In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on some browsers.) 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 (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 if and only if there exists a non-trivial elementary embedding j : L → L for the relativized Gödel constructible universe L, where U is an ultrafilter witnessing that some cardinal κ is measurable.
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.
Famous quotes containing the word dagger:
“Is this a dagger which I see before me,
The handle toward my hand? Come, let me clutch thee.
I have thee not, and yet I see thee still.
Art thou not, fatal vision, sensible
To feeling as to sight? Or art thou but
A dagger of the mind, a false creation
Proceeding from the heat-oppressèd brain?”
—William Shakespeare (1564–1616)