Mathematics Logic Large cardinal properties view content

# List of large cardinal properties

2014-3-16 10:03| view publisher: amanda| wiki(57883.com) 0 : 0

description: Main article: Large cardinalThis 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 c ...
 Main article: Large cardinalThis 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, ..., \aleph_0, \aleph_1,..., \kappa  = \aleph_{\kappa}, ... (see Aleph number)weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinalsweakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals.reflecting cardinalsweakly compact (= Π11-indescribable), Πmn-indescribable, totally indescribable cardinalsλ-unfoldable, unfoldable cardinals, ν-indescribable cardinals and λ-shrewd, shrewd cardinals [not clear how these relate to each other].ethereal cardinals, subtle cardinalsalmost ineffable, ineffable, n-ineffable, totally ineffable cardinalsremarkable cardinalsα-Erdős cardinals (for countable α), 0# (not a cardinal), γ-Erdős cardinals (for uncountable γ)almost Ramsey, Jónsson, Rowbottom, Ramsey, ineffably Ramsey cardinalsmeasurable cardinals, 0†λ-strong, strong cardinals, tall cardinalsWoodin, weakly hyper-Woodin, Shelah, hyper-Woodin cardinalssuperstrong cardinals (=1-superstrong; for n-superstrong for n≥2 see further down.)subcompact, strongly compact (Woodin< strongly compact≤supercompact), supercompact cardinalsη-extendible, extendible cardinalsVopěnka cardinalsn-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).

## related categories

GMT+8, 2014-3-16 10:03 , Processed in 0.113863 second(s), 17 queries .