TITLE: Infinite mathematics as non-standard finite mathematics?
QUESTION [24 upvotes]: I have in mind something like the following:

Start with some suitable version of "finite" mathematics. Some possibilities might be maybe ZFC with a suitable anti-infinity axiom, the topos $\mathbf{FinSet}$, Peano arithmetic, Turing machines... something whose objects are suitably "finite".
Then, posit the existence of both a standard and a non-standard model.
Now, in this setting, where we have access both to a standard model and a non-standard extension, use the non-standard objects as proxies for infinite objects (e.g. maybe some sort of set theory that has a set of natural numbers), and develop ordinary mathematics this way.

Has anybody worked on such a thing? Does anyone know of references of it being done? Or suggestions that it can't work out?
(P.S. I wasn't sure how to tag this....)

Edit: After more thought and reviewing the answers thus far, I think I can state an example of the sort of thing i was imagining. Define a first-order theory with two types $T_1$ and $T_2$, two binary relation symbols $\in_1, \in_2$ (one for each sort), and a map $\tau : T_1 \to T_2$ satisfying:

$(T_1, \in_1)$ satisfies the axioms of finite set theory
$(T_2, \in_2)$ satisfies the axioms of finite set theory
$\tau$ is injective
$\tau$ is not surjective
$\tau$ satisfies an axiom schema that says it's an elementary embedding

and the question is to what extent we can develop infinite set theory in this theory.

REPLY [6 votes]: You might want to look into the work of Vopenka and his collaborators on what they call alternative set theory.  The formal theory looks a bit strange, since it allows proper classes to be subclasses of sets, but one interpretation of the theory is that the sets are the internal sets of a nonstandard model of finite-set-theory while the classes are arbitrary (not necessarily internal) subsets.  There's a small book, "Mathematics in the Alternative Set Theory," by Vopenka, that should serve as a good introduction to the subject.