List of large cardinal properties
From Wikipedia, the free encyclopedia
This page is a list of some types of cardinals; 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 are unboundedly many 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.
- weakly compact (= Π11-indescribable) Πmn-indescribable , totally indescribable cardinals
- unfoldable, λ-unfoldable 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
- 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
- extendible, η-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)
- Reinhardt cardinals (not consistent with the axiom of choice)
- 0=1 is (somewhat jokingly) listed as the ultimate large cardinal axiom by some authors.
[edit] References
- 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, vol. 669 (typescript), Lecture Notes in Mathematics, Springer Berlin / Heidelberg, pp. 99-275, ISBN 978-3-540-08926-1, ISSN 1617-9692, DOI 10.1007/BFb0103104
- 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
