For another route to the phenomenon, consider the following theorem of Ressayre, which has always both fascinated and mystified me. Indeed, I find the conclusion a bit alarming and perhaps even bizarre, precisely because it seems to be a too-strong fulfillment of your requested phenomenon.

**Theorem.** If ${\cal M}=\langle M,\hat\in\rangle$ is a nonstandard model of finite set theory, such as the natural model arising from a nonstandard model of PA, and if $T$ is any consistent computably axiomatizable extension of ZF, such as ZFC or ZFC+$\exists$ supercompact cardinal, then there is a submodel $N\subset M$ such that ${\cal N}=\langle N,\hat\in\rangle$ is a model of $T$.

That is, even though $\cal M$ is a model of *finite* set theory, it has a substructure realizing the infinitary theory of ZFC or much more. In this way, the theorem fulfills your request, since we are enabled to find within the nonstandard finite part of the model a fully accurate copy of the infinitary set theory. The amazing thing, to me, is that we can do so in such a flexible way so as to realize large cardinals or any other consistent set theory.

Ali Enayat explains some of the details in his answer to Mirco Mannucci's question Set theory inside arithmetics via the Ackerman yoga, citing 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.

Update (11/20/2012).

My paper Every countable model of set theory embeds into its own constructible universe contains the following strengthening of Ressayre's theorem:

**Theorem.** If ${\cal M}$ is any nonstandard model of PA, then every countable model of set theory is isomorphic to a submodel of $\langle\text{HF}^{\cal M},{\in}^{\cal M}\rangle$. Indeed, this structure is universal for all countable acyclic binary relations.

Here, $\text{HF}$ refers to the natural model of finite set theory defined inside $\cal M$, the hereditary finite sets as coded in $\cal M$. The relation $\in^{\cal M}$ is the Ackerman relation, so that $n\in^{\cal M} m$ just in case the $n$ th bit in the binary expansion of $m$ is $1$.

This theorem eliminates the role of the theory $T$ in Ressayre's theorem, for not only do we get merely at least one model of $T$ as a submodel of $\cal M$, but indeed every countable model of $T$ arises as a submodel.

The point now---and the reason I look upon this as relevant for your question---is that any given countable model of ZFC, even one satisfying a very strong theory, can be found as a submodel of any given nonstandard model of the strictly finitary theory $\text{ZFC}^{\neg\infty}$, which thinks every set is finite. So this is exactly a situation where we have an entire universe of infinite mathematics arising precisely as a form of nonstandard finite mathematics.