List of large cardinal properties
This page includes a list of cardinals with large cardinal properties. 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, Vκ 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, ..., ,..., , ... (see Aleph number)
- weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals
- weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals.
- reflecting cardinals
- weakly compact (= Π1
n-indescribable, totally indescribable cardinals
- λ-unfoldable, unfoldable cardinals, ν-indescribable cardinals and λ-shrewd, shrewd cardinals [not clear how these relate to each other].
- ethereal cardinals, subtle cardinals
- almost ineffable, ineffable, n-ineffable, totally ineffable cardinals
- remarkable cardinals
- α-Erdős cardinals (for countable α), 0# (not a cardinal), γ-Erdős cardinals (for uncountable γ)
- almost Ramsey, Jónsson, Rowbottom, Ramsey, ineffably 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 cardinals
- η-extendible, extendible cardinals
- Vopěnka cardinals
- n-superstrong (n≥2), n-almost huge, n-super almost huge, n-huge, n-superhuge cardinals (1-huge=huge, etc.)
- rank-into-rank (Axioms I3, I2, I1, and I0)
Finally, if there were a nontrivial elementary embedding from the entire von Neumann universe V into itself, j:V→V, its critical point would be called a Reinhardt cardinal. It is provable in ZFC that there is no such embedding (and therefore no Reinhardt cardinals). However their existence has not yet been refuted in ZF alone (that is, without use of the axiom of choice).
- 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. Lecture Notes in Mathematics. 669 (typescript). 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". Annals of Mathematical Logic 13 (1): 73–116. doi:10.1016/0003-4843(78)90031-1