Answering a question of Kaye, we show that the compositional truth theory with the full collection scheme is conservative over Peano Arithmetic. We demonstrate it by showing that countable models of compositional truth which satisfy the internal induction or collection axioms can be end-extended to models of the respective theory.

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in \mathbb {N}$ and any countable model of $\mathrm {B}\Sigma _{n+2}$, we construct a proper $\Sigma _{n+2}$-elementary end extension satisfying $\mathrm {B}\Sigma _{n+1}$, which answers a question by Clote positively. We also give a characterization of the countable models of $\mathrm {I}\Sigma _{n+2}$ in terms of their end extendibility, similar to the case of $\mathrm {B}\Sigma _{n+2}$. Along the proof, we introduce a new type of regularity principle in arithmetic called the weak regularity principle, which serves as a bridge between the model’s end extendibility and the amount of induction or collection it satisfies.

The lattice problem for models of Peano Arithmetic ($\mathsf {PA}$) is to determine which lattices can be represented as lattices of elementary submodels of a model of $\mathsf {PA}$, or, in greater generality, for a given model $\mathcal {M}$, which lattices can be represented as interstructure lattices of elementary submodels $\mathcal {K}$ of an elementary extension $\mathcal {N}$ such that $\mathcal {M}\preccurlyeq \mathcal {K}\preccurlyeq \mathcal {N}$. The problem has been studied for the last 60 years and the results and their proofs show an interesting interplay between the model theory of PA, Ramsey style combinatorics, lattice representation theory, and elementary number theory. We present a survey of the most important results together with a detailed analysis of some special cases to explain and motivate a technique developed by James Schmerl for constructing elementary extensions with prescribed interstructure lattices. The last section is devoted to a discussion of lesser-known results about lattices of elementary submodels of countable recursively saturated models of PA.

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 the arithmetical part of models of $\text {CT}^-$ are recursively saturated. We also use this tool to provide a new proof of theorem from [8] that all models of $\text {CT}^-$ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for a set of formulae whose syntactic depth forms a nonstandard cut which cannot be extended to a full truth predicate satisfying $\text {CT}^-$.