Disjunctions with stopping condition

We introduce a tool for analysing models of ${\text{CT}}^{-}$, the compositional truth theory over Peano Arithmetic. We present a new proof of Lachlan’s theorem that arithmetical part of models of $\text{PA}$ are recursively saturated. We use this tool to provide a new proof that all models of ${\text{CT}}^{-}$ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for formulae from a nonstandard cut which cannot be extended to a full truth predicate satisfying ${\text{CT}}^{-}$.