0†

In set theory, 0^† is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s.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 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.
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.