Peano arithmetic and hyper-Ramsey logic

Abstract:

It is known that ${\text {PA}}({Q^2})$, Peano arithmetic in a language with the Ramsey quantifier, is complete and compact and that its first-order consequences are the same as those of $\Pi _1^1{\text {-CA}_0}$. A logic $\mathcal {H}{\mathcal {R}_\omega }$, called hyper-Ramsey logic, is defined; it is the union of an increasing sequence $\mathcal {H}{\mathcal {R}_1} \subseteq {\mathcal {H}_{\mathcal {R}2}} \subseteq \mathcal {H}{\mathcal {R}_3} \subseteq \cdots$ of sublogics, and $\mathcal {H}{\mathcal {R}_1}$ contains $L({Q^2})$. It is proved that ${\text {PA}}(\mathcal {H}{\mathcal {R}_n})$, which is Peano arithmetic in the context of $\mathcal {H}{\mathcal {R}_n}$, has the same first-order consequences as $\Pi _n^1{\text {-CA}_0}$. A by-product and ingredient of the proof is, for example, the existence of a model of ${\text {CA}}$ having the form $(\mathcal {N}, {\text {Class}}(\mathcal {N}))$.