${0}^{\u2020}$ is the (set of Gödel numbers of the formulae in the) theory of the structure $<L\left[U\right],\in ,\kappa ,U,{x}_{1},{x}_{2},{x}_{3},\xb7\xb7\xb7,{y}_{1},{y}_{2},{y}_{3},\xb7\xb7\xb7>$ where U is a normal measure on the cardinal $\kappa $, and $<{x}_{1},{x}_{2},{x}_{3},\xb7\xb7\xb7>$ and $<{y}_{1},{y}_{2},{y}_{3},\xb7\xb7\xb7>$ are increasing enumerations of sets X and Y of ordinals with $X<\kappa <Y$ which are indiscernibles for $<L\left[U\right],\in ,\kappa ,U>$, in the sense that the truth value of

is independent of the choice of the increasing sequences $<{\alpha}_{1},\xb7\xb7\xb7,{\alpha}_{m}>$ from X and $<{\beta}_{1},\xb7\xb7\xb7,{\beta}_{n}>$ from $Y\xb7$

${0}^{\u2020}$ was first defined by Solovay, who showed that if there were to measurable cardinals $\kappa $, $\lambda $ with $\kappa <\lambda $ then ${0}^{\u2020}$ exists [see *A. R. D. Mathias*, Period. Math. Hung. 10, 109-175 (1979; Zbl 0417.03021)]. The set ${0}^{\u2020}$ plays a similar role for inner models with a measurable cardinal as ${0}^{\#}$ plays for L.

Various results about ${0}^{\u2020}$ have appeared in the literature. However the survey paper under review is the first detailed presentation of the theory of ${0}^{\u2020}$. Section 1 defines ${0}^{\u2020}$ and contains the appropriate Ehrenfeucht-Mostowski theory and shows that sufficiently strong large cardinal hypotheses generate models L[U] with indiscernibles. In section 2 connections are established between classes of indiscernibles for various of these models. Finally, section 3 reviews various characterizations of the existence of ${0}^{\u2020}$.

##### MSC:

03E45 | Constructibility, ordinal definability, and related notions |

03E55 | Large cardinals |

03E10 | Ordinal and cardinal arithmetic |