In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s.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 no set of natural numbers satisfying the conditions. Specifically, if ZFC is consistent, then ZFC + "0† does not exist" is consistent. ZFC + "0† 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† 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.