In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the natural numbers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as Silver (1971), where it was denoted by Σ, and rediscovered by Solovay (1967, p.52), who considered it as a subset of the natural numbers and introduced the notation O# (with a capital letter O; this later changed to the numeral '0').

Roughly speaking, if 0# exists then the universe V of sets is much larger than the universe L of constructible sets, while if it does not exist then the universe of all sets is closely approximated by the constructible sets.

Definition

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Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols  ,  , ... for each nonzero natural number. Then   is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with   interpreted as the uncountable cardinal  . (Here   means   in the full universe, not the constructible universe.)

There is a subtlety about this definition: by Tarski's undefinability theorem it is not, in general, possible to define the truth of a formula of set theory in the language of set theory. To solve this, Silver and Solovay assumed the existence of a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible to define the truth of statements about the constructible universe. More generally, the definition of   works provided that there is an uncountable set of indiscernibles for some  , and the phrase "  exists" is used as a shorthand way of saying this.

A closed set   of order-indiscernibles for   (where   is a limit ordinal) is a set of Silver indiscernibles if:

  •   is unbounded in  , and
  • if   is unbounded in an ordinal  , then the Skolem hull of   in   is  . In other words, every   is definable in   from parameters in  .

If there is a set of Silver indiscernibles for  , then it is unique. Additionally, for any uncountable cardinal   there will be a unique set of Silver indiscernibles for  . The union of all these sets will be a proper class   of Silver indiscernibles for the structure   itself. Then,   is defined as the set of all Gödel numbers of formulae   such that

 

where   is any strictly increasing sequence of members of  . Because they are indiscernibles, the definition does not depend on the choice of sequence.

Any   has the property that  . This allows for a definition of truth for the constructible universe:

  only if   for some  .

There are several minor variations of the definition of  , which make no significant difference to its properties. There are many different choices of Gödel numbering, and   depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode   as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number.

Statements implying existence

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The condition about the existence of a Ramsey cardinal implying that   exists can be weakened. The existence of  -Erdős cardinals implies the existence of  . This is close to being best possible, because the existence of   implies that in the constructible universe there is an  -Erdős cardinal for all countable  , so such cardinals cannot be used to prove the existence of  .

Chang's conjecture implies the existence of  .

Statements equivalent to existence

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Kunen showed that   exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe   into itself.

Donald A. Martin and Leo Harrington have shown that the existence of   is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as  .

It follows from Jensen's covering theorem that the existence of   is equivalent to   being a regular cardinal in the constructible universe  .

Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of  .

Consequences of existence and non-existence

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The existence of   implies that every uncountable cardinal in the set-theoretic universe   is an indiscernible in   and satisfies all large cardinal axioms that are realized in   (such as being totally ineffable). It follows that the existence of   contradicts the axiom of constructibility:  .

If   exists, then it is an example of a non-constructible   set of natural numbers. This is in some sense the simplest possibility for a non-constructible set, since all   and   sets of natural numbers are constructible.

On the other hand, if   does not exist, then the constructible universe   is the core model—that is, the canonical inner model that approximates the large cardinal structure of the universe considered. In that case, Jensen's covering lemma holds:

For every uncountable set   of ordinals there is a constructible   such that   and   has the same cardinality as  .

This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that   is uncountable cannot be removed. For example, consider Namba forcing, that preserves   and collapses   to an ordinal of cofinality  . Let   be an  -sequence cofinal on   and generic over  . Then no set in   of  -size smaller than   (which is uncountable in  , since   is preserved) can cover  , since   is a regular cardinal.

If   does not exist, it also follows that the singular cardinals hypothesis holds.[1]p. 20

Other sharps

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If   is any set, then   is defined analogously to   except that one uses   instead of  , also with a predicate symbol for  . See the section on relative constructibility in constructible universe.

See also

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  • 0, a set similar to 0# where the constructible universe is replaced by a larger inner model with a measurable cardinal.

References

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  1. ^ P. Holy, "Absoluteness Results in Set Theory" (2017). Accessed 24 July 2024.
  • Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
  • Harrington, Leo (1978). "Analytic determinacy and 0 #". Journal of Symbolic Logic. 43 (4): 685–693. doi:10.2307/2273508. ISSN 0022-4812. MR 0518675.
  • Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.
  • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
  • Martin, Donald A. (1970). "Measurable cardinals and analytic games". Fundamenta Mathematicae. 66 (3): 287–291. doi:10.4064/fm-66-3-287-291. ISSN 0016-2736. MR 0258637.
  • Silver, Jack H. (1971). "Some applications of model theory in set theory". Annals of Mathematical Logic. 3 (1): 45–110. doi:10.1016/0003-4843(71)90010-6. MR 0409188.