In set theory, 0<sup>â </sup> (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. 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>â </sup> does not exist" is consistent. ZFC + "0<sup>â </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:
If 0<sup>â </sup> exists, then a careful analysis of the embeddings of into itself reveals that there is a closed unbounded subset of ú, and a closed unbounded proper class of ordinals greater than ú, which together are indiscernible for the structure , and 0<sup>â </sup> is defined to be the set of Gödel numbers of the true formulas about the indiscernibles in .
Solovay showed that the existence of 0<sup>â </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.