Among the basic results of logic that, simple as they are, never fail to intrigue me, is Ackermann's interpretation of $ZF$-Infinity in $PA$ (see for refs this MO question and here for an excellent overview)

Of course, the minus here is critical: $PA$ does not know anything about infinite objects.

Yet, my curiosity is by no means $\aleph_0$-bound:one could add to some fragment of $PA$ strong enough to verify all the axioms of $ZF$-infinity another axiom stating the existence of a number encoding an infinite countable set. That number would be non-standard in all models, but so what?

Perhaps the Ackermann Yoga can be pushed even further, attempting to add higher infinity axioms to the arithmetical theory, to "catch up" with Set Theory. The Ackermann's Yoga could give some insights on non standard models of arithmetics, by thinking of some nonstandard elements as large sets.

Has anything been investigated along these lines?