Concept in set theory
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 , where it was denoted by Σ, and rediscovered by , 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.
Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols formula_1, formula_2, ... for each nonzero natural number. Then formula_3 is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with formula_4 interpreted as the uncountable cardinal formula_5.
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 formula_3 works provided that there is an uncountable set of indiscernibles for some formula_9, and the phrase "formula_3 exists" is used as a shorthand way of saying this.
A closed set formula_11 of order-indiscernibles for formula_9 (where formula_13 is a limit ordinal) is a set of "Silver indiscernibles" if:
If there is a set of Silver indiscernibles for formula_24, then it is unique. Additionally, for any uncountable cardinal formula_25 there will be a unique set of Silver indiscernibles for formula_26. The union of all these sets will be a proper class formula_11 of Silver indiscernibles for the structure formula_28 itself. Then, formula_3 is defined as the set of all Gödel numbers of formulae formula_30 such that
formula_31
where formula_32 is any strictly increasing sequence of members of formula_11. Because they are indiscernibles, the definition does not depend on the choice of sequence.
Any formula_34 has the property that formula_35. This allows for a definition of truth for the constructible universe:
formula_36 only if formula_37 for some formula_34.
There are several minor variations of the definition of formula_3, which make no significant difference to its properties. There are many different choices of Gödel numbering, and formula_3 depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode formula_3 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.
The condition about the existence of a Ramsey cardinal implying that formula_3 exists can be weakened. The existence of formula_43-Erdős cardinals implies the existence of formula_3. This is close to being best possible, because the existence of formula_3 implies that in the constructible universe there is an formula_13-Erdős cardinal for all countable formula_13, so such cardinals cannot be used to prove the existence of formula_3.
Chang's conjecture implies the existence of formula_3.
Statements equivalent to existence.
Kunen showed that formula_3 exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe formula_28 into itself.
Donald A. Martin and Leo Harrington have shown that the existence of formula_3 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 formula_3.
It follows from Jensen's covering theorem that the existence of formula_3 is equivalent to formula_55 being a regular cardinal in the constructible universe formula_28.
Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of formula_3.
Consequences of existence and non-existence.
The existence of formula_3 implies that every uncountable cardinal in the set-theoretic universe formula_59 is an indiscernible in formula_28 and satisfies all large cardinal axioms that are realized in formula_28 (such as being totally ineffable). It follows that the existence of formula_3 contradicts the "axiom of constructibility": formula_63.
If formula_3 exists, then it is an example of a non-constructible formula_65 set of natural numbers. This is in some sense the simplest possibility for a non-constructible set, since all formula_66 and formula_67 sets of natural numbers are constructible.
On the other hand, if formula_3 does not exist, then the constructible universe formula_28 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 formula_70 of ordinals there is a constructible formula_71 such that formula_72 and formula_71 has the same cardinality as formula_70.
This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that formula_70 is uncountable cannot be removed. For example, consider Namba forcing, that preserves formula_43 and collapses formula_77 to an ordinal of cofinality formula_78. Let formula_79 be an formula_78-sequence cofinal on formula_81 and generic over formula_28. Then no set in formula_28 of formula_28-size smaller than formula_81 (which is uncountable in formula_59, since formula_43 is preserved) can cover formula_79, since formula_77 is a regular cardinal.
If formula_3 does not exist, it also follows that the singular cardinals hypothesis holds.p. 20
Other sharps.
If formula_70 is any set, then formula_92 is defined analogously to formula_3 except that one uses formula_94 instead of formula_28, also with a predicate symbol for formula_70. See Constructible universe#Relative constructibility.
References.
Citations.
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