set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

, 0^† (zero dagger) is a particular subset of the natural numbers, first defined by

Robert M. Solovay Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory. Biography Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on '' ...

in unpublished work in the 1960s. (The superscript † should be a

dagger A dagger is a fighting knife with a very sharp point and usually two sharp edges, typically designed or capable of being used as a thrusting or stabbing weapon.State v. Martin, 633 S.W.2d 80 (Mo. 1982): This is the dictionary or popular-use de ...

, 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 In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

, 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 In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least � ...

for a discussion). It is usually formulated as follows: :0^† exists

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

there exists a non-trivial

elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often ...

''j'' : ''L ' → ''L ' for the relativized

Gödel constructible universe In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It ...

''L ', where ''U'' is an

ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...

witnessing that some cardinal κ is

measurable In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...

. 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 In mathematical logic, indiscernibles are objects that cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered. Examples If ''a'', ''b'', and ''c'' are distinct and is a set ...

'' for the structure

(L,\in,U)

, 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 In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...

, although it is not a large cardinal, nor indeed a cardinal at all.

References

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External links

Definition by "Zentralblatt math database" (PDF)
Large cardinals {{mathlogic-stub

See also

References

External links