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We analyze Ekelands variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $Pi^1_1$-${sf CA}_0$, a strong theory of second-order arithmetic, while natural restrictions (e.g.~to compact spaces or continuous functions) yield statements equivalent to weak Konigs lemma (${sf WKL}_0$) and to arithmetical comprehension (${sf ACA}_0$). We also find that the localized version of Ekelands variational principle is equivalent to $Pi^1_1$-${sf CA}_0$ even when restricting to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.
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Simpson showed that every countable model $mathcal{M} models mathsf{PA}$ has an expansion $(mathcal{M}, X) models mathsf{PA}^*$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a non-prime model in which the definable elements coincide with those of the underlying model. Enayat showed that this is impossible by proving that there is $mathcal{M} models mathsf{PA}$ such that for each undefinable class $X$ of $mathcal{M}$, the expansion $(mathcal{M}, X)$ is pointwise definable. We call models with this property Enayat models. In this paper, we study Enayat models and show that a model of $mathsf{PA}$ is Enayat if it is countable, has no proper cofinal submodels and is a conservative extension of all of its elementary cuts. We then show that, for any countable linear order $gamma$, if there is a model $mathcal{M}$ such that $mathrm{Lt}(mathcal{M}) cong gamma$, then there is an Enayat model $mathcal{M}$ such that $mathrm{Lt}(mathcal{M}) cong gamma$.
A subset of a model of ${sf PA}$ is called neutral if it does not change the $mathrm{dcl}$ relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non-existence of neutral sets in various models of ${sf PA}$. We show that cofinal extensions of prime models are neutrally expandable, and $omega_1$-like neutrally expandable models exist, while no recursively saturated model is neutrally expandable. We also show that neutrality is not a first-order property. In the last section, we study a local version of neutral expandability.
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