# List of large cardinal properties

﻿
List of large cardinal properties

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, ..., $aleph_0, aleph_1$,..., $kappa = aleph_\left\{kappa\right\}$, ... (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 (= Π11-indescribable) Πmn-indescribable , totally indescribable cardinals
* λ-unfoldable, 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"&ge;2 see further down.)
* subcompact, strongly compact (Woodin< strongly compact&le;supercompact), supercompact cardinals
* η-extendible, extendible cardinals
* Vopěnka cardinals
* "n"-superstrong ("n"&ge;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.

References

*
*
*citation|last=Kanamori|first=Akihiro|first2=M. |last2=Magidor
chapter=The evolution of large cardinal axioms in set theory
series= Lecture Notes in Mathematics
publisher =Springer Berlin / Heidelberg
ISSN = 1617-9692
volume =669 ( [http://math.bu.edu/people/aki/e.pdf typescript] )
title=Higher Set Theory
DOI 10.1007/BFb0103096
year=1978
ISBN =978-3-540-08926-1
DOI =10.1007/BFb0103104
pages= 99-275

*

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Large cardinal property — In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very large (for example, bigger than aleph zero …   Wikipedia

• List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

• List of set theory topics — Logic portal Set theory portal …   Wikipedia

• Cardinal number — This article describes cardinal numbers in mathematics. For cardinals in linguistics, see Names of numbers in English. In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of… …   Wikipedia

• List of first-order theories — In mathematical logic, a first order theory is given by a set of axioms in somelanguage. This entry lists some of the more common examples used in model theory and some of their properties. PreliminariesFor every natural mathematical structure… …   Wikipedia

• Large numbers — This article is about large numbers in the sense of numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive… …   Wikipedia

• Cardinal direction — Cardinal point redirects here. For other uses of the term, see Cardinal point (disambiguation). A compass rose showing the four cardinal directions, the four ordinal directions, plus eight further divisions. The four cardinal directions or… …   Wikipedia

• List of abbeys and priories in England — Contents 1 Overview 1.1 Article layout 2 Abbreviations and key …   Wikipedia

• List of important publications in mathematics — One of the oldest surviving fragments of Euclid s Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5. This is a list of important publications in mathematics, organized by field. Some… …   Wikipedia

• List of Metalocalypse characters — The following is a list of characters in the American animated television series Metalocalypse. Survival from near fatal injuries in the show is a common running gag; a character may be burned, maimed or hurt in some manner and still be able to… …   Wikipedia