List of large cardinal properties
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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".
- "Small" cardinals: 0, 1, 2, ...,
,..., 
, ... (see Aleph number) - weakly and strongly inaccessible cardinals, α-inaccessible cardinals, and hyper inaccessible cardinals
- weakly and strongly Mahlo cardinals, α-Mahlo cardinals, and hyper Mahlo cardinals.
- weakly compact cardinals (same as Π11-indescribable cardinals)
- Πmn-indescribable cardinals, totally indescribable cardinals
- unfoldable cardinals and λ-unfoldable cardinals
- subtle cardinals
- almost ineffable cardinals, ineffable cardinals, n-ineffable cardinals, totally ineffable cardinals
- remarkable cardinals
- α-Erdős cardinals (for countable α)
- 0# (not a cardinal, but proves the existence of transitive models with the cardinals above)
- γ-Erdős cardinals (for uncountable γ)
- Almost Ramsey cardinals
- Jónsson cardinals (equiconsistent with Ramsey)
- Rowbottom cardinals (every Rowbottom cardinal is Jónsson, and every Ramsey is Rowbottom)
- Ramsey cardinals (= a cardinal κ that is κ-Erdős)
- Ineffably Ramsey cardinals
- measurable cardinals
- 0†
- λ-strong cardinals
- strong cardinals
- Woodin cardinals
- weakly hyper-Woodin cardinals
- Shelah cardinals
- hyper-Woodin cardinals
- superstrong cardinals
- subcompact cardinals
- strongly compact cardinals (Warning: the exact strength is not known. Not stronger than supercompact, stronger than Woodin cardinals)
- supercompact cardinals
- extendible cardinals and η-extendible cardinals
- Vopěnka cardinals
- almost huge cardinals, super almost huge cardinals, huge cardinals (same as 1-huge cardinals), and superhuge cardinals
- n-huge cardinals for n≥ 2 (and similar analogues for superstrong, almost huge, super almost huge, and super huge)
- rank-into-rank (Axioms I3, I2, I1, and I0)
- Reinhardt cardinals (not consistent with the axiom of choice)
- 1=0 is jokingly listed as the ultimate large cardinal axiom by some authors, to suggest that the very large cardinals are "close" to being inconsistent.
[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.
- Solovay, Robert M.; William N. Reinhardt, and Akihiro Kanamori (1978). "Strong axioms of infinity and elementary embeddings". Annals of Mathematical Logic 13 (1): 73–116.
