Zero sharp

$0^{\#}$ is a $\Sigma_3^1$ real number which cannot be proven to exist in $\text{ZFC}$. It's existence contradicts the Axiom of constructibility, $\text{V=L}$. In fact, it's existence is somewhat equivalent to $\text{L}$ being completely different from $\text{V}$.

Definition

$0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $\text{L}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$. Because of the stability of $\aleph_\omega$, $0^{\#}$ is equivalent to the set of all Gödel numberings of first-order formula $\varphi$ such that $\text{L}_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$. This definition implies the existence of Silver Indiscernables. Moreover, it implies:

• Given any set $X\in\text{L}$ which is first-order definable in $\text{L}$, $X\in\text{L}_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $\text{L}$, because $\aleph_1\not\in\text{L}_{\omega_1}$. This is already a disproof of $\text{V=L}$ (because $\aleph_1$ is first-order definable).
• For every $\alpha\in\omega_1^\text{L}$, every uncountable cardinal is $\alpha$-iterable, $\geq$ an $\alpha$-Erdős, and totally ineffable in $\text{L}$.
• There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in\text{L}$).

The existence of $0^\#$ is implied by:

• Chang's Conjecture.
• The existence of an $\omega_1$-iterable cardinal.
• The negation of the singular cardinal hypothesis ($\text{SCH}$).
• The axiom of determinacy ($\text{AD}$).

$0^{\#}$ cardinal

$0^{\#}$ exists iff there is a nontrivial elementary embedding $j:\text{L}\rightarrow\text{L}$ (by a theorem of Kunen). The critical point of such an embedding is sometimes called a $0^{\#}$ cardinal, and sometimes called a $j:\text{L}\rightarrow\text{L}$ cardinal. These cardinals do not coincide with measurable cardinals for a long time. While the least measurable cardinal is $\Sigma_1^2$-describable, each of these cardinals is totally indescribable. Furthermore, the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of a measurable cardinal is not a $j:\text{L}\rightarrow\text{L}$ cardinal, and the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of such a cardinal is not a $j:\text{L}\rightarrow\text{L}$ cardinal, and so on.

However, the existence of a measurable suffices to prove the existence and consistency of a $j:\text{L}\rightarrow\text{L}$ cardinal.