# Indescribable cardinal
An **indescribable cardinal** is a large cardinal κ with the property that any statement about the structure \( (V_\kappa, \in, A) \), for a predicate \( A \subseteq V_\kappa \), expressed in a higher-order language of restricted complexity, already holds in some smaller structure \( (V_\alpha, \in, A \cap V_\alpha) \) with \( \alpha < \kappa \). The notion was introduced by William Hanf and [Dana Scott](https://www.edgepedia.org/dana-scott) in a 1961 abstract as a way of classifying inaccessible cardinals by their reflection behavior, and Azriel Lévy gave the systematic analysis of the resulting hierarchy in 1971.[1](https://www.ams.org/journals/notices/196110/196110FullIssue.pdf) • [2](https://math.bu.edu/people/aki/intro.pdf) • [3](https://math.stanford.edu/~feferman/papers/Indes%20Cards%20%26%20Admiss.pdf)
| Key fact | Detail |
|---|---|
| Origin | Formulated by W. P. Hanf and Dana Scott, University of California, Berkeley, in "Classifying inaccessible cardinals", Notices of the American Mathematical Society, vol. 8 (1961), p. 445[1](https://www.ams.org/journals/notices/196110/196110FullIssue.pdf) |
| Defining property | For a class Ω of higher-order formulas, κ is Ω-indescribable if every sentence Φ in Ω true of \( (V_\kappa, \in, A) \), for \( A \subseteq V_\kappa \), is true of some \( (V_\alpha, \in, A \cap V_\alpha) \), \( \alpha < \kappa \)[4](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/indescribable-cardinals-and-elementary-embeddings/9B20079D282895E0C410486037562B6C) |
| First-order case | \( \Pi^1_0 \)-indescribability (reflection for first-order sentences with a predicate) is equivalent to inaccessibility[5](https://arxiv.org/abs/1701.04358) |
| Weak compactness | \( \Pi^1_1 \)-indescribability coincides with weak compactness[6](https://thesis.caltech.edu/1949/3/hauser-k_1989.pdf) |
| Hierarchy order | The least \( \Sigma^n_m \)-indescribable cardinal is less than the least \( \Pi^n_{m+1} \)-indescribable cardinal[7](https://digicoll.lib.berkeley.edu/nanna/record/140220/files/2019Spring_Russ-Navarro_Kyle.pdf?registerDownload=1&version=1&withMetadata=0&withWatermark=0) |
| Relation to measurability | Below any measurable cardinal there are many totally indescribable cardinals (Vaught, 1963)[6](https://thesis.caltech.edu/1949/3/hauser-k_1989.pdf) |
| Ideals | For \( n < \omega \), the collection of non-\( \Pi^1_n \)-indescribable subsets of a \( \Pi^1_n \)-indescribable κ is a normal proper ideal (Lévy, 1971)[5](https://arxiv.org/abs/1701.04358) |
## Definition
The definition builds on the [reflection principle](https://www.edgepedia.org/reflection-principle) of [Zermelo–Fraenkel set theory](https://www.edgepedia.org/zermelo-fraenkel-set-theory). The universe of sets stratifies into a cumulative hierarchy \( (V_\alpha : \alpha \in \mathrm{On}) \), and ZF proves that any first-order formula holding in the universe already holds in some stage \( V_\alpha \). Hanf and Scott observed that a large cardinal notion arises if the reflecting formulas are allowed second-order free variables, assigned subsets of \( V_\alpha \).[4](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/indescribable-cardinals-and-elementary-embeddings/9B20079D282895E0C410486037562B6C)
Formally, for a collection Ω of formulas in the language of set theory with higher-type variables and a unary predicate symbol, an ordinal α is Ω-indescribable if for every sentence Φ in Ω and every \( A \subseteq V_\alpha \), whenever \( (V_\alpha, \in, A) \models \Phi \), there is some \( \beta < \alpha \) with \( (V_\beta, \in, A \cap V_\beta) \models \Phi \). Because a sufficient coding apparatus is available, this is equivalent to allowing finite sequences of relations over \( V_\alpha \), some possibly k-ary.[4](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/indescribable-cardinals-and-elementary-embeddings/9B20079D282895E0C410486037562B6C) The standardized classes \( \Sigma^n_m \) and \( \Pi^n_m \) consist of formulas whose prenex normal form has n alternating blocks of quantifiers of type m, meaning quantifiers over objects of order \( m+1 \), starting with existential (universal, respectively) quantifiers and no quantifiers of higher type.[4](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/indescribable-cardinals-and-elementary-embeddings/9B20079D282895E0C410486037562B6C)
## Place in the large cardinal hierarchy
The hierarchy calibrates several familiar notions. Hanf and Scott showed that in ZFC, \( \Pi^1_0 \)-indescribability is equivalent to inaccessibility and \( \Pi^1_1 \)-indescribability coincides with weak compactness; consequently the existence of \( \Pi^1_1 \)-indescribable cardinals is unprovable in ZFC.[6](https://thesis.caltech.edu/1949/3/hauser-k_1989.pdf) In the same spirit, the Stanford Encyclopedia's account identifies strong inaccessibility with reflection of \( \Sigma^1_1 \) (existential second-order) sentences and weak compactness with reflection of \( \Pi^1_1 \) (universal second-order) sentences in \( (V_\kappa, \in, A) \).[8](https://plato.stanford.edu/archives/sum2016/entries/set-theory/)
**Strength above weak compactness.** Allowing reflection for more complex second-order or higher-order sentences yields large cardinal notions stronger than weak compactness.[8](https://plato.stanford.edu/archives/sum2016/entries/set-theory/) Weakly compact cardinals sit a few steps beyond the inaccessible and Mahlo cardinals, well below the indescribable cardinals, the unfoldable, subtle, and ineffable cardinals, and far below the Ramsey and measurable cardinals.[9](https://mathoverflow.net/questions/109205/existence-of-weakly-compact-cardinals/109212) Every weakly compact cardinal is Mahlo, and the set of Mahlo cardinals below it is stationary.[8](https://plato.stanford.edu/archives/sum2016/entries/set-theory/)
**Strength below measurability.** Indescribable cardinals are comparatively tame. Vaught showed in 1963 that below any measurable cardinal there are many totally indescribable cardinals, meaning \( \Sigma^n_m \)-indescribable for all m and n, and Jensen's results imply that below \( \kappa(\omega) \), the last cardinal with \( \kappa \to (\omega)^{<\omega} \), there are stationarily many totally indescribable cardinals. Moreover, \( \Sigma^1_1 \) and \( \Pi^1_1 \)-indescribability relativize down to the constructible universe L.[6](https://thesis.caltech.edu/1949/3/hauser-k_1989.pdf) In the original abstract, Hanf and Scott noted that if κ is measurable, carrying a nontrivial κ-additive two-valued measure, then κ is \( \Pi^2_1 \)-indescribable, and the least measurable cardinal is strictly greater than the least \( \Pi^1_1 \)-indescribable cardinal.[1](https://www.ams.org/journals/notices/196110/196110FullIssue.pdf)
**Ordering within the hierarchy.** The hierarchy satisfies the expected strict order: the least \( \Sigma^n_m \)-indescribable cardinal is less than the least \( \Pi^n_{m+1} \)-indescribable cardinal.[7](https://digicoll.lib.berkeley.edu/nanna/record/140220/files/2019Spring_Russ-Navarro_Kyle.pdf?registerDownload=1&version=1&withMetadata=0&withWatermark=0) Lévy computed the complexity of truth definitions for these formula classes to establish large gaps between the least cardinals at successive levels.[6](https://thesis.caltech.edu/1949/3/hauser-k_1989.pdf)
## Ideals, reflection, and the constructible universe
Lévy proved that if κ is \( \Pi^1_n \)-indescribable, then the collection \( \Pi^1_n(\kappa) \) of subsets of κ that are not \( \Pi^1_n \)-indescribable is a normal proper ideal on κ; for \( n = 0 \) this is the nonstationary ideal, and for \( n = 1 \) it is called the weakly compact ideal.[5](https://arxiv.org/abs/1701.04358) Hellsten showed that this ideal is nowhere κ-saturated on a \( \Pi^1_n \)-indescribable cardinal, and that if κ is \( \kappa^+ \)-\( \Pi^1_n \)-indescribable then the ideal is not \( \kappa^+ \)-saturated.[5](https://arxiv.org/abs/1701.04358)
Reflection characterizations connect indescribability to stationary set behavior. Jensen's theorem says that inside L, a cardinal is weakly compact exactly when it is regular and satisfies the stationary reflection principle, i.e., every stationary subset of κ has a stationary proper initial segment. Bagaria, Magidor, and Sakai generalized this to a characterization of the \( \Pi^1_{n+1} \)-indescribable cardinals in L.[5](https://arxiv.org/abs/1701.04358)
## Extensions and independence results
The finite hierarchy has been extended transfinitely. Sharpe and Welch, and independently Joan Bagaria, extended \( \Pi^1_n \)-indescribability for \( n < \omega \) to \( \Pi^1_\xi \)-indescribability for \( \xi \geq \omega \), and Bagaria used generalized logic for the same purpose; these extensions support refined Ramsey-like hierarchies in which homogeneous sets are required to be indescribable.[10](https://arxiv.org/abs/1907.13540) • [11](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12940)
The relative sizes of the least \( \Sigma^n_m \)- and \( \Pi^n_m \)-indescribable cardinals are not fixed by ZFC for \( m \geq 2 \) and \( n \geq 1 \). Kai Hauser, a set theorist whose 1989 Caltech doctoral thesis under W. Hugh Woodin studied these questions, proved consistency results showing that the ordering can be forced either way, and an Easton-type theorem giving substantial freedom in arranging the relative sizes of the least \( \Sigma \)- and \( \Pi \)-indescribable cardinals simultaneously across levels, assuming indescribables exist at all those levels.[6](https://thesis.caltech.edu/1949/3/hauser-k_1989.pdf)
On the proof-theoretic side, Toshiyasu Arai, a proof theorist at Hiroshima University, showed that the existence of a weakly compact cardinal over ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations, and related work reduces the existence of \( \Pi^1_N \)-indescribable cardinals to iterations of lower Mahlo operations.[12](https://arxiv.org/abs/1111.0462) • [13](https://arxiv.org/abs/1409.1336)
## References
1. [W. P. Hanf and D. Scott (1961). Classifying inaccessible cardinals. Notices of the American Mathematical Society, vol. 8, p. 445.](https://www.ams.org/journals/notices/196110/196110FullIssue.pdf)
2. [introduction xiii](https://math.bu.edu/people/aki/intro.pdf)
3. [Akihiro Kanamori, The Higher Infinite, references on indescribable cardinals (Lévy 1971, Aczel 1974).](https://math.stanford.edu/~feferman/papers/Indes%20Cards%20%26%20Admiss.pdf)
4. [Kai Hauser (1991). Indescribable cardinals and elementary embeddings. Journal of Symbolic Logic 56(2), 439–457.](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/indescribable-cardinals-and-elementary-embeddings/9B20079D282895E0C410486037562B6C)
5. [Brent Cody (2017). Adding a non-reflecting weakly compact set. arXiv:1701.04358.](https://arxiv.org/abs/1701.04358)
6. [Kai Hauser (1989). Independence Results for Indescribable Cardinals. Ph.D. thesis, California Institute of Technology.](https://thesis.caltech.edu/1949/3/hauser-k_1989.pdf)
7. [Russell Navarro and Kyle (2019). A description of the large cardinal hierarchy up to measurability, UC Berkeley.](https://digicoll.lib.berkeley.edu/nanna/record/140220/files/2019Spring_Russ-Navarro_Kyle.pdf?registerDownload=1&version=1&withMetadata=0&withWatermark=0)
8. [Set Theory, Stanford Encyclopedia of Philosophy (Summer 2016 edition).](https://plato.stanford.edu/archives/sum2016/entries/set-theory/)
9. [MathOverflow: Existence of weakly compact cardinals.](https://mathoverflow.net/questions/109205/existence-of-weakly-compact-cardinals/109212)
10. [A refinement of the Ramsey hierarchy via indescribability. arXiv:1907.13540.](https://arxiv.org/abs/1907.13540)
11. [Benhamou (2024). Transferring compactness. Journal of the London Mathematical Society.](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12940)
12. [Toshiyasu Arai (2011). Proof theory of weak compactness. arXiv:1111.0462.](https://arxiv.org/abs/1111.0462)
13. [Toshiyasu Arai. Lifting proof theory to the countable ordinals II: second-order indescribable cardinals. arXiv:1409.1336.](https://arxiv.org/abs/1409.1336)
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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 › Large cardinals › Inaccessible and reflecting cardinals*
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