http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Laver_function 2026-05-06T14:47:44Z Revision history for this page on the wiki MediaWiki 1.43.1 http://debianws.lexgopc.com/wiki143/index.php?title=Laver_function&diff=7249042&oldid=prev 2024-07-17T01:17:14Z

<p>Added short description</p> <p><b>New page</b></p><div>{{short description|Mathematical function in set theory}}<br /> In [[set theory]], a &#039;&#039;&#039;Laver function&#039;&#039;&#039; (or &#039;&#039;&#039;Laver diamond&#039;&#039;&#039;, named after its inventor, [[Richard Laver]]) is a function connected with [[supercompact cardinal]]s.<br /> <br /> ==Definition==<br /> If κ is a supercompact cardinal, a Laver function is a function &#039;&#039;ƒ&#039;&#039;:κ&amp;nbsp;→&amp;nbsp;&#039;&#039;V&#039;&#039;&lt;sub&gt;κ&lt;/sub&gt; such that for every set &#039;&#039;x&#039;&#039; and every cardinal λ&amp;nbsp;≥&amp;nbsp;|TC(&#039;&#039;x&#039;&#039;)|&amp;nbsp;+&amp;nbsp;κ there is a supercompact measure &#039;&#039;U&#039;&#039; on [λ]&lt;sup&gt;&lt;κ&lt;/sup&gt; such that if &#039;&#039;j&#039;&#039;&lt;sub&gt;&amp;nbsp;&#039;&#039;U&#039;&#039;&lt;/sub&gt; is the associated elementary embedding then &#039;&#039;j&#039;&#039;&lt;sub&gt;&amp;nbsp;&#039;&#039;U&#039;&#039;&lt;/sub&gt;(&#039;&#039;ƒ&#039;&#039;)(κ) = &#039;&#039;x&#039;&#039;. (Here &#039;&#039;V&#039;&#039;&lt;sub&gt;κ&lt;/sub&gt; denotes the κ-th level of the [[cumulative hierarchy]], TC(&#039;&#039;x&#039;&#039;) is the [[transitive set|transitive closure]] of &#039;&#039;x&#039;&#039;)<br /> <br /> ==Applications==<br /> The original application of Laver functions was the following theorem of Laver. <br /> If κ is supercompact, there is a κ-c.c. [[forcing (mathematics)|forcing]] notion (&#039;&#039;P&#039;&#039;,&amp;nbsp;≤) such after forcing with (&#039;&#039;P&#039;&#039;,&amp;nbsp;≤) the following holds: κ is supercompact and remains supercompact after forcing with any κ-directed closed forcing.<br /> <br /> There are many other applications, for example the proof of the consistency of the [[proper forcing axiom]].<br /> <br /> ==References==<br /> {{refbegin}}<br /> *{{cite journal | zbl=0381.03039 | first=Richard | last=Laver | authorlink=Richard Laver | title=Making the supercompactness of κ indestructible under κ-directed closed forcing | journal=[[Israel Journal of Mathematics]] | volume=29 | year=1978 | issue=4 | pages=385–388 | doi=10.1007/bf02761175 | doi-access=}}<br /> {{refend}}<br /> <br /> [[Category:Set theory]]<br /> [[Category:Large cardinals]]<br /> [[Category:Functions and mappings]]<br /> <br /> <br /> {{settheory-stub}}</div>

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