List of large cardinal properties

This page includes a list of large cardinal properties in the mathematical field of set theory. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, V_κ satisfies "there is an unbounded class of cardinals satisfying φ".

The following table usually arranges cardinals in order of consistency strength, with size of the cardinal used as a tiebreaker. In a few cases (such as strongly compact cardinals) the exact consistency strength is not known and the table uses the current best guess.

"Small" cardinals: 0, 1, 2, ..., $\aleph _{0},\aleph _{1}$ ,..., $\kappa =\aleph _{\kappa }$ , ... (see Aleph number); the height of the minimal transitive model of ZFC
worldly cardinals
weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals
weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals.
reflecting cardinals
pseudo uplifting cardinals, uplifting cardinals^[1]
weakly compact (= Π¹
₁-indescribable), Π^m
_n-indescribable, totally indescribable cardinals, ν-indescribable cardinals
λ-unfoldable, unfoldable cardinals, λ-shrewd, shrewd cardinals, strongly uplifting cardinals (not clear how these relate to each other).
ethereal cardinals, subtle cardinals
almost ineffable, ineffable, n-ineffable, totally ineffable cardinals
1-iterable cardinal, remarkable cardinal, virtually extendible cardinal, γ-iterable cardinal for 2≤γ≤ω, virtually rank-into-rank cardinal, ω-Erdős cardinal
λ-iterable cardinal, λ-Erdős cardinal (for ω+1≤λ<ω₁)
Silver cardinal, 0^# (not a cardinal), ω₁-iterable cardinal, ω₁-Erdős cardinal
almost Ramsey, Jónsson, Rowbottom, Ramsey, ineffably Ramsey, completely Ramsey, strongly Ramsey, super Ramsey cardinals
measurable cardinals, 0^†
λ-strong, strong cardinals, tall cardinals
Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin cardinals
superstrong cardinals (=1-superstrong; for n-superstrong for n≥2 see further down.)
subcompact, strongly compact (Woodin< strongly compact≤supercompact), supercompact, hypercompact cardinals
η-extendible, extendible cardinals
almost high jump cardinals
Vopěnka cardinals, Shelah for supercompactness, high jump cardinals, super high jump cardinals
n-superstrong (n≥2), n-almost huge, n-super almost huge, n-huge, n-superhuge cardinals (1-huge=huge, etc.)
exacting cardinals, ultraexacting cardinals^[2]
Wholeness axiom, rank-into-rank (Axioms I3, I2, I1, and I0)

The following even stronger large cardinal properties are not consistent with the axiom of choice, but their existence has not yet been refuted in ZF alone (that is, without use of the axiom of choice).

weakly Reinhardt cardinal, Reinhardt cardinal, proto-Berkeley cardinal, Berkeley cardinal, super Reinhardt cardinal, totally Reinhardt cardinal, club Berkeley cardinal, limit club Berkeley cardinal^[3]

Many of these large cardinals axioms also have virtual versions.^[4]

References

^ Hamkins, Joel David; Johnstone, Thomas A. (2014-05-01). "Resurrection axioms and uplifting cardinals". Archive for Mathematical Logic. 53 (3): 463–485. doi:10.1007/s00153-014-0374-y. ISSN 1432-0665.
^ Aguilera, Juan Pablo; Bagaria, Joan; Goldberg, Gabriel; Lücke, Philipp (2025-09-12). "Large cardinals beyond HOD". arXiv.org. Retrieved 2026-04-22.
^ Bagaria, Joan; Koellner, Peter; Woodin, W. Hugh (2019). "Large Cardinals Beyond Choice". The Bulletin of Symbolic Logic. 25 (3): 283–318. ISSN 1079-8986.
^ Gitman, Victoria; Schindler, Ralf (2018-12-01). "Virtual large cardinals". Annals of Pure and Applied Logic. Logic Colloquium 2015. 169 (12): 1317–1334. doi:10.1016/j.apal.2018.08.005. ISSN 0168-0072.

Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
Kanamori, Akihiro; Magidor, M. (1978). "The evolution of large cardinal axioms in set theory". Higher Set Theory (PDF). Lecture Notes in Mathematics. Vol. 669. Springer Berlin / Heidelberg. pp. 99–275. doi:10.1007/BFb0103104. ISBN 978-3-540-08926-1.
Solovay, Robert M.; Reinhardt, William N.; Kanamori, Akihiro (1978). "Strong axioms of infinity and elementary embeddings" (PDF). Annals of Mathematical Logic. 13 (1): 73–116. doi:10.1016/0003-4843(78)90031-1.

External links

[1] Hamkins, Joel David; Johnstone, Thomas A. (2014-05-01). "Resurrection axioms and uplifting cardinals". Archive for Mathematical Logic. 53 (3): 463–485. doi:10.1007/s00153-014-0374-y. ISSN 1432-0665.

[2] Aguilera, Juan Pablo; Bagaria, Joan; Goldberg, Gabriel; Lücke, Philipp (2025-09-12). "Large cardinals beyond HOD". arXiv.org. Retrieved 2026-04-22.

[3] Bagaria, Joan; Koellner, Peter; Woodin, W. Hugh (2019). "Large Cardinals Beyond Choice". The Bulletin of Symbolic Logic. 25 (3): 283–318. ISSN 1079-8986.

[4] Gitman, Victoria; Schindler, Ralf (2018-12-01). "Virtual large cardinals". Annals of Pure and Applied Logic. Logic Colloquium 2015. 169 (12): 1317–1334. doi:10.1016/j.apal.2018.08.005. ISSN 0168-0072.

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