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Topic: Strongly inaccessible cardinal

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Large cardinals

Weakly inaccessible cardinal

	NationMaster - Encyclopedia: Strongly inaccessible cardinal
		In mathematics, a strongly inaccessible cardinal is an uncountable cardinal number κ that is regular and a strong limit cardinal.
		Assuming that ZFC is consistent, the existence of strongly inaccessible cardinals provably cannot be proved in ZFC.
		Strongly inaccessible cardinals are therefore a type of large cardinal.
	www.nationmaster.com /encyclopedia/Strongly-inaccessible-cardinal (522 words)

	tScholars.com \| Inaccessible cardinal (Site not responding. Last check: 2007-09-10)
		In set theory, a cardinal number is called weakly inaccessible if it is an uncountable regular weak limit cardinal and strongly inaccessible, or just inaccessible, if it is an uncountable regular strong limit cardinal.
		The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.
		Thus this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom).
	www.tscholars.com /encyclopedia/Strongly_inaccessible_cardinal (822 words)

	Inaccessible cardinal - (Site not responding. Last check: 2007-09-10)
		Every transfinite cardinal number, except for aleph-null which meets those two conditions (but is not weakly inaccessible because it is countable), is either regular or a limit; however, only a rather large cardinal number can be both.
		Assuming that ZFC is consistent, the existence of (strongly or weakly) inaccessible cardinals provably cannot be proved in ZFC; inaccessible cardinals are therefore a type of large cardinal.
		In fact, ZFC cannot even prove that the existence of inaccessible cardinals is consistent with ZFC (because ZFC+"there exists an inaccessible cardinal" proves the consistency of ZFC); however, the assumption that there is no inaccessible cardinal is provably consistent with ZFC (assuming the consistency of ZFC).
	www.gurgaongrid.com /mediawiki/index.php/Strongly_inaccessible_cardinal (273 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		Let's be careful: there's "strongly inaccessible" and "weakly inaccessible", and the default meaning of "inaccessible" is "strongly inaccessible".
		A weakly inaccessible cardinal is a regular limit cardinal.
		A (strongly) inaccessible cardinal is a regular strong limit cardinal; m is a strong limit cardinal if n < m implies 2^n < m.
	www.math.niu.edu /~rusin/known-math/00_incoming/GCH (283 words)

	Inaccessible cardinal (Site not responding. Last check: 2007-09-10)
		Such a cardinal is called a regular cardinal.
		Every transfinite cardinal number is either regular or a limit; however, only a rather large cardinal number can be both.
		In fact, assuming that ZFC is consistent, the existence of inaccessible cardinals provably cannot be proven in ZFC.
	pedia.newsfilter.co.uk /wikipedia/i/in/inaccessible_cardinal.html (109 words)

	Limit cardinal
		With the cardinal successor operation defined, we can define a limit cardinal in analogy to that for limit ordinals: λ is a (weak) limit cardinal if λ is not a successor cardinal, i.e.
		is a successor cardinal if and only if α is a successor ordinal, hence also a limit cardinal if and only if α is a limit ordinal.
		The preceding defines a notion of "inaccessibility": we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above.
	www.xasa.com /wiki/en/wikipedia/l/li/limit_cardinal.html (454 words)

	CATHOLIC ENCYCLOPEDIA: Council of Constance
		In the meantime the treachery and violence of Ladislaus of Naples made John XXIII quite dependent politically on the new Emperor-elect Sigismund whose anxiety for a general council on German territory was finally satisfied by the pope, then an exile from Rome.
		The famous Dominican Cardinal John of Ragusa (Johannes Dominici), friend and adviser of Gregory XII, and since 19 Dec., 1414, the pope's representative at Constance, convoked anew the council in the pope's name and authorized its future acts.
		The highest figures reached were: 29 cardinals, 3 patriarchs, 33 archbishops, 150 bishops, 100 abbots, 50 provosts, 300 doctors (mostly of theology).
	www.newadvent.org /cathen/04288a.htm (4993 words)

	Infinite Ink: Cardinal Numbers
		A cardinal that is not a finite cardinal is an infinite cardinal.
		In ZF it is possible to have a Dedekind-finite cardinal that is not a finite cardinal.
		1 is the ordinal successor of a, is the cardinal successor of aleph
	www.ii.com /math/cardinals (1276 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		If one can prove that the existence of a weakly inaccessible cardinal is consistent with ZFC (arguing in ZFC), then we can prove in ZFC, by G"odel's Completeness theorem, that there is a model for ZFC+there exists a weakly inaccessible cardinal.
		Then within this model, construct L. L satisfies GCH and the weakly inaccessible cardinal is still weakly inaccessible in L, but in the presence of GCH weakly inaccessible and strongly inaccessible cardinals are the same.
		Now this model has a strongly inaccessible cardinal, but the existence of a strongly inaccessible cardinal implies the consistency of ZFC.
	www.math.niu.edu /~rusin/known-math/98/inaccessible (510 words)

	Strongly inaccessible cardinal: Definition and Links by Encyclopedian.com
		is called strongly inaccessible iff the following conditions hold:
		κ is weakly inaccessible; that is, cf(κ) = κ.
		Post a link to definition / meaning of " Strongly inaccessible cardinal " on your site.
	www.encyclopedian.com /st/Strongly-inaccessible-cardinal.html (99 words)

	Online Encyclopedia and Dictionary - Grothendieck universe
		In mathematics, if κ is a strongly inaccessible cardinal, the corresponding Grothendieck universe is the set of all sets with rank less than κ.
		Since it cannot be proven in ZFC that inaccessible cardinals exist, using a Grothendieck universe in the place of the class of all sets suffices for normal mathematical purposes, rendering proper classes unnecessary.
		Grothendieck universes can be characterized by certain axiomatic properties, and it is possible to prove that any set satisfying such a property has cardinality equal to a strongly inaccessible cardinal.
	www.fact-archive.com /encyclopedia/Grothendieck_universe (180 words)

	[No title]
		The axiom of inaccessible cardinals was first suggested by Ernst Zermelo in 1930, formulated by Alfred Tarski in 1938, and defended by Kurt Gödel in 1947.
		An uncountable cardinal l is strongly inaccessible if and only if (i) for all cardinals a < l, l is not the sum of a cardinals less than l, or (ii) if a is any set with cardinality less than l, then the cardinality of P(a) is also less than l.
		Postulating the axiom of inaccessible cardinals to ZF is quite analogous to adding the axiom of infinity to ZF.
	www.sunysb.edu /philosophy/faculty/gmar/cantor.txt (6245 words)

	PlanetMath: inaccessible cardinals
		is a strong limit cardinal if for any
		is called weakly inaccessible, and a regular strong limit cardinal is called inaccessible.
		This is version 3 of inaccessible cardinals, born on 2002-07-27, modified 2004-09-01.
	planetmath.org /encyclopedia/InaccessibleCardinals.html (51 words)

	Antimeta: Large Cardinals and their Justifications
		It might also be equivalent to some already-known large cardinal axiom (my friend Adam Booth suggested to me that it sounds like it could be a consequence of the existence of an inaccessible that is the limit of a set of measurables).
		Anyway, all this is just one means of justifying large cardinal axioms, but it seems to make sense to me. It also has the added benefit of not requiring a platonist view of mathematics, but works also on a fictionalist view, and probably a variety of other views as well.
		I suppose any such a has to be a strong limit cardinal, because if it wasn't, then there would be something with rank less than a whose powerset would have actual cardinality greater than a, and this submodel could only assign it a larger cardinality, not a smaller one.
	www.ocf.berkeley.edu /~easwaran/blog/2005/12/large_cardinals_and_their_just.html (5467 words)

	Large Cardinals
		In the mathematical theory of the infinite, classes of cardinals which are very remote have been considered, with names such as inaccessible cardinals, strongly inaccessible cardinals, Mahlo cardinals, hyper-Mahlo cardinals, and so on.
		A few simple axioms suffice to define a set the number of whose elements must be a strongly inaccessible cardinal, unless it has either no elements or aleph-null elements.
		But it appears that the fourth axiom will not allow that, as it vastly increases the size of the universe; and, indeed, aleph-omega is too small to be a strongly inaccessible cardinal.
	www.quadibloc.com /math/inf02.htm (363 words)

	Atlas: Universal Sets, Tarski Sets, and Inaccessible von Neumann Sets by Valeri Zakharov
		The crises arisen in naive set theory in the beginning of the 20th century brought to the origin of some strict axiomatic theories.
		Developing the ideas of Tarski, they proposed to strengthen the theory ZF by additional axioms on existence of strongly inaccessible cardinals, because it was already known then that the von Neumann sets V
		f[X] For axiomatic constructing of strongly inaccessible cardinal numbers Tarski introduced into the theory ZF the Tarski axiom, which postulates that every set is an element of some Tarski set.
	atlas-conferences.com /cgi-bin/abstract/cajy-02 (733 words)

	[No title]
		Then there is an inaccessible cardinal gamma and a model N of NFUB such that the strongly Cantorian sets of this model are isomorphic to V_gamma.
		The map which assigns to an inaccessible cardinal which is not n+1-Mahlo a closed cofinal subset which is disjoint from the n-Mahlo cardinals.
		A cardinal is 0-Mahlo iff it is inaccessible.
	math.boisestate.edu /~holmes/holmes/solovay.txt (16517 words)

	Large cardinal Other types of large cardinals axiomatic set theory ZFC strongly inaccessible cardinal Mahlo cardinal ... (Site not responding. Last check: 2007-09-10)
		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.
		The following is a list of some types of large cardinals; it is arranged in order of the consistency strength.
		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 all listed cardinal descriptions φ of lesser consistency strength, V(κ) satisfies "there are unboundedly many cardinals satisfying φ".
	en.powerwissen.com /08WOxwzBQRFCI7XcNuuGfg%3D%3D_Large_cardinal.html (279 words)

	Infinite Ink: The Continuum Hypothesis by Nancy McGough
		This "Trichotomy of Cardinals" tells us that any two cardinals are comparable and all cardinals can be lined up in one (very!) long well-ordered sequence, called the "aleph sequence." Trichotomy of Cardinals is actually equivalent to the Axiom of Choice.
		inaccessible (this follows immediately from the definition of strongly inaccessible).
		Examples of large cardinals are inaccessible cardinals, hyperinaccessible cardinals, Mahlo cardinals, measurable cardinals, and supercompact cardinals.
	www.ii.com /math/ch (4563 words)

	Infinitary Logic (Stanford Encyclopedia of Philosophy)
		Given a pair κ, λ of infinite cardinals such that λ ≤ κ, we define a class of infinitary languages in each of which we may form conjunctions and disjunctions of sets of formulas of cardinality < κ, and quantifications over sequences of variables of length < λ.
		It was quickly realized that a measurable cardinal must be inaccessible, but the falsity of the converse was not established until the 1960s when Tarski showed that measurable cardinals are weakly compact and his student Hanf showed that the first, second, etc. inaccessibles are not weakly compact (cf.
		Although the conclusion that measurable cardinals must be monstrously large is now normally proved without making the detour through weak compactness and infinitary languages, the fact remains that these ideas were used to establish the result in the first instance.
	plato.stanford.edu /entries/logic-infinitary (6676 words)

	Logic Colloquium Archive
		In a core model induction argument, one produces canonical inner models which are correct for statements at a given level of complexity, using core model theory together with the existence of models which are correct at lower levels.
		The general theory is concerned with the relationship between, on the one hand the form of an inductive operator, and on the other hand the form of the set and closure ordinal defined by that operator.
		This study naturally leads to a variety of statements equivalent to the 1-consistency of Mahlo cardinals of finite order, even in concrete settings in the integers which are subject to known quantifier elimination.
	www.math.ucla.edu /~hbe/logic-old.html (10494 words)

	[No title]
		w+w Transfinite cardinals An ordinal a is identified with the set { b : b
		Cardinality card S is the least ordinal a such that card a = card S. aleph_0 = w aleph_1 is the first uncountable cardinal.
		strongly inaccessible = inaccessible cardinals inaccessible Mahlo indescribable ineffable partition Ramsey measurable strongly compact supercompact extendible 0 1 2 w aleph_1 0 w aleph_1 theta rho 0 k lambda The first measurable cardinal is k.
	www.chez.com /log/text/logic/infini.txt (262 words)

	Strongly inaccessible cardinal (Site not responding. Last check: 2007-09-10)
		In mathematics, a cardinal number κ > ‭א‬
		κ is a strong limit cardinal, that is, 2
		Under the Generalized Continuum Hypothesis, a cardinal is strongly inaccessible iff it is weakly inaccessible.
	www.fact-index.com /s/st/strongly_inaccessible_cardinal.html (98 words)

	AskPhilosophers.org
		Such people would hold that the statement that there is a strongly inaccessible cardinal shouldn't be accepted as mathematically correct.
		If you are interested in determining how many things of some particular kind there are, then the appropriate numbers to use are the cardinal numbers, and as Richard has explained, there are indeed infinite cardinal numbers.
		On the other hand, if you're a student in a calculus class, then the numbers you are using are probably the real numbers, and all of the real numbers are finite.
	www.amherst.edu /askphilosophers/topic/Mathematics&page=2 (4650 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		Reasonably large cardinals do not exist in the constructible
		> inaccessible cardinal (I believe) is still consistent with ZF+(V=L).
		> And of course strongly inaccessible = weakly inaccessible in this
	www.mathforum.com /kb/plaintext.jspa?messageID=4937650 (137 words)

	Publications of Paul Erdos
		-measure for the first uncountable inaccessible cardinal (In English)
		is strongly inaccessible and does not have property
		is an enumeration of all strongly inaccessible cardinals, then (i) and (iv) imply that, if
	www.zblmath.fiz-karlsruhe.de /MATH/text/general/general/erdos/cit/13401602.htm (124 words)

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