Set theory inside arithmetics via the Ackermann yoga 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 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?
 A: You might want to look into the notion of the standard system of a nonstandard model $M$ of PA.  Any nonstandard member $x$ of $M$, determines, via your favorite coding, a subset $X$ of $M$ that is finite in the internal sense of $M$ but may be infinite when seen from outside $M$.  Intersecting this $X$ with the standard part $\mathbb N$ of $M$, we get some subset of $\mathbb N$, and the family of all the intersections obtainable in this way, as $x$ varies over $M$, is called the standard system of $M$.  With this definition, it's a collection of subsets of $\mathbb N$, but it corresponds, via Ackermann coding, to a collection of subsets of the set $HF$ of hereditarily finite sets (the standard model of ZF minus infinity).  
A: This question reminds me of a magical little-known theorem of Jean Pierre Ressayre that shows that every nonstandard model of $PA$ has a model of $ZF$ as a submodel of its Ackermann intepretation, more specifically:
Theorem. [Ressayre] Suppose $(M, +, \cdot)$ is a nonstandard model of $PA$, and $\in_{Ack}$ is the Ackermann epsilon on $M$, i.e., $a\in_{Ack}b$ iff $\mathcal{M}$ satisfies "the $a$-th digit in the binary expansion of $b$ is 1".  Then for every consistent recursive extension $T$ of $ZF$ there is a subset $A$ of $M$ such that $(A,\in_{Ack})$ is a model of $T$.
Proof Outline: By Löwenheim-Skolem, it suffices to consider the case when $M$ is countable. Choose a nonstandard integer $k$ in $M$, and consider the submodel $M_k$ of $(M,\in_{Ack})$ consisting of sets of ordinal rank less than $k$ [as computed within $(M,\in_{Ack})$]. "Usual arguments" show that $(M,\in_{Ack})$ has a Tarskian truth-definition for $M_k$, which in turn implies that $(M_k,\in_{Ack})$ is recursively saturated. Since $M_k$ is also countable, $(M_k,\in_{Ack})$ must be resplendent [which means that it has an expansion to every recursive $\Sigma^1_1$ theory that its elementary diagram  is consistent with]. 
Now add a new unary predicate symbol $A$ to the language ${\in}$ of set theory and consider the (recursive) theory $T^A$ consisting of sentences of the form $\phi^A$, where $\phi \in T$, and $\phi^A$ is obtained by relativizing every quantifier of $\phi$ to $A$. It is not hard to show that $T^A$ is consistent with the the elementary diagram of $(M_k,\in_{Ack})$, so by replendence the desired $A$ can be produced. 
[I will be glad to add clarifications]
Ressayre's theorem appears in the following paper:
J. P. Ressayre, Introduction aux modèles récursivement saturés, Séminaire Général de Logique 1983–1984 (Paris, 1983–1984), 53–72, Publ. Math. Univ. Paris VII, 27, Univ. Paris VII, Paris, 1986. 
A: I wrote a paper proposing a solution to this. https://www.mdpi.com/2075-1680/8/1/31
I am working on a newer version,
https://www.preprints.org/manuscript/202006.0098/v3
