Among the basic results of logic which, simple as they are, never fail to intrigue me, is Ackermann's interpretation of ZF-Infinity in PA (see for refs this MO [question][1] and [here][2] 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? 


  [1]: https://mathoverflow.net/questions/63887/non-standard-models-of-finite-set-theory
  [2]: http://www.math.nus.edu.sg/~matwtl/papers/finitesettheory.pdf