Zero dagger

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.)

ee also

*zero sharp: a real number defined in a similar fashion, but much simpler.

External links

*Definition by "Zentralblatt math database" PDF file [http://www.zentralblatt-math.org/zmath/en/search/?q=an:04170902&type=pdf&format=complete]

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