In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s.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 definition is a bit awkward, because there might be no set of natural numbers satisfying the conditions.
ZFC + "0† exists" is not known to be inconsistent (and most set theorists believe that it is consistent).
It is usually formulated as follows: 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].