# ⓘ Zero sharp. In the mathematical discipline of set theory, 0 # is the set of true formulae about indiscernibles and order-indiscernibles in the Godel constructib ..

## ⓘ Zero sharp

In the mathematical discipline of set theory, 0 # is the set of true formulae about indiscernibles and order-indiscernibles in the Godel constructible universe. It is often encoded as a subset of the integers, 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 Silvers 1966 thesis, later published as Silver, where it was denoted by Σ, and rediscovered by Solovay, who considered it as a subset of the natural numbers and introduced the notation O #.

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.

## 1. Definition

Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols c 1, c 2. for each positive integer. Then 0 # is defined to be the set of Godel numbers of the true sentences about the constructible universe, with c i interpreted as the uncountable cardinal ℵ i. Here ℵ i means ℵ i in the full universe, not the constructible universe.

There is a subtlety about this definition: by Tarskis 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 0 # works provided that there is an uncountable set of indiscernibles for some L α, and the phrase "0 # exists" is used as a shorthand way of saying this.

There are several minor variations of the definition of 0 #, which make no significant difference to its properties. There are many different choices of Godel numbering, and 0 # depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode 0 # as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number.

## 2. Statements implying existence

The condition about the existence of a Ramsey cardinal implying that 0 # exists can be weakened. The existence of ω 1 -Erdos cardinals implies the existence of 0 #. This is close to being best possible, because the existence of 0 # implies that in the constructible universe there is an α-Erdos cardinal for all countable α, so such cardinals cannot be used to prove the existence of 0 #.

Changs conjecture implies the existence of 0 #.

## 3. Statements equivalent to existence

Kunen showed that 0 # exists if and only if there exists a non-trivial elementary embedding for the Godel constructible universe L into itself.

Donald A. Martin and Leo Harrington have shown that the existence of 0 # 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 0 #.

It follows from Jensens covering theorem that the existence of 0 # is equivalent to ω being a regular cardinal in the constructible universe L.

Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of 0 #.

## 4. Consequences of existence and non-existence

Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L and satisfies all large cardinal axioms that are realized in L such as being totally ineffable. It follows that the existence of 0 # contradicts the axiom of constructibility: V = L.

If 0 # exists, then it is an example of a non-constructible Δ 1 3 set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ 1 2 and Π 1 2 sets of integers are constructible.

On the other hand, if 0 # does not exist, then the constructible universe L is the core model - that is, the canonical inner model that approximates the large cardinal structure of the universe considered. In that case, Jensens covering lemma holds:

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

This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannot be removed. For example, consider Namba forcing, that preserves ω 1 {\displaystyle \omega _{1}} and collapses ω 2 {\displaystyle \omega _{2}} to an ordinal of cofinality ω {\displaystyle \omega }. Let G {\displaystyle G} be an ω {\displaystyle \omega } -sequence cofinal on ω 2 L {\displaystyle \omega _{2}^{L}} and generic over L. Then no set in L of L -size smaller than ω 2 L {\displaystyle \omega _{2}^{L}} which is uncountable in V, since ω 1 {\displaystyle \omega _{1}} is preserved can cover G {\displaystyle G}, since ω 2 {\displaystyle \omega _{2}} is a regular cardinal.

## 5. Other sharps

If x is any set, then x # is defined analogously to 0 # except that one uses L instead of L. See the section on relative constructibility in constructible universe.

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