Block matrix pseudoinverse - Revision history

98.230.192.179 at 06:35, 8 January 2015

2015-01-08T06:35:11Z

← Older revision		Revision as of 08:35, 8 January 2015
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	~~In [[set theory]], '''Easton's theorem'''~~ is ~~a result on~~ the ~~possible [[cardinal number]]s of [[powerset]]s. {{harvtxt\|Easton\|1970}} (extending a result~~ of ~~[[Robert M~~. ~~Solovay]]) showed via [[forcing (mathematics)\|forcing]] that~~		The author is known by the title of Figures Wunder. Years in the past we moved to Puerto Rico and my family members enjoys it. To collect cash is a factor that I'm completely addicted to. For years he's been operating as a meter reader and it's something he really appreciate.<br><br>my page ... at home std testing ([http://Www.blaze16.com/blog/257588 similar internet site])

	~~: <math> \kappa < \operatorname{cf}(2^\kappa)\,</math>~~

	~~and, for <math> \kappa < \lambda\,</math>, that~~

	~~:<math> 2^\kappa\le 2^\lambda\,</math>~~

	~~are the only constraints on permissible values for 2<sup>κ</sup> when κ is a [[regular cardinal]].~~

	~~== Statement of the theorem ==~~

	~~'''Easton's theorem''' states that if ''G'' is a class function whose domain consists of [[ordinal number\|ordinals]] and whose range consists of ordinals such that~~
	~~# ''G'' is non-decreasing,~~
	~~# the [[cofinality]] of <math>\aleph_{G(\alpha)}</math> is greater than <math>\aleph_{\alpha}</math> for each α~~ in the ~~domain of G,~~ and
	~~# <math>\aleph_{\alpha}</math> is regular for each α in the domain of G,~~

	~~then there~~ is a ~~model of ZFC such~~ that

	~~:<math>2^{\aleph_{\alpha}} = \aleph_{G(\alpha)}\,</math>~~

	~~for each <math>\alpha</math> in the domain of ''G'~~'.

	~~The proof of Easton~~'s ~~theorem uses [[forcing (mathematics)\|forcing]] with~~ a ~~[[proper class]] of forcing conditions.~~

	~~All conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, while condition 2 follows from [[König's theorem (set theory)\|König~~'s ~~theorem]]~~.

	~~== No extension to singular cardinals ==~~

	{{harvtxt\|Silver\|1975}} proved that a singular cardinal of uncountable cofinality cannot be the smallest cardinal for which the [[generalized continuum hypothesis]] fails. This shows that Easton's theorem cannot be extended to the class of all cardinals.
	~~The program of [[PCF theory]] gives results on the possible values of~~
	<~~math~~>~~2^\lambda~~<~~/math> for [[singular cardinal]]s <math>\lambda</math~~>. PCF theory shows that the values of the [[continuum function]] on singular cardinals are strongly influenced by the values on smaller cardinals, whereas Easton's theorem shows that the values of the continuum function on [[regular cardinal]]s are only weakly influenced by the values on smaller cardinals.

	~~== See also ==~~

	* [~~[Singular cardinal hypothesis]]~~
	* [[Aleph number]]
	* [[Beth number]]

	~~== References ==~~
	*{{citation\|authorlink=William Bigelow Easton\|first= W.~~\|last= Easton\|title= Powers of regular cardinals\|journal=Ann~~. ~~Math. Logic\|volume=1\|year=1970\|issue=2\|pages= 139–178~~
	~~\|doi=10.1016~~/~~0003-4843(70)90012-4 }}~~
	*{{citation\|mr=0429564\|authorlink=Jack Silver\|last= Silver\|first= Jack\|chapter=On the singular cardinals problem\|title= Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974)\|volume= 1\|pages= 265–268\|publisher= Canad. Math. Congress\|publication-place= Montreal, Que.\|year= 1975}}

	~~[[Category:Set theory]~~]
	~~[[Category:Theorems in the foundations of mathematics]]~~
	~~[[Category:Cardinal numbers]]~~
	~~[[Category:Forcing (mathematics~~)]]
	~~[[Category:Independence results]]~~

129.10.63.90: /* External links */

2014-01-04T23:29:36Z

External links

← Older revision		Revision as of 01:29, 5 January 2014
Line 1:		Line 1:
	~~Nice to satisfy you~~, ~~my name~~ is ~~Refugia~~. ~~Playing baseball~~ is the ~~pastime he will by no means stop performing. Her family members life~~ in ~~Minnesota. Managing individuals~~ is ~~his occupation.~~<br><br>~~My web page~~ ... ~~at home std test~~ ([~~http:~~//~~www~~.~~ddaybeauty~~.~~com~~/~~node/14766 Go At this site~~])		In [[set theory]], '''Easton's theorem''' is a result on the possible [[cardinal number]]s of [[powerset]]s. {{harvtxt\|Easton\|1970}} (extending a result of [[Robert M. Solovay]]) showed via [[forcing (mathematics)\|forcing]] that

			: <math> \kappa < \operatorname{cf}(2^\kappa)\,</math>

			and, for <math> \kappa < \lambda\,</math>, that

			:<math> 2^\kappa\le 2^\lambda\,</math>

			are the only constraints on permissible values for 2<sup>κ</sup> when κ is a [[regular cardinal]].

			== Statement of the theorem ==

			'''Easton's theorem''' states that if ''G'' is a class function whose domain consists of [[ordinal number\|ordinals]] and whose range consists of ordinals such that
			# ''G'' is non-decreasing,
			# the [[cofinality]] of <math>\aleph_{G(\alpha)}</math> is greater than <math>\aleph_{\alpha}</math> for each α in the domain of G, and
			# <math>\aleph_{\alpha}</math> is regular for each α in the domain of G,

			then there is a model of ZFC such that

			:<math>2^{\aleph_{\alpha}} = \aleph_{G(\alpha)}\,</math>

			for each <math>\alpha</math> in the domain of ''G''.

			The proof of Easton's theorem uses [[forcing (mathematics)\|forcing]] with a [[proper class]] of forcing conditions.

			All conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, while condition 2 follows from [[König's theorem (set theory)\|König's theorem]].

			== No extension to singular cardinals ==

			{{harvtxt\|Silver\|1975}} proved that a singular cardinal of uncountable cofinality cannot be the smallest cardinal for which the [[generalized continuum hypothesis]] fails. This shows that Easton's theorem cannot be extended to the class of all cardinals.
			The program of [[PCF theory]] gives results on the possible values of
			<math>2^\lambda</math> for [[singular cardinal]]s <math>\lambda</math>. PCF theory shows that the values of the [[continuum function]] on singular cardinals are strongly influenced by the values on smaller cardinals, whereas Easton's theorem shows that the values of the continuum function on [[regular cardinal]]s are only weakly influenced by the values on smaller cardinals.

			== See also ==

			* [[Singular cardinal hypothesis]]
			* [[Aleph number]]
			* [[Beth number]]

			== References ==
			*{{citation\|authorlink=William Bigelow Easton\|first= W.\|last= Easton\|title= Powers of regular cardinals\|journal=Ann. Math. Logic\|volume=1\|year=1970\|issue=2\|pages= 139–178
			\|doi=10.1016/0003-4843(70)90012-4 }}
			*{{citation\|mr=0429564\|authorlink=Jack Silver\|last= Silver\|first= Jack\|chapter=On the singular cardinals problem\|title= Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974)\|volume= 1\|pages= 265–268\|publisher= Canad. Math. Congress\|publication-place= Montreal, Que.\|year= 1975}}

			[[Category:Set theory]]
			[[Category:Theorems in the foundations of mathematics]]
			[[Category:Cardinal numbers]]
			[[Category:Forcing (mathematics)]]
			[[Category:Independence results]]

en>FoCuSandLeArN: /* Notes and references */

2012-06-17T22:35:21Z

Notes and references

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