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In set theory, **0 ^{†}** (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on some browsers.) The definition is a bit awkward, because there might be

- 0
^{†}exists if and only if there exists a non-trivial elementary embedding*j*:*L[U]*→*L[U]*for the relativized Gödel constructible universe*L[U]*, where*U*is an ultrafilter witnessing that some cardinal κ is measurable.

If 0^{†} exists, then a careful analysis of the embeddings of *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 , and 0^{†} is defined to be the set of Gödel numbers of the true formulas about the indiscernibles in *L[U]*.

Solovay showed that the existence of **0 ^{†}** 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.

- 0
^{#}: a set of formulas (or subset of the integers) defined in a similar fashion, but simpler.

- Kanamori, Akihiro; Awerbuch-Friedlander, Tamara (1990). "The compleat 0
^{†}".*Zeitschrift für Mathematische Logik und Grundlagen der Mathematik*.**36**(2): 133–141. doi:10.1002/malq.19900360206. ISSN 0044-3050. MR 1068949. - Kanamori, Akihiro (2003).
*The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings*(2nd ed.). Springer. ISBN 3-540-00384-3.

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