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Topic: Large cardinals

Related Topics

Cardinal

Ramsey cardinal

Extendible cardinal

Subtle cardinal

Transfinite number

Shelah cardinal

Cardinal number

Huge cardinal

Large numbers

History of large numbers

69 (number)

Small number

List of publications in mathematics

Grothendieck topology

Georg Cantor

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	Large cardinal - Wikipedia, the free encyclopedia
		In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent.
		Therefore the discussion of large cardinals takes place in a realm of conditional proofs, which (according to the consensus view of logicians) will remain so.
		Erdős cardinals (The existence of the Aleph-1-Erdős cardinal implies the existence of 0#, which implies the consistency of Erdős cardinals for all countable ordinals.)
	en.wikipedia.org /wiki/Large_cardinal (203 words)

	Strong cardinal - Wikipedia, the free encyclopedia
		In mathematical set theory, a strong cardinal or λ-strong cardinal is a type of large cardinal.
		Specifically, a λ-strong cardinal is a cardinal number κ such that exists an elementary embedding j from V into a transitive inner model M with critical point κ and
		It should be noted that the least strong cardinal is larger than the least Woodin, superstrong, etc. cardinals, but that the consistency strength of strong cardinals is lower: For example, if κ is Woodin, then V
	en.wikipedia.org /wiki/Strong_cardinal (150 words)

	Infinite Ink: Cardinal Numbers
		A cardinal that is not a finite cardinal is an infinite cardinal.
		1 is the ordinal successor of a, is the cardinal successor of aleph
		Large cardinals are cardinals whose existence cannot be proved in ZFC.
	ii.best.vwh.net /math/cardinals (1276 words)

	Weakly hyper-Woodin cardinal - Wikipedia, the free encyclopedia
		In axiomatic set theory, weakly hyper-Woodin cardinals are a kind of large cardinals.
		A cardinal κ is called weakly hyper-Woodin iff for every set S there exists a normal measure U on κ such that the set {λ < κ
		The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of U does not depend on the choice of the set S for hyper-Woodin cardinals.
	en.wikipedia.org /wiki/Weakly_hyper-Woodin_cardinal (122 words)

	Cardinal Mobile, garden art , yard art, nature gifts
		The Cardinal is a medium-sized bird, measuring approximately eight and a half inches in length.
		Cardinals love sunflower seeds, and are commonly encouraged to live near houses by people placing sunflower seed feeders in their yards.
		Cardinals are nonmigratory and are permanent residents throughout their range but some movement does occur in the later summer and fall.
	www.flyingmobiles.com /html/mobiles/bd12.htm (575 words)

	USATODAY.com - Pope installs new cardinals, adds to list of papal hopefuls (Site not responding. Last check: 2007-11-05)
		As he watched the new cardinals greet one another, all dressed in scarlet to show their willingness to give their lives for the church, John Paul appeared visibly pleased, and at one point seemed to wipe away a tear.
		Since it may be John Paul's last consistory, the current roster of cardinals is considered to be the list of possible popes, since cardinals largely choose from among themselves for the top job in the Roman Catholic Church.
		He also announced a 31st cardinal whose name was kept secret, or "in pectore." That's a Vatican formula often used when the pope wants to name a cardinal in a country where the church is oppressed.
	www.usatoday.com /news/religion/2003-10-21-pope-cardinals_x.htm (1132 words)

	Set Theory
		Cantor's discovery of uncountable sets led him to the subsequent development of ordinal and cardinal numbers, with their underlying order and arithmetic, as well as to a plethora of fundamental questions that begged to be answered (such as the Continuum Hypothesis).
		Along with the theory of large cardinals it is used to gauge the consistency strength of mathematical statements.
		Large cardinal axioms (also referred to as axioms of strong infinity) form a hierarchy whereby a stronger axiom not only implies a weaker axiom but also proves its consistency.
	plato.stanford.edu /entries/set-theory (3302 words)

	Pope installs 30 new cardinals (Site not responding. Last check: 2007-11-05)
		Cardinals take their positions for a consistory led by Pope John Paul II in Saint Peter's Square at the Vatican, on Tuesday.
		As a result, the current roster of cardinals is considered to be the list of possible popes, since cardinals largely choose from among themselves for the top job in the Roman Catholic Church.
		In fact, the college has gotten so large that cardinals wore tags with their names and home dioceses during meetings at the Vatican last week.
	www.msnbc.com /news/982030.asp?0cv=CB10 (789 words)

	Harvey Friedman colloquium talk (Site not responding. Last check: 2007-11-05)
		Large cardinal axioms have been the main proposal for new axioms, going back to Godel.
		It is a theorem of set theory that ) requires a suitably large cardinal** that goes well beyond the usual axioms of mathematics.
		The main result is that its consequence, ), also requires use of the same suitably large* cardinal despite the fact that *) lives in discrete mathematics.
	math.berkeley.edu /~ribet/Colloquium/friedman.html (422 words)

	The Future of Set Theory by S.Shelah
		To a large extent I was attracted first to mathematics and subsequently to mathematical logic by their generality, anticipating that this is the normal attitude; it seems I was mistaken.
		So the existence of a large cardinal is a very natural statement (and an interesting one) and theorems on large cardinals are very interesting as implications, not as theorems (whereas proving you can use less than ZFC does not seem to me very interesting).
		It is to a large extent the duty of new generation to solve the problems of the older one.
	shelah.logic.at /E16/E16.html (4649 words)

	Cardinals Not Named Pujols, Womack, or Edmonds: The Genus Cheilodipterus by Henry C. Schultz III - Reefkeeping.com
		In addition to their low-key persona, and if their large eyes didn't already tell the story, the vast majority of Cardinalfish are nocturnal.
		In the case of the Cardinalfish it is both the association with, and similar color pattern to, the poisonous, fang-toothed blenny that gives the cardinal a hall pass on the reef crests.
		The many benefits these Cardinals have in their favor, however, may tempt many aquarists to set up an aquarium with its overall design shaped around these fish.
	www.reefkeeping.com /issues/2004-10/hcs3/index.php (2794 words)

	BioKIDS: Northern cardinal (Cardinalis cardinalis) : Information (Site not responding. Last check: 2007-11-05)
		Northern cardinals are less selective during winter in the colder climates.
		Because Northern Cardinals eat large quantities of seeds and fruits they may sometimes act to disperse seeds, they may also influence the plant community composition through eating seeds.
		Young Northern Cardinals remain near their parents until they are from 25 to 56 days old, they are still occasionally fed by their parents during this time.
	www.biokids.umich.edu /critters/information/Cardinalis_cardinalis (662 words)

	MTH- : (Site not responding. Last check: 2007-11-05)
		Among the topics included are the axioms of set theory, ordinals and cardinals, measure and category on the real line, infinitary combinatorics, large cardinals, and some independence results.
		Infinite combinatorics naturally leads to the development of large cardinals, which will be our next topic.
		Toward the end of the course we shall discuss the idea of independence, as well as its connection with large cardinals.
	www.mth.uea.ac.uk /maths/syllabuses/0001/4E1700.html (309 words)

	[No title]
		Working up the large cardinal hierarchy this is the pattern for a long time, and it is easy to expect it to always happen.
		ZFC + there exists a huge cardinal + there exists a supercompact cardinal - the least huge cardinal < the least supercompact cardinal So, in a model with both supercompacts and huges, letting nu be the least huge, we have exactly that situtation, that all the V_nu - "supercompacts" below nu were fake supercompacts.
		Ie the V_rho2 's are smart enought to properly identify the small large cardinal properties.
	www.math.niu.edu /~rusin/known-math/00_incoming/inaccessibles (932 words)

	Arizona Cardinals Licensed Logo Fan Merchandise Store (Site not responding. Last check: 2007-11-05)
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	CNN.com - Pope names new cardinals - Oct. 21, 2003
		CNN's Jim Bittermann says the views of some cardinals recently named by Pope John Paul II are not necessarily those of the pope himself.
		His health failing, Pope John Paul II added 31 names to the list of his possible successors, installing a diverse collection of cardinals in a consistory some say may be his last.
		In a ritual-filled ceremony on Tuesday at the steps of St. Peter's Basilica, the 83-year-old pope was giving the new "princes of the church" their red hats and declaring them members of the elite band of churchmen who will elect the next pope.
	cnn.com /2003/WORLD/europe/10/21/pope.anniversary.ap (855 words)

	Amazon.com: Books: Set Theory (Perspectives in Mathematical Logic) (Site not responding. Last check: 2007-11-05)
		This introduction to modern set theory covers all aspects of its two main general areas: classical set theory including large cardinals, infinitary combinatorics, desriptive set theory, and independence proofs starting with Goedel's proof around 1938 followed by Cohen's proof in 1963, whereby Cohen's method of forcing probably had a greater influence on mathematics.
		The author's primary emphasis is on forcing and large cardinals (on which he has collected an enormous amount of material which had previously been available only through scattered journal articles or private communication) but there is a substantial discussion of descriptive set theory and infinitary combinatorics as well.
		Nonetheless, Jech's wide variety of forcing applications, his in-depth treatment of large cardinals, and his compact surveys of saturated ideals and descriptive set theory make his work really an outstanding contribution.
	www.amazon.com /exec/obidos/tg/detail/-/3540630481?v=glance (1357 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Work address: UCLA Electronic mail: ketchers@math.ucla.edu Abstract: Ideals on small cardinals are known to exist in forcing extensions where some large cardinal is collapsed.
		Conversely, the ideals themselves imply the existence of inner models for various large cardinals.
		Unfortunately the gap between the large cardinals required to get the ideals and the large cardinals that the ideals give rise to is enormous.
	www.unlv.edu /math/asl/abstracts/ketchersid.txt (85 words)

	JS Online: A little small-ball keeps Cardinals living large
		The Cardinals combined eight singles, two doubles and one triple, and the closest they came to a home run was a drive to the wall by winning pitcher Dan Haren.
		Less prominent players in St. Louis' high-powered lineup were largely responsible with the 6-7-8 slots in the order going a combined 8 for 10 with five RBI and five runs scored.
		Matheny, the No. 8 hitter, became the first Cardinals player to get four RBI in a division series game with two-run singles in the fifth and seventh.
	www.jsonline.com /sports/brew/oct04/264992.asp (790 words)

	Cornell Math - Math 787, Spring 2003 (Site not responding. Last check: 2007-11-05)
		It will be taught at an introductory level, not assuming any previous knowledge of the subject.
		Topics in this course include the axioms of Zermelo-Frankel (ZF) set theory; ordinals and cardinals; Von Neumann rank and reflection principles; absoluteness; inner models; infinitary combinatorics and Martin's Axiom; Suslin's Hypothesis; Godel's universe of constructible sets (L); and basic Cohen forcing.
		If time permits, we may also discuss iterated forcing and/or results on large cardinals.
	www.math.cornell.edu /Courses/GradCourses/SP03/787.html (214 words)

	Arthur W. Apter (Site not responding. Last check: 2007-11-05)
		Hamkins) "Indestructible Weakly Compact Cardinals and the Necessity of Supercompactness for Certain Proof Schemata", Mathematical Logic Quarterly 47, 2001, 563-571.
		Sargsyan) "Can A Large Cardinal Be Forced From A Condition Implying Its Negation?", Proceedings of the American Mathematical Society 133, 2005, 3103-3108.
		Koepke) "The Consistency Strength of $\aleph_\omega$ and $\aleph_{\omega_1}$ being Rowbottom Cardinals without the Axiom of Choice", submitted for publication to the Journal of Symbolic Logic.
	faculty.baruch.cuny.edu /apter (1418 words)

	Math 180 (Site not responding. Last check: 2007-11-05)
		Large Cardinals: One way to extend the axioms of ZFC is to claim the existence of large cardinals that can not be proved to exist in ZFC.
		We will look at measurable cardinal and examine their effect on the real numbers and the constructible universe.
		Jech is the best reference for large cardinals and determinacy.
	www.math.uic.edu /~marker/512-s99.html (203 words)

	Logic and Semantics Seminar - Monday, 27th October, 1997: Peter Dybjer
		This talk is based on joint work (in progress) with Anton Setzer, Uppsala, on the syntax and semantics of large sets in Martin-Lof type theory.
		Then I show that when we generalize this to strong functional programming with dependent types, we are naturally led into the world of analogues of large cardinals.
		I will say something about the relationship between such "large" recursive data types and some notions of large cardinals in set theory (inaccessible cardinals, hyperinaccessible cardinals, and Mahlo cardinals).
	www.cl.cam.ac.uk /Research/LS/Talks/1997_98/97_10_27.Abstract.html (188 words)

	A Dilemma in the Philosophy of Set Theory, Ralf-Dieter Schindler
		More generally, we indicate the ontological commitments of any theory that exploits reflection principles in order to yield large cardinals.
		The disappointing conclusion will be that our only apparently good arguments for the existence of large cardinals have bad presuppositions.
		[6] Reinhardt, W., ``Reflection Principles, Large Cardinals, and Elementary Embeddings," Proceedings of the Symposium on Pure Mathematics, vol.
	projecteuclid.org /getRecord?id=euclid.ndjfl/1040511351 (210 words)

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	The Infography about Large Cardinals (Mathematics)
		The following sources are recommended by a professor whose research specialty is large cardinals in mathematics.
		Zeman, M. (2002) Inner Models and Large Cardinals.
		Kanamori A., and Magidor M. (1978) The Evolution of Large Cardinal Axioms in Set Theory.
	www.infography.com /content/938266789511.html (82 words)

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