- Zero dagger
In mathematical
set theory , 0† (zero dagger) is defined to be a particularreal number satisfying certain conditions. The definition is a bit awkward, because there might be "no" real number satisfying the conditions. Specifically, ifZFC 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 (seelarge cardinal for a discussion). It is usually formulated as follows::0† exists
iff there exists a non-trivialelementary embedding "j" : "L [U] " → "L [U] " for the relativizedGödel constructible universe "L [U] ", where "U" is anultrafilter 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 theGödel number s 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
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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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