Block matrix pseudoinverse

2014-01-04T23:29:36Z

129.10.63.90: /* External links */

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]]

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Block matrix pseudoinverse