---
tag: HCTNBZZQ0W
kind: object
name: Zero sharp
msc: ["03E45", "03E55"]
uses: [6TF2JBTPPJ, 0YSCSX5YZC]
sources:
  - url: https://en.wikipedia.org/wiki/Zero_sharp
same_as:
  wikidata: Q8069610
  wikipedia: Zero sharp
---

## Description

Let $L$ be the [[constructible universe|6TF2JBTPPJ]].
Suppose that there is a closed unbounded class $I$ of [[ordinals|0YSCSX5YZC]] with the following property for every uncountable cardinal $\kappa$.
The set $I \cap \kappa$ has order type $\kappa$ and is a set of indiscernibles for the structure $(L_\kappa, \in)$.
Every element of $L_\kappa$ is definable in $L_\kappa$ from finitely many elements of $I \cap \kappa$.
Such a class is unique when it exists, and its elements are the *Silver indiscernibles*.
In this case *zero sharp* is the set
$$0^\sharp = \set{\ulcorner \varphi \urcorner : L_{\aleph_\omega} \models \varphi[\aleph_1, \ldots, \aleph_n]}.$$
Here $\varphi(x_1, \ldots, x_n)$ runs over the formulas of set theory, and $\ulcorner \varphi \urcorner$ is the Gödel number of $\varphi$.
One says that $0^\sharp$ exists when such a class $I$ exists.

## Remarks

Since $0^\sharp$ is a set of natural numbers, it is often treated as a real number.
If $0^\sharp$ exists, then every uncountable cardinal of $V$ is a Silver indiscernible, and hence an inaccessible cardinal in $L$.
If $0^\sharp$ exists, then $0^\sharp \notin L$, so $V \neq L$.
Silver proved that $0^\sharp$ exists if there is a [[measurable cardinal|AKBWT323RQ]].
If $0^\sharp$ does not exist, then [[Jensen's covering theorem|RJE1W5NRE3]] shows that $L$ is close to $V$.
If ZFC is consistent, it does not prove that $0^\sharp$ exists, since $0^\sharp$ does not exist in $L$.

## Explanations

### L1 human

Zero sharp is a single set of natural numbers that, if it exists, describes all truths about the constructible universe.
It exists only if the full universe is much richer than the constructible universe.
Its existence marks a sharp line: either the constructible universe is close to everything, or zero sharp exists and the constructible universe is very thin.
Its existence follows from large cardinals such as a measurable cardinal, but it cannot be proved from the usual axioms.