{{Short description|Subset of the natural numbers in set theory}} {{More footnotes|date=June 2014}} In set theory, '''0<sup>†</sup>''' ('''zero dagger''') is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s.<ref>{{Cite journal | last1=Kanamori | first1=Akihiro | author1-link=Akihiro Kanamori | last2=Awerbuch-Friedlander | first2=Tamara | author2-link = Tamara Awerbuch-Friedlander| title=The compleat 0<sup>†</sup> | doi=10.1002/malq.19900360206 |mr=1068949 | year=1990 | journal=Zeitschrift für Mathematische Logik und Grundlagen der Mathematik | issn=0044-3050 | volume=36 | issue=2 | pages=133–141}}</ref> The definition is a bit awkward, because there might be ''no'' set of natural numbers satisfying the conditions. Specifically, if ZFC is consistent, then ZFC + "0<sup>†</sup> does not exist" is consistent. ZFC + "0<sup>†</sup> exists" is not known to be inconsistent (and most set theorists believe that it is consistent). In other words, it is believed to be independent (see large cardinal for a discussion). It is usually formulated as follows:

:0<sup>†</sup> exists if and only if there exists a non-trivial elementary embedding &nbsp;''j'' : ''L''[''U''] → ''L''[''U''] for the relativized Gödel constructible universe {{tmath|L[U]}}, where ''U'' is an ultrafilter witnessing that some cardinal ''κ'' is measurable.

If 0<sup>†</sup> exists, then a careful analysis of the embeddings of {{tmath|L[U]}} into itself reveals that there is a closed unbounded subset of ''κ'', and a closed unbounded proper class of ordinals greater than ''κ'', which together are ''indiscernible'' for the structure <math>(L,\in,U)</math>, and 0<sup>†</sup> is defined to be the set of Gödel numbers of the true formulas about the indiscernibles in {{tmath|L[U]}}.

Solovay showed that the existence of 0<sup>†</sup> follows from the existence of two measurable cardinals. It is traditionally considered a large cardinal axiom, although it is not a large cardinal, nor indeed a cardinal at all.

== See also ==

*0<sup>#</sup>: a set of formulas (or subset of the integers) defined in a similar fashion, but simpler.

== References == {{Reflist}} * {{cite book|last=Kanamori|first=Akihiro|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|title-link=The Higher Infinite|edition=2nd|isbn=3-540-00384-3}}

== External links == *[http://www.zentralblatt-math.org/zmath/en/search/?q=an:04170902&type=pdf&format=complete Definition by "Zentralblatt math database" (PDF)]

Category:Large cardinals

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