**April 1**

Whan Ki Lee,
CUNY

**Solid bases and AH-sets**

When $A \subseteq \mathcal{M} \vDash PA^{\ast}$, we say that $A$ is a *basis* for $\mathcal{M}$ if for all $X \subseteq A$, the submodel $M_{X}$ generated by $X$ is the unique $\mathcal{N} \prec \mathcal{M}$ such that $X = \mathcal{N} \cap A$, and that such $A$ is *solid* if for all finite $X,Y \subseteq A$, whenever $f: \mathcal{M}_{X} \to \mathcal{M}_{Y}$ is an isomorphism, then $f \upharpoonright_{X}$ is one-to-one onto $Y$. We will discuss what role a solid basis can play in controlling the amount of indiscernibility and the automorphisms of a model. Also, using a set of types called an 'AH-set', we will construct a solid basis.