Undefinable classes and definable elements in models of set theory and arithmetic

Abstract:

Every countable model ${\mathbf {M}}$ of PA or ZFC, by a theorem of S. Simpson, has a "class" $X$ which has the curious property: Every element of the expanded structure $({\mathbf {M}},X)$ is definable. Here we prove: Theorem A. Every completion $T$ of PA has a countable model ${\mathbf {M}}$ (indeed there are ${{\mathbf {2}}^\omega }$ many such ${\mathbf {M}}$’s for each $T$) which is not pointwise definable and yet becomes pointwise definable upon adjoining any undefinable class $X$ to ${\mathbf {M}}$. Theorem B. Let ${\mathbf {M}} \vDash {\text {ZF}} + ''V = {\text {HOD''}}$ be a well-founded model of any cardinality. There exists an undefinable class $X$ such that the definable points of ${\mathbf {M}}$ and $({\mathbf {M}},X)$ coincide. Theorem C. Let ${\mathbf {M}} \vDash {\text {PA}}$ or ${\text {ZF}} + ''V = {\text {HOD''}}$. There exists an undefinable class $X$ such that the definable points of ${\mathbf {M}}$ and $({\mathbf {M}},X)$ coincide if one of the conditions below is satisfied. (A) The definable elements of ${\mathbf {M}}$ are cofinal in ${\mathbf {M}}$. (B) ${\mathbf {M}}$ is recursively saturated and $\operatorname {cf}(\mathbf {M}) = \omega$.