Zero dagger


In set theory, 0† is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s.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† does not exist" is consistent. ZFC + "0† exists" is not known to be inconsistent. In other words, it is believed to be independent. It is usually formulated as follows:
If 0† exists, then a careful analysis of the embeddings of L 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.
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.