Zero dagger

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In mathematical set theory, 0^† (zero dagger) is defined to be a particular real number satisfying certain conditions. The definition is a bit awkward, because there might be no real number 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 iff 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 $(L,\in,U)$ , and 0^† is defined to be the real number that codes in the canonical way the Gödel numbers of the true formulas about the indiscernibles in L[U].

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, or indeed a cardinal at all.

(Note, the superscript † should be a dagger, but it appears as a plus sign on some browsers.)