfrak{d}, \mathfrak{a}, \mathfrak{s} \), the only nontrivial inequalities between distinct elements of \( \{\aleph_1, \mathfrak{c}, \mathfrak{b}, \mathfrak{a}, \mathfrak{d}, \mathfrak{s}\} \) that ZFC proves beyond the trivial bounds \( \aleph_1 \leq \cdot \leq \mathfrak{c} \) are \( \mathfrak{b} \leq \mathfrak{d} \), \( \mathfrak{b} \leq \mathfrak{a} \), and \( \mathfrak{s} \leq \mathfrak{d} \).<sup>[9](https://www.logic.univie.ac.at/~vfischer/on_four_cardinal_characteristics.pdf)</sup><sup> • </sup><sup>[12](https://msp.org/pjm/2025/337-2/pjm-v337-n2-s.pdf)</sup>

## The Cichoń diagram and ZFC inequalities

**Cichoń's diagram** lists twelve cardinal characteristics, and the provable inequalities between them, associated with four ideals: the null sets \( \mathcal{N} \), the meager sets \( \mathcal{M} \), the countable sets, and the σ-compact subsets of the irrationals.<sup>[4](https://www.dmg.tuwien.ac.at/kellner/publications/CMWLC.pdf)</sup> Two entries are determined by the others in ZFC: \( \operatorname{add}(\mathcal{M}) = \min(\mathfrak{b}, \operatorname{cov}(\mathcal{M})) \) and \( \operatorname{cof}(\mathcal{M}) = \max(\mathfrak{d}, \operatorname{non}(\mathcal{M})) \).<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup>

The **Bartoszyński–Raisonnier–Stern theorem** gives \( \operatorname{add}(\mathcal{N}) \leq \operatorname{add}(\mathcal{M}) \), and dually \( \operatorname{cof}(\mathcal{M}) \leq \operatorname{cof}(\mathcal{N}) \).<sup>[1](https://eprints.whiterose.ac.uk/id/eprint/118463/1/cichon-large%28revised%29.pdf)</sup>

The diagram is complete in a strong sense: any assignment of the values \( \aleph_1 \) and \( \aleph_2 \) to its entries that does not contradict the diagram, including the two equalities above, is consistent with ZFC, proved using countable support iterations of forcing.<sup>[5](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2042-01.pdf)</sup> Equivalently, all assignments of \( \aleph_1 \) and \( \aleph_2 \) to the nine independent characteristics (excluding \( \aleph_1 \) and including \( \mathfrak{c} \)) that honor the arrows are consistent with ZFC.<sup>[4](https://www.dmg.tuwien.ac.at/kellner/publications/CMWLC.pdf)</sup> A classical series of theorems by Bartoszyński, Blass, Weiss, Cichoń, Judah, Shelah, Kamke, Miller, and Raisonnier and Stern shows these are the only provable relations.<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup>

## Independence and forcing models

Forcing, the method introduced by [Paul Cohen](https://www.edgepedia.org/paul-cohen) in the early 1960s after Gödel's constructible universe, shows that ZFC does not determine the value of \( \mathfrak{c} \) or of the characteristics.<sup>[9](https://www.logic.univie.ac.at/~vfischer/on_four_cardinal_characteristics.pdf)</sup> For example, \( \operatorname{cov}(\mathcal{N}) = \aleph_1 < 2^{\aleph_0} \) is consistent, as is \( \aleph_1 < \operatorname{cov}(\mathcal{N}) = 2^{\aleph_0} \), so \( \operatorname{cov}(\mathcal{N}) \) is independent of ZFC and of the continuum's size.<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup> Likewise both \( \aleph_0 < \operatorname{cov}(\mathcal{M}) < \mathfrak{c} \) and \( \aleph_1 < \operatorname{cov}(\mathcal{M}) = \mathfrak{c} \) are relatively consistent with ZFC.<sup>[9](https://www.logic.univie.ac.at/~vfischer/on_four_cardinal_characteristics.pdf)</sup>

**Named models and techniques.** The relative consistency of \( \mathfrak{s} < \mathfrak{b} \) was obtained by Baumgartner and Dordal in their study of the Hechler model, one of the first uses of a rank argument in forcing; Shelah later introduced creature forcing to establish the consistencies of \( \mathfrak{b} = \aleph_1 < \mathfrak{s} = \aleph_2 \) and \( \mathfrak{b} = \aleph_1 < \mathfrak{a} = \aleph_2 \).<sup>[9](https://www.logic.univie.ac.at/~vfischer/on_four_cardinal_characteristics.pdf)</sup> Kellner, Shelah, and Tanăsie introduced a forcing \( \widetilde{\mathbb{E}} \) with fam-limits that adds eventually different reals, enabling separation of the left side of [Cichoń's diagram](https://www.edgepedia.org/cichons-diagram) with \( \mathfrak{b} < \operatorname{cov}(\mathcal{N}) \).<sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup> Technique matters: only four instances of Cichoń's maximum can be forced with finite-support iterations of ccc posets, because such iterations add Cohen reals at limit steps and thereby force \( \operatorname{non}(\mathcal{M}) \leq \operatorname{cov}(\mathcal{M}) \).<sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup>

A further application of forcing with preserved splitting families produced a model in which 15 classical characteristics are pairwise different: the 10 non-dependent entries of Cichoń's diagram, \( \mathfrak{m} \) (2-Knaster), \( \mathfrak{p} \), \( \mathfrak{h} \), \( \mathfrak{s} \), and \( \mathfrak{r} \).<sup>[7](https://link.springer.com/article/10.1007/s11856-021-2237-7)</sup>

## By the numbers: Cichoń's maximum and its status

**Cichoń's maximum** is the situation in which all non-dependent cardinals of the diagram are pairwise different.<sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup> The first proof of its consistency with ZFC is due to Goldstern, Kellner, and Shelah (GKS19), using Boolean ultrapowers, a method that uses large cardinals; specifically, assuming four strongly compact cardinals, it is consistent that

\[ \aleph_1 < \operatorname{add}(\mathcal{N}) < \operatorname{cov}(\mathcal{N}) < \mathfrak{b} < \operatorname{non}(\mathcal{M}) < \operatorname{cov}(\mathcal{M}) < \mathfrak{d} < \operatorname{non}(\mathcal{N}) < \operatorname{cof}(\mathcal{N}) < 2^{\aleph_0}. \]

<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup><sup> • </sup><sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup> A later line of work (GKMS, 2022 onward) found a method to force the constellations of Cichoń's maximum without using large cardinals.<sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup>

Some constellations remain of unknown consistency: no instance of Cichoń's maximum with \( \operatorname{cov}(\mathcal{M}) < \operatorname{non}(\mathcal{M}) \) is known to be consistent with ZFC, even under large cardinals, and two constellations compatible with finite-support iterations are of unknown consistency.<sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup>

## How it compares with related notions

The continuum hypothesis is the statement \( 2^{\aleph_0} = \aleph_1 \), a special case of the [generalized continuum hypothesis](https://www.edgepedia.org/generalized-continuum-hypothesis) \( 2^{\aleph_{\alpha}} = \aleph_{\alpha+1} \).<sup>[10](https://plato.stanford.edu/entries/continuum-hypothesis/)</sup> Under CH every characteristic collapses to \( \aleph_1 \), so the entire separation theory presupposes \( \mathfrak{c} > \aleph_1 \).<sup>[2](https://arxiv.org/html/math/9407203)</sup> Large cardinals enter in two ways: as hypotheses used in early maximality proofs (four strongly compact cardinals in GKS19, later eliminated), and in Woodin's program, where in the maximal theory \( T^{*} \), assuming the Strong Ω Conjecture, CH fails and the continuum has size \( \aleph_2 \).<sup>[3](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)</sup><sup> • </sup><sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup><sup> • </sup><sup>[10](https://plato.stanford.edu/entries/continuum-hypothesis/)</sup>

The theory also generalizes beyond \( \omega \). For an uncountable regular cardinal \( \kappa \), many ZFC results carry over, including \( \kappa^{+} \leq \mathfrak{b}_{\kappa} = \operatorname{cf}(\mathfrak{b}_{\kappa}) \leq \operatorname{cf}(\mathfrak{d}_{\kappa}) \leq \mathfrak{d}_{\kappa} \leq 2^{\kappa} \).<sup>[1](https://eprints.whiterose.ac.uk/id/eprint/118463/1/cichon-large%28revised%29.pdf)</sup> The **higher Cichoń diagram** at \( \kappa \) is built from three ideals: the meager sets in the \( <\kappa \)-box product topology, the "null" sets in Shelah's sense, and the nowhere stationary subsets of a naturally defined stationary set \( S^{\kappa}_{\mathrm{pr}} \subseteq \kappa \).<sup>[11](https://shelah.logic.at/files/204987/fm666-4-2020.pdf)</sup>

## History and recent developments

Cantor proved in 1874 that the continuum is uncountable, and Hilbert's first problem asked whether it is the smallest uncountable cardinal; Gödel (1939) and Cohen (1963) showed that CH is independent of ZFC.<sup>[8](https://shelah.logic.at/files/95517/E74.pdf)</sup> Hausdorff proved in 1936 that \( \aleph_1 \leq \mathfrak{p} \), and Rothberger proved in 1948 that \( \mathfrak{p} = \aleph_1 \) implies \( \mathfrak{p} = \mathfrak{t} \); van Douwen's 1984 survey focused on six primary invariants, \( \mathfrak{b}, \mathfrak{p}, \mathfrak{t}, \mathfrak{d}, \mathfrak{a}, \mathfrak{s} \).<sup>[8](https://shelah.logic.at/files/95517/E74.pdf)</sup> The question whether \( \mathfrak{p} = \mathfrak{t} \), the oldest problem on cardinal invariants of the continuum, was solved in ZFC through work on Keisler's order in model theory by Malliaris and Shelah.<sup>[8](https://shelah.logic.at/files/95517/E74.pdf)</sup>

The maximality program then moved to Cichoń's diagram: the GKS19 proof of Cichoń's maximum with large cardinals was followed by the large-cardinal-free GKMS method and by the 15-characteristic separation of 2021.<sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup><sup> • </sup><sup>[7](https://link.springer.com/article/10.1007/s11856-021-2237-7)</sup> Work on the anatomy of \( \widetilde{\mathbb{E}} \) (2024) continues to map which constellations the new forcing can realize, and the higher Cichoń diagram extends the framework to uncountable cardinals.<sup>[6](https://ar5iv.labs.arxiv.org/html/2402.04706)</sup><sup> • </sup><sup>[11](https://shelah.logic.at/files/204987/fm666-4-2020.pdf)</sup>

## References

1. [Cichoń's diagram for uncountable cardinals (Brendle et al.)](https://eprints.whiterose.ac.uk/id/eprint/118463/1/cichon-large%28revised%29.pdf)
2. [Reductions Between Cardinal Characteristics of the Continuum (Blass)](https://arxiv.org/html/math/9407203)
3. [Cichoń's maximum, Annals of Mathematics 190 (Goldstern, Kellner, Shelah)](https://annals.math.princeton.edu/wp-content/uploads/annals-v190-n1-p02-s.pdf)
4. [Cichoń's maximum without large cardinals (Goldstern, Kellner, Mejía, Shelah)](https://www.dmg.tuwien.ac.at/kellner/publications/CMWLC.pdf)
5. [Some problems in forcing theory: large continuum and generalized cardinal invariants (Brendle, RIMS Kokyuroku)](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2042-01.pdf)
6. [Anatomy of 𝔼̃ (2024, arXiv)](https://ar5iv.labs.arxiv.org/html/2402.04706)
7. [Preservation of splitting families and cardinal characteristics of the continuum, Israel Journal of Mathematics](https://link.springer.com/article/10.1007/s11856-021-2237-7)
8. [Malliaris and Shelah, PNAS: p = t](https://shelah.logic.at/files/95517/E74.pdf)
9. [On four cardinal characteristics of the continuum (Fischer)](https://www.logic.univie.ac.at/~vfischer/on_four_cardinal_characteristics.pdf)
10. [The Continuum Hypothesis, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/continuum-hypothesis/)
11. [The higher Cichoń diagram (Shelah, Fundamenta Mathematicae)](https://shelah.logic.at/files/204987/fm666-4-2020.pdf)
12. [Pjm V337 N2 S.pdf, msp.org](https://msp.org/pjm/2025/337-2/pjm-v337-n2-s.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Ordinal and cardinal numbers › Cardinal numbers › Cardinal characteristics of the continuum*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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