en>Joule36e5: Use stronger statement: not just irrational, but transcendental.

2013-08-29T10:04:38Z

Use stronger statement: not just irrational, but transcendental.

← Older revision		Revision as of 12:04, 29 August 2013
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	~~The writer~~'~~s title~~ is ~~Andera and she thinks it seems fairly great~~. ~~Office supervising~~ is ~~exactly where her primary earnings comes from but she~~'~~s already applied~~ for ~~another 1~~. ~~My wife~~ and ~~I reside~~ in ~~Kentucky~~. ~~The favorite pastime for him and his children~~ is to ~~play lacross and he would by no means give it up~~.<br><br>~~Feel free~~ to ~~visit my web page~~ ... [~~http~~:~~//netwk.hannam.ac~~.kr/~~xe/data_2/85669 psychic phone~~]		In [[set theory]], a '''Jónsson cardinal''' (named after [[Bjarni Jónsson]]) is a certain kind of [[large cardinal]] number.

			An [[uncountable set\|uncountable]] [[cardinal number]] κ is said to be '''''Jónsson''''' if for every function ''f'': [κ]<sup><ω</sup> → κ there is a set ''H'' of order type κ such that for each ''n'', ''f'' restricted to ''n''-element subsets of ''H'' omits at least one value in κ.

			Every [[Rowbottom cardinal]] is Jónsson. By a theorem of Eugene M. Kleinberg, the theories ZFC + “there is a [[Rowbottom cardinal]]” and ZFC + “there is a Jónsson cardinal” are equiconsistent. [[William Mitchell (mathematician)\|William Mitchell]] proved, with the help of the Dodd-Jensen [[core model]] that the consistency of the existence of a Jónsson cardinal implies the consistency of the existence of a [[Ramsey cardinal]].

			In general, Jónsson cardinals need not be large cardinals in the usual sense: they can be [[singular cardinal\|singular]]. But the existence of a singular Jónsson cardinal is equiconsistent to the existence of a [[measurable cardinal]]. Using the [[axiom of choice]], a lot of small cardinals (the <math>\aleph_n</math>, for instance) can be proved to be not Jónsson. Results like this need the axiom of choice, however: The [[axiom of determinacy]] does imply that for every positive natural number ''n'', the cardinal <math>\aleph_n</math> is Jónsson.

			A '''Jónsson algebra''' is an algebra with no proper subalgebras of the same cardinality. (Here an algebra means
			a model for a language with a countable number of function symbols, in other words a set with a countable number of functions from finite products of the set to itself.) A cardinal is a Jónsson cardinal if and only if there are no Jónsson algebras of that cardinality. The existence of [[Jónsson function]]s shows that if algebras are allowed to have infinitary operations, then there are no analogues of Jónsson cardinals.

			==References==

			* {{cite book\|last=Kanamori\|first=Akihiro\|year=2003\|authorlink=Akihiro Kanamori\|publisher=Springer\|title=The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings\|edition=2nd ed\|isbn=3-540-00384-3}}
			*{{Citation \| last1=Jónsson \| first1=Bjarni \| title=Topics in universal algebra \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Mathematics \| doi=10.1007/BFb0058648 \| id={{MathSciNet \| id = 0345895}} \| year=1972 \| volume=250}}

			{{DEFAULTSORT:Jonsson Cardinal}}
			[[Category:Large cardinals]]


			{{settheory-stub}}

en>Luckas-bot: robot Adding: uk:Експоненційний факторіал

2010-07-16T19:14:39Z

robot Adding: uk:Експоненційний факторіал

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The writer's title is Andera and she thinks it seems fairly great. Office supervising is exactly where her primary earnings comes from but she's already applied for another 1. My wife and I reside in Kentucky. The favorite pastime for him and his children is to play lacross and he would by no means give it up.<br><br>Feel free to visit my web page ... [http://netwk.hannam.ac.kr/xe/data_2/85669 psychic phone]

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en>Joule36e5: Use stronger statement: not just irrational, but transcendental.

en>Luckas-bot: robot Adding: uk:Експоненційний факторіал